# Ergodicity for the hyperbolic $P(\Phi)_{2}$-model

Leonardo Tolomeo  Leonardo Tolomeo
Leonardo Tolomeo
School of Mathematics
The University of Edinburgh
and The Maxwell Institute for the Mathematical Sciences
James Clerk Maxwell Building
The King’s Buildings
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
United Kingdom
and Mathematical Institute
Hausdorff Center for Mathematics
Universität Bonn
Bonn
Germany<EMAIL_ADDRESS>
###### Abstract.

We consider the problem of ergodicity for the $P(\Phi)_{2}$ measure of quantum
field theory under the flow of the singular stochastic (damped) wave equation
$u_{tt}+u_{t}+(1-\Delta)u+{:}\,p(u)\mspace{2.0mu}{:}=\sqrt{2}\xi$, posed on
the two-dimensional torus $\mathbb{T}^{2}$. We show that the $P(\Phi)_{2}$
measure is ergodic, and moreover that it is the unique invariant measure for
(the Markov process associated to) this equation which belongs to a fairly
large class of probability measures over distributions.

The main technical novelty of this paper is the introduction of the new
concepts of asymptotic strong Feller and asymptotic coupling restricted to the
action of a group. We first develop a general theory that allows us to deduce
a suitable support theorem under these hypotheses, and then show that the
stochastic wave equation satisfies these properties when restricted the action
of translations by shifts belonging to the Sobolev space
$H^{1-\varepsilon}\times H^{-\varepsilon}$. We then exploit the newly
developed theory in order to conclude ergodicity and (conditional) uniqueness
for the $P(\Phi)_{2}$ measure.

###### Key words and phrases:

stochastic nonlinear wave equation; nonlinear wave equation; damped nonlinear
wave equation; ergodicity; renormalization; Wick renormalization; white noise;
Gibbs measure

###### 2020 Mathematics Subject Classification:

35L15, 37A25, 60H15

###### Contents

  1. 1 Introduction
    1. 1.1 The strong Feller property and its failure for stochastic wave equations
    2. 1.2 Asymptotic couplings restricted to the action of a group
    3. 1.3 Structure of the paper
    4. 1.4 Further remarks
  2. 2 Restricted asymptotic strong Feller and restricted asymptotic coupling properties
    1. 2.1 Assumptions
    2. 2.2 Definitions
    3. 2.3 Support theorems
    4. 2.4 Proof of the support theorems
  3. 3 The hyperbolic $P(\Phi)_{2}$-model
    1. 3.1 On of the $P(\Phi)_{2}$ measure and Wick renormalisation
    2. 3.2 Local and global theory for (SDNLW)
    3. 3.3 Construction of the Girsanov shift
    4. 3.4 Restricted asymptotic strong Feller and restricted coupling properties
    5. 3.5 Ergodicity of the $P(\Phi)_{2}$ measure
    6. 3.6 Conditional uniqueness of the $P(\Phi)_{2}$ measure

##  1\. Introduction

In this paper, we consider the (massive) $P(\Phi)_{2}$ measure of quantum
field theory, formally given by

$dP(\Phi)_{2}(u)=\frac{1}{Z}\exp\Big{(}-\int_{\mathbb{T}^{2}}{:}\,P(u)\mspace{2.0mu}{:}-\frac{1}{2}\int_{\mathbb{T}^{2}}u(m^{2}-\Delta)u\Big{)}du.$
($P(\Phi)_{2}$)

Here $P$ is a polynomial of even degree $2k\in\mathbb{N}$ with positive
leading coefficient, i.e. $a_{2k}>0$ and

$P(x)=a_{2k}x^{2k}+a_{2k-1}x^{2k-1}+\dotsb+a_{0},$

and ${:}\,P\mspace{2.0mu}{:}$ denotes the Wick renormalisation (which will be
rigorously introduced in Section 3). The construction of these measure has
firstly been achieved by Guerra, Rosen and Simon in [23], and has been one of
the major milestones in the program of constructive quantum field theory.

In [37], Parisi and Wu suggested a new approach to the construction of
measures such as ($P(\Phi)_{2}$), kickstarting the project of stochastic
quantisation. In short, this project consists of the following. If we see
($P(\Phi)_{2}$) (formally) as a measure of the form

$\sigma=\frac{1}{Z}\exp(-V(u))du,$ (1.1)

then we can write the (overdamped) Langevin equation for such a measure, i.e.
the stochastic differential equation (SDE)

$u_{t}=-\nabla V(u)+\sqrt{2}\xi,$ (1.2)

where $\xi$ is a space-time white noise. In the finite dimensional setting,
the measure $\sigma$ is invariant for (1.2). If then one shows that the
equation (1.2) admits a unique invariant measure, we can exploit this
uniqueness to _redefine_ $\sigma$ as the unique invariant measure for (1.2).
From the point of view of numerics, this definition/program has the benefit
that then one can generate samples of ($P(\Phi)_{2}$) by firstly solving (1.2)
and then performing a Markov Chain Montecarlo (MCMC) procedure.

When specialised to the case of ($P(\Phi)_{2}$), the equation (1.2) becomes
the stochastic partial differential equation (SPDE)

$u_{t}=-(m^{2}-\Delta)u-{:}\,p(u)\mspace{2.0mu}{:}+\sqrt{2}\xi,$ (SQE)

which in this context has the name of stochastic quantisation equation (SQE)
for the measure ($P(\Phi)_{2}$). Here $p(x)=P^{\prime}(x)$ denotes the
derivative of $P$. To this day, the stochastic quantisation program for (SQE)
has been successfully completed, with local well posdeness for (SQE) being
shown by Da Prato and Debussche in [11], global well posedness shown by
Mourrat and Weber in [31] for $2k=4$ and by Tsatsoulis and Weber in [41] for
higher values of $k$, and unique ergodicity being shown by Tsatsoulis and
Weber in [41]. We would like to remark here that the efforts of concluding the
program of stochastic quantisation go well beyond the study of the measure
($P(\Phi)_{2}$) and its associated overdamped Langevin dynamics. Indeed, the
Langevin dynamics for measures of the form

$d\Phi^{4}_{d}=\frac{1}{Z}\exp\Big{(}-\frac{1}{4}\int_{\mathbb{T}^{d}}u^{4}-\text{
renormalisation }-\frac{1}{2}\int_{\mathbb{T}^{d}}u(m^{2}-\Delta)u\Big{)}$

has been constructed in the whole subcritical regime thanks to the theory of
regularity structures developed by Hairer and collaborators [25, 7, 6].
Moreover, thanks to the work by Hairer and Mattingly [30], we now possess a
general theory for showing the strong Feller property for equations such as
(SQE). We will enter more in details about what this entails in Section 1.1,
but as observed in [30], this property automatically implies uniqueness for
the invariant measure as soon as an invariant measure with full support is
known to exist. This is indeed the case in most situations in which the
$\Phi^{4}_{d}$ measure can be constructed explicitly.

Despite the success of the project of stochastic quantisation, the choice of
the overdamped Langevin dynamics as the model equation to sample the measure
($P(\Phi)_{2}$) is somewhat arbitrary. Indeed, in the context of sampling
measures of the form of (1.1), one can consider many other models, with the
only restriction being that the measure $\sigma$ is an ergodic measure for the
flow, and a set of initial data whose flow will converge to $\sigma$ (in the
sense of Birkhoff’s ergodic theorem) is known. If we focus our attention to
the finite dimensional setting, in recent years the following kinetic Langevin
equation has attracted particular attention

$\begin{cases}u_{t}=v,\\\ v_{t}=-v-\nabla V(u)+\sqrt{2}\xi.\end{cases}$ (1.3)

The unique invariant measure for (1.3) is given by

$\frac{1}{Z^{\prime}}\exp\Big{(}-V(u)-\frac{|v|^{2}}{2}\Big{)}dudv,$ (1.4)

so one can sample the measure $\frac{1}{Z}\exp(-V(u))du$ by sampling the law
of the first component of the solution of (1.3). This procedure has the name
of Halmitonian Montecarlo (HCM). It has numerically been observed that HCM
converges faster than MCMC in many situations. While the author is not an
expert in this field, we can refer the interested reader to [14] (and
references within), which contains a rigorous justification of these faster
convergence rates for a class of potentials $V$.

With this point in mind, it would be interesting to complete the project of
stochastic quantisation for the analogous of (1.3), which is given by

$\begin{cases}u_{tt}+u_{t}+(1-\Delta)u+{:}\,p(u)\mspace{2.0mu}{:}=\sqrt{2}\xi,\\\
(u(0),u_{t}(0))=\mathbf{u}_{0}.\end{cases}$ (SDNLW)

Here we fixed $m^{2}=1$ for simplicity of notation, and we will keep this
choice for the rest of this paper. In the context of stochastic quantisation,
this is the so-called _canonical stochastic quantisation_ equation for the
measure ($P(\Phi)_{2}$). The invariant measure for this equation is (formally)
given by

$\displaystyle\rho(u,u_{t})$
$\displaystyle=P(\Phi)_{2}(u)\otimes\mu_{0}(u_{t})$ (1.5)
$\displaystyle``=\frac{1}{Z}\exp\Big{(}-\int_{\mathbb{T}^{2}}{:}\,P(u)\mspace{2.0mu}{:}-\frac{1}{2}\int_{\mathbb{T}^{2}}u(1-\Delta)u\Big{)}\exp\Big{(}-\frac{1}{2}\int_{\mathbb{T}^{2}}u_{t}^{2}\Big{)}dudu_{t}".$

The main result of this paper is the first step in the resolution of the
stochastic quantisation program applied to equation (SDNLW), which can be
summarised in the following statement.

######  Theorem 1.1.

The measure $\rho$ as in (1.5) is an ergodic meausure for the Markov semigroup
generated by the flow of (SDNLW). Moroever, for
$0<\varepsilon\leq\varepsilon_{k}\ll 1$, the measure $\rho$ is the unique
invariant measure belonging to the class

$W^{1}_{{:}\,p\mspace{2.0mu}{:}}:=\Big{\\{}\mu\in\mathcal{P}(H^{-\varepsilon}\times
H^{-\varepsilon-1}):\int\|{:}\,p(u)\mspace{2.0mu}{:}\|_{\mathcal{H}^{-\varepsilon}}d\mu(u,u_{t})<\infty\Big{\\}}.$
(1.6)

While the uniqueness part of the statement is conditional to the first moment
of $\|{:}\,p(u)\mspace{2.0mu}{:}\|$ being finite, Theorem 1.1 still suggests
an algorithm for sampling the measure $P(\Phi)_{2}$ according to a HCM
procedure. More specifically, one could pick a (random) initial data
$\mathbf{u}_{0}\in\mathcal{H}^{1-\varepsilon}$, and a sample of the noise
$\xi$. We can then compute $\Phi_{t}(\mathbf{u}_{0},\xi_{j})$ by solving the
equation (SDNLW). We then consider the statistical average of
$\|{:}\,p(u)\mspace{2.0mu}{:}\|_{\mathcal{H}^{-\varepsilon}}$ at some large
time $T\gg 1$,

$[\|{:}\,p\mspace{2.0mu}{:}\|]_{T}=\frac{1}{M}\sum_{j=1}^{M}\frac{1}{T}\int_{0}^{T}\|{:}\,p(\Phi_{t}(\mathbf{u}_{0},\xi))\mspace{2.0mu}{:}\|_{\mathcal{H}^{-\varepsilon}}dt.$

Then if $[\|{:}\,p\mspace{2.0mu}{:}\|]_{T}\leq K$ for some (appropriately
chosen) constant $K$, we “accept” the sample and use the flow
$\\{\Phi_{t}(\mathbf{u}_{0},\xi)\\}_{t\geq 0}$ to study the measure
($P(\Phi)_{2}$). Otherwise, we pick a different (randomly chosen) initial data
$\mathbf{u}_{0}^{\prime}$ and restart the procedure. Since the measure $\rho$
is absolutely continuous with respect to the following Gaussian measure,

$d\rho_{0}(u,u_{t})=\frac{1}{Z}\exp\Big{(}-\frac{1}{2}\int_{\mathbb{T}^{2}}u(m^{2}-\Delta)u-\frac{1}{2}u_{t}^{2}\Big{)}dudu_{t},$
(1.7)

if one chooses the initial data $\mathbf{u}_{0}$ as a random sample of the
measure $\rho_{0}$, then the quantity $[\|{:}\,p\mspace{2.0mu}{:}\|]_{T}$ is
going be finite almost surely, and in principle there should be no need to
sample a different initial data $\mathbf{u}_{0}^{\prime}$. However, the
techniques of this paper cannot exclude the situation in which the solution
“escapes” the invariant measure due to numerical errors.

###  1.1. The strong Feller property and its failure for stochastic wave
equations

It turns out that from a technical point of view, the stochastic quantisation
program for wave equations such as (SDNLW) is much harder to achieve than in
the parabolic case. While important milestones for the local well posedness
theory for canonical stochastic quantisation equations are progressively been
achieved, global well posedness and ergodicity results are both very rare.
Indeed, local well posedness for equation (SDNLW) has been proven in [20] on
$\mathbb{T}^{2}$, in [40, 35] on $\mathbb{R}^{2}$, and in [34] on a general
2-dimensional compact manifold. A series of 3-dimensional results have been
proven in [21] and [33] for the equation with quadratic nonlinearity, in [32,
3, 36] for cubic nonlinearities under the addition of some smoothing in the
equation, and finally in [4] for the canonical stochastic quantisation
equation for the $\Phi^{4}_{3}$ measure.

When an invariant measure is available, often global well-posedness for a.e.
initial data sampled according to the invariant measure follows via an
application of Bourgain’s invariant measure argument [2] (see also [17,
Theorem 6.1] for a general formulation). However, this is more-or-less the
only globalisation argument that has been shown to work for _singular_
stochastic wave equations. The only exceptions that the author is aware of, in
which it is actually possible to show some appropriate (pathwise) energy
estimates for the solutions, are the results by the author [40] and Gubinelli,
Koch, Oh and the author [22]. While in principle good energy estimates are not
necessary in order to prove (unique) ergodicity, they are a fundamental tool
in many applications. Nevertheless, the main difficulty in showing ergodicity
for stochastic wave equations comes from a different issue, which is the
failure of the strong Feller property.

Before we can discuss in details the techniques developed in this paper to get
around this problem, it is instructive to move back to the case of parabolic
SPDEs (such as (SQE)), and describe the general strategy to proving unique
ergodicity in that case. Let us denote the solution of a SPDE at time $t\geq
0$ with initial data $u_{0}$ and driven by a noise $\xi$ by
$\Phi_{t}(u_{0},\xi)$. Suppose that on an appropriate space $X$ of initial
data, the solution $\Phi_{t}(u_{0},\xi)$ exists for every $t>0$. Under
reasonable assumptions on the equation, the noise $\xi$, and the local well
posedeness theory, this defines a Markov process on the space $X$. In
particular, for any bounded measurable function $F:X\to\mathbb{R}$, we can
define

$P_{t}F:=\mathbb{E}[F(\Phi_{t}(u_{0},\xi))],$

and $P_{t}$ will be a Markov semigroup. We denote its dual by $P_{t}^{*}$.
Unique ergodicity for the SPDE then corresponds to having a unique invariant
measure for the semigroup $P_{t}$. Typically, the main ingredients for showing
such a statement are the following.

(Long time estimates):

Show good long time estimates for the flow, i.e. estimates of the form

$\mathbb{E}\|\Phi_{t}(u_{0},\xi)\|\leq C(u_{0})\text{ for every }u_{0}\in X,$

where $\|\cdot\|$ is some appropriate norm of the solution111There is actually
no need for this to be a norm, and one can consider situations in which
$\|\cdot\|$ is replaced by an appropriate “size function” $r:X\to\mathbb{R}$
such that $r(\Phi_{t}(u_{0},\xi))\to\infty$ as $t\uparrow t^{*}$ implies
blowup at time $t^{*}$ (or more typically, is the _definition_ of blowup at
time $t^{*}$). , and the constant $C(u_{0})$ is allowed to depend on the
initial data $u_{0}$, but not on time.

(Irreducibility):

Fix a base point $u_{*}$, and show that for every small ball
$B_{\varepsilon}(u_{*})$, for every $R>0$, and for every $u_{0}$ belonging to
$B_{R}(u_{*})$, we have that

$\sup_{t\geq 0}\mathbb{P}(\\{\Phi_{t}(u_{0},\xi)\in B_{\varepsilon}\\})\geq
2\varepsilon_{0}(\varepsilon,R)>0.$

(Coupling):

Show that for every $\delta>0$ and for some appropriate distance on
probability measures $d$, there exists some $\varepsilon>0$, such that for
every $u_{0}\in B_{\varepsilon}(u_{*})$,

$\limsup_{t\to\infty}d(P_{t}^{*}\delta_{u_{0}},P_{t}^{*}\delta_{u_{*}})\leq\delta.$

The proof of (unique) ergodicity then goes roughly as follows: starting from
$u_{0}$, by irreducibility, after some time $t_{1}>0$, we have that
$\mathbb{P}(\Phi_{t_{1}}(u_{0},\xi)\in
B_{\varepsilon})\geq\varepsilon_{0}(\varepsilon,\|u_{0}\|)$. Then from the
coupling property, the evolution starting from the ball $B_{\varepsilon}$ will
be “close in law” to the evolution of ${u_{*}}$. For the part of the evolution
that at time $t_{1}$ is outside the ball $B_{\varepsilon}$, we repeat the same
process: after a (random) time $t_{2}$, we have that
$\mathbb{P}(\Phi_{t_{2}}(u_{0},\xi)\in
B_{\varepsilon}|\Phi_{t_{1}}(u_{0},\xi)\not\in
B_{\varepsilon})\geq\varepsilon_{0}(\varepsilon,\|\Phi_{t_{1}}(u_{0},\xi)\|)$.
Iterating this process, we obtain that

$\mathbb{P}(\\{\Phi_{t_{j}}(u_{0},\xi)\not\in B_{\varepsilon}\text{ for every
}j\leq
J)\leq\prod_{j=1}^{J}(1-\varepsilon_{0}(\varepsilon,\|\Phi_{t_{j-1}}(u_{0},\xi)\|)).$

This is where the long time estimates come into play: up to possibly extending
the times $t_{1},t_{2},\dotsc$, we can guarantee that
$\|\Phi_{t_{j-1}}(u_{0},\xi)\|$ remains under control, and so
$\sum_{j}\varepsilon_{0}(\varepsilon,\|\Phi_{t_{j-1}}(u_{0},\xi)\|)$ diverges.
Therefore, we obtain that the evolution starting from $u_{0}$ is “close in
law” (with respect to some appropriate distance) to the evolution starting
from $u_{*}$ with high probability, and this allows us to conclude uniqueness
of the invariant measure. If one then has better control over the quantities
$C(u_{0}),\varepsilon_{0}(\varepsilon,R),$ and $\varepsilon$ as a function of
$\delta$, it is also possible to extract a convergence rate from this
argument, which will typically be exponential. It is also possible to slightly
weaken the three properties above, at the cost of getting a worse convergence
rate.

In practice, however, one needs not to perform this complicated analysis and
show tight control over the various quantities, but there are instead a number
of pre-confectioned results that can be used in order to obtain exponential
convergence to equilibrium. A classical example of such results is Harris
theorem (see [29] for a satisfying proof of this result). In this context of
“pre-pacakged” results, a very successful approach has been to rely on the
_strong Feller property_. In short, we say that a semigroup has the strong
Feller property if, for some time $t>0$,

$F\text{ measurable and bounded}\Rightarrow P_{t}F\text{ is continuous.}$

This is essentially an infinitesimal smoothing property of the semigroup
$P_{t}$, and in the case of finite dimensional systems of SDEs, checking this
property is in most situation a simple consequence of Hörmander’s
hypoellipticity theorem. It is easy to check that, in the finite dimensional
setting, both (1.2) and (1.3) always satisfy Hörmander’s condition, hence
under extremely general assumptions on the potential $V$, the associated
semigroups do possess the strong Feller property. Since the invariant measures
(1.1), (1.4) trivially have full support, the strong Feller property
automatically implies unique ergodicity (see [30, Corollary 3.9]). This
approach has had incredible success in infinite dimension as well, with the
result in [30] showing that an incredibly large class of parabolic SPDEs has
the strong Feller property.

Even in situations where the strong Feller property fails or it is otherwise
hard to prove, we do possess an alternative theory, developed by Hairer and
Mattingly in their seminal work [26]. In their work, they introduced the
notion of “asymptotic strong Feller property”. Morally spealing, this property
does not require $P_{t}F$ to be continuous for any positive time $t>0$, but
requires the continuity to hold in the limit $t\to\infty$.222For the actual
definition, we refer the reader to [26, Section 3.2]. The strength of both the
strong Feller property and the asymptotic strong Feller property is
encapsulated in the following support theorem.

######  Proposition 1.2 (Theorem 3.16 in [26]).

Suppose that the semigroup $P_{t}$ has the (asymptotic) strong Feller
property. Let $\mu,\nu$ be two invariant measures with $\mu\perp\nu$. Then we
have

$\operatorname*{supp}(\mu)\cap\operatorname*{supp}(\nu)=\emptyset.$

It is fairly easy to see that the conclusion of this proposition and the
irreducibility property above are in contradiction. Therefore, we have that
irreducibility and the (asymptotic) strong Feller property imply unique
ergodicity, if at least one invariant measure is shown exists.333More
precisely, without this extra assumption, they imply that there exists _at
most_ one invariant measure. The combined toolbox of strong Feller property
and asymptotic strong Feller property has been extremely successful in showing
ergodicity for several classes of SPDEs, including the above mentioned results
for singular parabolic SPDEs [41, 30], Navier-Stokes equations [15, 26, 28],
and in many situations with degenerate noise [27, 8, 10, 19].

However, such techniques do not seem to be easily applicable to wave
equations, and more in general dispersive SPDEs. Indeed, the only results
known to the author that prove ergodicity for stochastic dispersive PDEs are
[1, 5, 13, 18, 39, 16]. While some advances towards the low-regularity regime
has been achieved in the results by the author and by Forlano and the author
[39, 16], none of these results can deal with the singular regime.

The main observation to explain this discrepancy is that dispersive
(stochastic) PDEs seem to _never_ have the strong Feller property on a
connected state space, as it was firstly observed by the author in [39]. The
reason is the following. By expressing the solution of a stochastic dispersive
PDE using the Duhamel/variation of constants formula, we obtain that

$u(t)=S(t)u_{0}+\sqrt{2}\int_{0}^{t}S(t-t^{\prime})\xi(t^{\prime})dt^{\prime}+\mathcal{N}_{t}(u),$

where $S(t)$ denotes the linear propagator for the equation at hand, $\xi$ is
the particular choice of the noise and $\mathcal{N}_{t}(u)$ denotes a
nonlinear remainder. We expect the regularity of the solutions (and hence the
space in which the invariant measures are concentrated) to be dictated by the
stochastic convolution
$\psi=\sqrt{2}\int_{0}^{t}S(t-t^{\prime})\xi(t^{\prime})dt^{\prime},$ which
will typically belong to some Sobolev space $H^{s_{0}-\varepsilon}$ for some
$s_{0}\in\mathbb{R}$, and every $\varepsilon>0$, _but not_ belong to
$H^{s_{0}}$. This forces us to take a state space $X$ for the Markov semigroup
to be rich enough to contain functions that belong to
$H^{s_{0}-\varepsilon}\setminus H^{s_{0}}$, but not are not any smoother. As
it is common in this business, we expect the nonlinear remainder to be
smoother than the linear solution.444As far as the author is aware, this is
the case for every (stochastc) dispersive equation that has a satisfactory
local well posedness theory. The main difference with the parabolic case, is
that the linear propagator is _invertible_ in the Sobolev spaces $H^{\sigma}$
for every $\sigma\in\mathbb{R}$. In particular, the linear propagator
preserves the regularity of the initial data. Therefore, one can test the
definition of the strong Feller property on the following indicator functions

$F(u)=\mathbb{1}_{H^{s_{0}-\varepsilon}\setminus H^{s_{0}}}(u).$ (1.8)

From the discussion abobe, one obtains that

$P_{t}F(u)=F(S(t)u)=F(u),$

which is also an indicator function, hence it is not continuous (as long as
the space is connected, and there exists at least one element of the state
space that also belongs to $H^{s_{0}}$).

In principle, one could try to shrink the state space in order to avoid this
kind of counterexamples, but in many applications this seems to be a fool’s
errand. The reason is that one can replicate the counterexample above by
replacing $H^{s_{0}-\varepsilon}\setminus H^{s_{0}}$ with any set $S$ which is
both invariant for the linear propagator $S(t)$ and by (relatively) smooth
perturbations. In the case of dispersive equations, this is an extremely rich
family.

Of course, as per the discussion above, one could try to completely avoid
relying on the strong Feller property, and instead attempt building the theory
using the asymptotic strong Feller property instead (or the related concept of
asymptotic couplings, see [28, 19, 8]). In principle, this seems to be a
reasonable approach, since the linear equation associated to (SDNLW)

$\begin{cases}u_{t}=v\\\ v_{t}=-v+(1-\Delta)u\end{cases}$

has a propagator $S(t):\begin{pmatrix}u\\\ v\end{pmatrix}\mapsto
S(t)\begin{pmatrix}u\\\ v\end{pmatrix}$ which satisfies

$\|S(t)\|_{H^{s}\times H^{s-1}\righttoleftarrow}\lesssim e^{-\frac{t}{2}}.$

However, in order to show such property, one would need a good long-time
estimate on the difference of two solutions
$\Phi_{t}(\mathbf{u}_{0},\xi_{1})-\Phi_{t}(\mathbf{u}_{1},\xi_{2})$, where
$\xi_{1},\xi_{2}$ are two copies of the noise with the same law as
$\xi$.555One actually just needs some kind of control of the signed measures
$\operatorname{Law}(\xi_{j})-\operatorname{Law}(\xi)$ in total variation. This
generalisation is extremely useful in many applications, and we will make use
of similar ideas in the following sections. This is how Forlano and the author
achieved the ergodicity result for $2k=4$ in the non-singular case [16].
However, dealing with the singular case seems to be beyond the current
technology. The reason is that, since the nonlinearity $u^{3}$ has a
controlled modulus of continuity only on bounded set, one would need to show
some good global estimates for a _single_ solution
$\Phi_{t}(\mathbf{u}_{0},\xi)$ to begin with. As discussed earlier in the
introduction, this seems to be extremely hard in the singular case, and such
an estimate is not known for any singular wave equation666The estimates in
[40, 22] grow with a double exponential in time, which is way too fast for
this argument to work - and their proof works only in the case $2k=4$.. The
main reason is that, contrarily to the parabolic case, the only “useful”
quantity to control the global evolution for wave equations is the energy

$E(u,u_{t})=\frac{a_{2k}}{2k}\int u^{2k}+\frac{1}{2}\int|\nabla
u|^{2}+\frac{1}{2}\int u^{2}+\frac{1}{2}\int|u_{t}|^{2},$

up to some refinements. It should not be a surprise that when solutions became
rougher and rougher, the quantity above gives progressively less and less
information on the growth of solutions, up to a point in which the argument
breaks down completely. It is interesting that for $2k=4$, the threshold of
regularity for obtaining energy estimates corresponds exactly to the threshold
for singularity of the equation (SDNLW). It is unclear to the author if this
is just an accident due to the proof techniques, or there is a deeper
connection between the two.

###  1.2. Asymptotic couplings restricted to the action of a group

In view of the discussion in Section 1.1, one might wonder how it is possible
to get a positive result in Theorem 1.1 to begin with. The starting point is
the following observation by the author in [39], that now we adapt to (SDNLW).
By the analysis in [20, 22], we know that the solution of (SDNLW) can be
written as

$\mathbf{u}(t)=S(t)\mathbf{u}_{0}+\vec{\psi}(t,\xi)+\mathbf{v}(t),$

where $S(t)\mathbf{u}_{0}+\vec{\psi}(t,\xi)$ denotes the (vector) solution
$(\psi,\psi_{t})$ to the _linear_ equation

$\begin{cases}\psi_{tt}+\psi_{t}=\sqrt{2}\xi,\\\
(u,u_{t})(0)=\mathbf{u}_{0},\end{cases}$ (1.9)

and $\mathbf{v}(t)\in H^{1-\varepsilon}\times
H^{-\varepsilon}=:\mathcal{H}^{1-\varepsilon}$ is a smoother nonlinear
reminder. For simplicity of notation, denote
$\mathcal{H}^{s}:=H^{s}\times\mathcal{H}^{s-1}$. Inspired by the form of the
functionals (1.8) that were used to disprove the strong Feller property, we
consider

$F(\mathbf{u})=\mathbb{1}_{\mathcal{H}^{-\varepsilon}\setminus\mathcal{H}^{1-\varepsilon}}.$

We remark here that $\mathcal{H}^{-\varepsilon}\setminus\mathcal{H}^{0}$ is
the typical regularity of solutions with initial data sampled according to
(1.5). As discussed in Section 1.1, we have that

$P_{t}F(\mathbf{u}_{0})=F(\Phi_{t}(\mathbf{u}_{0},\xi))=F(S(t)\mathbf{u}_{0})=F(\mathbf{u}_{0}).$

As a consequence, $P_{t}F$ is not continuous in the topology of
$\mathcal{H}^{-\varepsilon}$. However, if
$\mathbf{u}_{0}^{1},\mathbf{u}_{0}^{2}$ are such that
$\mathbf{u}_{0}^{1}-\mathbf{u}_{0}^{2}\in\mathcal{H}^{1-\varepsilon}$, then we
have

$P_{t}F(\mathbf{u}_{0}^{1})-P_{t}F(\mathbf{u}_{0}^{2})=0.$

In particular, if we consider the distance

$d_{\mathcal{H}^{1-\varepsilon}}(\mathbf{u}_{0}^{1},\mathbf{u}_{0}^{2})=\|\mathbf{u}_{0}^{1}-\mathbf{u}_{0}^{2}\|_{\mathcal{H}^{1-\varepsilon}},$

one has that $P_{t}F$ is _continuous_ in the topology induced by
$d_{\mathcal{H}^{1-\varepsilon}}$. While in this way the state space is not
connected, the change of topology removes the main obstruction to showing the
strong Feller property. If then we are able to show a (stronger) version of
Proposition 1.2, we can deduce that if two invariant measures
$\nu_{1},\nu_{2}$ are such that $\nu_{1}\perp\nu_{2}$, then there exists a set
$E$ such that

$\nu_{1}(E)=1,\hskip 10.0pt\nu_{2}(E)=0,\hskip
25.0ptE=E+\mathcal{H}^{1-\varepsilon}$

Finally, one can show that such properties are in contradiction with the extra
property $\nu_{1},\nu_{2}\ll\rho$. From this, we deduce that the measure
$\rho$ must be ergodic. The reason why the contradiction holds, is that the
family $\\{E=E+\mathcal{H}^{1-\varepsilon}\\}$ is contained in the
$\sigma$-algebra generated by sure events for the Gaussian measure $\rho_{0}$
(see Lemma 3.15 and [39, Remark 5.8]), and so for any such set $E$ we must
have $\nu_{j}(E)=\rho_{0}(E)$.

The strategy described above is essentially how ergodicity for (SDNLW) when
$2k=4$ was shown on the one dimensional torus $\mathbb{T}$ in the previous
work by the author [39]. However, there are a series of issues in extending
this strategy to the 2 dimensional case. The most important of these, is that
it is unclear if the strong Feller property holds after the change of topology
induced by $d_{\mathcal{H}^{1-\varepsilon}}$. The technical reason for this,
is the fact that the space $\mathcal{H}^{1-\varepsilon}$ is strictly bigger
than the Cameron-Martin space for the Gaussian measure $\rho_{0}$, as opposed
to the analogous property for the 1-dimensional flow, which does hold. This
element was used crucially in [39].

Nevertheless, we can still show the following coupling property: for every
$\mathbf{u}_{0}$ such that the solution $\Phi_{t}(\mathbf{u}_{0},\xi)$ does
not grow too fast, and for every $\mathbf{u}_{0}^{2}$ such that
$\mathbf{u}_{0}-\mathbf{u}_{0}^{2}\in\mathcal{H}^{1-\varepsilon}$, there exist
a noise $\xi^{\prime}$ which satisfies
$\operatorname{Law}(\xi^{\prime})\ll\operatorname{Law}(\xi)$ and

$\mathbb{P}\big{(}\\{\|\Phi_{t}(\mathbf{u}_{0}^{1},\xi)-\Phi_{t}(\mathbf{u}_{0}^{2},\xi^{\prime})\|_{\mathcal{H}^{1-\varepsilon}}\lesssim
e^{-\frac{t}{4}}\\}\big{)}>\frac{1}{2}.$

This is essentially the content of Lemma 3.9 below, and can be shown via a
carefully chosen Girsanov shift argument. While we cannot show that the growth
assumption on $\Phi_{t}(\mathbf{u}_{0},\xi)$ holds for every initial data
$\mathbf{u}_{0}$ due to the difficulties described in Section 1.1, we can show
that it holds almost surely according to any invariant measure in the class
$W^{1}_{{:}\,p\mspace{2.0mu}{:}}$. The main novelty of this paper then
consists in codifying the correct support theorem that holds under the very
weak coupling assumption above.

To this scope, it is convenient to put everything into an abstract framework,
and show a support theorem that holds in a general setting. We consider the
space $Y$ of “good” initial data (such that the solution has controlled growth
as $t\to\infty$), and we define the state space to be

$X=Y+\mathcal{H}^{1-\varepsilon}\subset\mathcal{H}^{-\varepsilon}.$

Then we see the space $\mathcal{H}^{1-\varepsilon}$ as a group $\mathscr{G}$
acting on $X$ by translations, i.e.

$\mathscr{G}\times
X\ni(\mathbf{v}_{0},\mathbf{u}_{0})\mapsto\tau_{\mathbf{v}_{0}}(\mathbf{u}_{0})=\mathbf{u}_{0}+\mathbf{v}_{0}.$

This action allows us to define a new topology on the space $X$, which is
induced by the distance

$d_{\mathscr{G}}(\mathbf{u}_{1},\mathbf{u}_{2})=\inf\\{1\wedge\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}:\tau_{\mathbf{v}_{0}}(\mathbf{u}_{1})=\mathbf{u}_{2}\\}=1\wedge\|\mathbf{u}_{1}-\mathbf{u}_{2}\|_{\mathcal{H}^{1-\varepsilon}}$

Under this distance, the space $X$ loses most of the “good” measure-
theoretical properties of Polish spaces, namely, the space
$(X,d_{\mathscr{G}})$ is not separable, it has uncountably many connected
components, and the invariant measure $\rho$ will not be a Radon measure on
this space. Nevertheless, thanks to this definition, the coupling property
above can be codified in the following way: for every $\mathbf{u}_{0}\in X$,
for every $\mathbf{v}_{0}\in\mathscr{G}$ and for every
$\varphi:(X,d_{\mathscr{G}})\to\mathbb{R}$ Lipschitz, we have that

$\big{|}P_{t}\varphi(\mathbf{u}_{0})-P_{t}\varphi(\tau_{\mathbf{v}_{0}}(\mathbf{u}_{0}))\big{|}\leq
2\varepsilon_{0}(\mathbf{u}_{0},\mathbf{v}_{0})\|\varphi\|_{\infty}+e^{-\frac{t}{4}}C(\mathbf{u}_{0},\mathbf{v}_{0})\|\varphi\|_{\operatorname{Lip}}$

for some $\varepsilon_{0}<1$. This is the basis for the concept of _asymptotic
coupling property restricted to the action of a group_ that we will introduce
in Section 2. Similarly, when $\varepsilon_{0}\to 0$ as $\|\mathbf{v}_{0}\|\to
0$, we say that the semigroup has the asymptotic _strong Feller_ property
restricted to the action of the group $\mathscr{G}$. One needs to be very
careful with measurability issues here, since for a function
$\varphi:(X,d_{\mathscr{G}})\to\mathbb{R}$, being Lipschitz does not
automatically imply being measurable. Nevertheless, we can still deduce a
support theorem in the guise of Proposition 1.2, which we formulate in Theorem
2.6 and Theorem 2.7. The latter states that under an appropriate asymptotic
coupling assumption, if $\mu,\nu$ are two invariant measures with
$\mu\perp\nu$, and $X/\mathscr{G}$ is the space of orbits of the action of
$\mathscr{G}$, then $\pi_{\sharp}\mu\perp\pi_{\sharp}\nu$ as well, where
$\pi:X\to X/\mathscr{G}$ is the canonical projection. Together with properties
of the linear evolution for the equation (1.9), this support theorem allows us
to conclude the result of Theorem 1.1.

###  1.3. Structure of the paper

Section 2:

In this section, we introduce the abstract theory and prove the main support
theorems (Theorem 2.6 and 2.7). More specifically, in the various subsections
we will do the following.

  1. 2.1.:

We introduce the assumptions on the measurable space $X$, the semigroup
$P_{t}$ and the group action $\tau:\mathscr{G}\times X\to X$.

  2. 2.2.:

We introduce the definitions of the concepts of asymptotic strong Feller
property restricted to the action of a group (Definition 2.2) and asymptotic
coupling property restricted to the action of a group (Definition 2.3).

  3. 2.3.:

We state our main support theorems, Theorem 2.6 and Theorem 2.7, and in
Example 2.8 and Remark 2.9, we discuss how they relate to the existing theory.

  4. 2.4.:

We perform the proof of the main theorems of this section.

Section 3:

In this section, we focus our attention to the hyperbolic $P(\Phi)_{2}$ model,
and perform the proof of Theorem 1.1. More specifically, in the various
subsections we will do the following.

  1. 3.1.:

We rigorously define the $P(\Phi)_{2}$ measure and the Wick renormalisation,
and discuss a number of properties of each that are relevant for the proof of
Theorem 1.1.

  2. 3.2.:

We collect the existing local and global theory for equation (SDNLW), and use
them to build a Markov process on the space $\mathcal{H}^{-\varepsilon}$.

  3. 3.3.:

We define the space $Y$ of “good” initial data $\mathbf{u}_{0}$, and via a
Girsanov shift argument, show the main estimates conclusive to the coupling
property for (SDNLW).

  4. 3.4.:

We build the Markov semigroup $P_{t}$ on the space
$X=Y+\mathcal{H}^{1-\varepsilon}\cup\\{\infty\\}$, and show that it satisfies
the asymptotic coupling property restricted to the action of
$\mathcal{H}^{1-\varepsilon}$ on $X$. We also show that the asymptotic strong
Feller property restricted to the action of $\mathcal{H}^{1-\varepsilon}$
holds on $Y$.

  5. 3.5.:

We show the $0-1$ property for the measure $\rho_{0}$, and combine it with
Theorem 2.7 to deduce that the $P(\Phi)_{2}$ measure $\rho$ must be ergodic.

  6. 3.6.:

We focus our analysis to the class $W^{1}_{{:}\,p\mspace{2.0mu}{:}}$, and show
that for every invariant measure $\mu$ in this class, one must have that
$\pi_{\sharp}\mu=\pi_{\sharp}\rho$. We then use this, together with the
support theorem 2.7, to conclude uniqueness in the class
$W^{1}_{{:}\,p\mspace{2.0mu}{:}}$ and hence the proof of Theorem 1.1.

###  1.4. Further remarks

######  Remark 1.3.

The main part of the analysis for equation (SDNLW) will happen on the state
space $X$ defined in (3.34). Unfortunately, at this stage we are not able to
show that this space is a Borel subset of $\mathcal{H}^{-\varepsilon}$ (even
if we believe it should be), but only that it has full measure according to
any invariant measure in $W^{1}_{{:}\,p\mspace{2.0mu}{:}}$. While this seems
to be a minor point, due to the unusual setting of the theory in Section 2, we
choose to take an extremely cautious approach, and slowly check that by
performing all the straightforward modifications to the definition of the
measures and of the semigroup, one is still able to apply the theory of
Section 2 to conclude Theorem 1.1. This is the role of the map $\iota^{\ast}$
introduced in Section 3.5. Unfortunately, the addition of this map makes the
proofs of Section 3.5 and 3.6 notationally heavy. On a first reading, the
author would suggest that the reader assumes that the set $X$ is a Borel
subset of $\mathcal{H}^{-\varepsilon}$, in which case $\iota^{\ast}$ is simply
the identity map (after restricting the measure to $X$).

######  Remark 1.4.

In the result of Theorem 1.1, the particular choice of $\varepsilon_{k}$ is
such that the Sobolev inequality

$\|u^{2k-1}\|_{H^{\varepsilon}(\mathbb{T}^{2})}\lesssim\|u\|_{H^{1-\varepsilon}}^{2k-1},$

holds, and does not play a major role in the proof (see Lemma 3.4). It is
likely possible to push the value of $\varepsilon$ to
$\varepsilon<\min(\frac{2}{2k-2},\frac{1}{4}+\frac{1}{2k-2})$, which
corresponds to the local well posedness theory for the equation (SDNLW) (see
[20]). However, such an extension would make the technical part of the proof
significantly harder to digest, without substantially affecting the result of
Theorem 1.1. For sake of exposition, we decided to avoid this further
complication.

######  Remark 1.5.

The (conditional) uniqueness result in Theorem 1.1 depends on the particular
definition of the flow $\Phi_{t}(\mathbf{u}_{0},\xi)$ of (SDNLW). More
precisely, whenever the initial data $\mathbf{u}_{0}$ of (SDNLW) does not
satisfy the hypotheses of Theorem 3.3, the equation does not admit a
satisfying local well posedness theory, and so the definition of the flow
starting from these data is somewhat arbitrary. The choice that we make in
this paper is to declare that if $\mathbf{u}_{0}$ does not satisfy such
hypotheses, then the flow “blows up immediately”, which prevents the existence
of pathological invariant measures which are concentrated on a set where the
flow is not well defined. See (3.14) for the precise definition.

######  Acknowledgements.

The author wishes to thank Tadahiro Oh for his continuous encouragement and
support during the preparation of this paper.

The author was partially supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy-
EXC-2047/1-390685813, through the Collaborative Research Centre (CRC) 1060.

##  2\. Restricted asymptotic strong Feller and restricted asymptotic coupling
properties

In this Section, we introduce the abstract concepts of asymptotic strong
Feller and asymptotic coupling restricted to the action of group, and show how
they imply a support theorem in the same vein as Proposition 1.2. The main
results of this section are Theorem 2.6 and Theorem 2.7.

###  2.1. Assumptions

Throughout this section, we will assume the following.

######  Assumption 1.

Let $X$ be a metric space, with distance $d_{X}$. Let $\mathscr{B}(X)$ be its
Borel sigma-algebra, and let

$\mathscr{L}^{\infty}(X):=\\{f:X\to\mathbb{R}\text{ Borel}:\sup_{x\in
X}|f(x)|<\infty\\},$

equipped with the sup-norm, denoted by $\|f\|_{\infty}$.

######  Assumption 2.

It is given a Markov semigroup $(P_{t})_{t\geq 0}$ on
$\mathscr{L}^{\infty}(X)$. More precisely,

  1. (i)

For every $t\geq 0$, $P_{t}:\mathscr{L}^{\infty}(X)\to\mathscr{L}^{\infty}(X)$
is linear and bounded.

  2. (ii)

For every $t,s\geq 0$, $P_{t+s}=P_{t}P_{s}=P_{s}P_{t}$.

  3. (iii)

For every $t\geq 0$ and for every $f$ Borel with $f(x)\geq 0$ $\forall x\in
X$, then $P_{t}f(x)\geq 0$ $\forall x\in X$.

  4. (iv)

Denoting by $\mathbb{1}$ the constant function $\mathbb{1}(x)=1$ $\forall x\in
X$, for every $t\geq 0$ we have that $P_{t}\mathbb{1}=\mathbb{1}$.

######  Assumption 3.

We have a topological group $\mathscr{G}$ with identity $e$, whose topology is
induced by a left-invariant distance. More precisely, there exists a function
$|\cdot|:\mathscr{G}\to\mathbb{R}$ such that

  1. (i)

for every $g\in\mathscr{G}$, $|g|\geq 0$, and $|g|=0$ if and only if $g=e$,

  2. (ii)

for every $g\in\mathscr{G}$, $|g^{-1}|=|g|$,

  3. (iii)

for every $g_{1},g_{2}\in\mathscr{G}$,

$|g_{1}g_{2}|\leq|g_{1}|+|g_{2}|,$

and the distance $d_{\mathscr{G}}(g_{1},g_{2})$ between two elements
$g_{1},g_{2}$ is given by

$d_{\mathscr{G}}(g_{1},g_{2})=|g_{1}^{-1}g_{2}|=|g_{2}^{-1}g_{1}|.$

Given $r\geq 0$, we denote by $B_{r}$ the closed ball with centre $e$ and
radius $r$:

$B_{r}:=\\{g\in\mathscr{G}:|g|\leq r\\}.$

We remark that Assumption 3 holds if and only if the group $\mathscr{G}$ is
metrisable. See [24, Theorem 8.3].

######  Assumption 4.

We have a group action $\tau:\mathscr{G}\times X\to X$. More precisely, by
denoting $\tau_{g}(u):=\tau(g,u)$, we have that

  1. (i)

for every $x\in X$, $\tau_{e}(x)=x$,

  2. (ii)

for every $g_{1},g_{2}\in\mathscr{G},x\in X$,
$\tau_{g_{1}g_{2}}(x)=\tau_{g_{1}}(\tau_{g_{2}}(x))$.

######  Assumption 5.

For every compact set $K\subseteq X$, the map $g\mapsto\tau_{g}(x)$ is
equicontinuous in $e$ for $x\in K$. More precisely,

$\lim_{r\to 0}\sup_{x\in K}\sup_{g\in B_{r}}d_{X}(x,\tau_{g}(x))=0.$ (2.1)

Moreover, for every $r\geq 0$ and for every compact set $K\subset X$, we have
that

$\tau(B_{r}\times K)\in\mathscr{B}(X).$ (2.2)

We note that the assumptions above are very general, and we do not require
many of the usual properties of the space $X$ and the action $\tau$. For
instance, the space $X$ does not need to be complete, nor separable, and the
action $\tau$ does not need to be continuous in the $X$-variable. The only
compatibility conditions between the topologies of $X$ and $\mathscr{G}$
respectively is delineated in Assumption 5. However, we will only work with
Radon probabilities, which recovers a number of the usual properties of
measures on Polish spaces.

The goal of being so general in the settings is not (only) being able to
provide the most comprehensive statement possible. In order to obtain the
result in Theorem 1.1, we will need to consider a space $X$ which is merely a
subset of the Banach space $\mathcal{H}^{-\varepsilon}$, without any clear
connection with the topology of $\mathcal{H}^{-\varepsilon}$ (and actually, it
is not even clear if the state space is going to be a Borel subset of
$\mathcal{H}^{-\varepsilon}$). This prevents us from exploiting most of the
“usual” assumptions on the space $X$. While the group action we will consider
is going to be continuous with respect to the topology of
$\mathcal{H}^{\varepsilon}$, it is convenient to allow for discontinuous
actions in order to relate the results of this section with the existing
theory, hence our choice of “minimal” compatibility conditions in Assumption
5. See Example 2.8 for more details.

###  2.2. Definitions

As discussed in Section 1.2, the group action will induce a new distance on
the space $X$.

######  Definition 2.1.

The action $\tau$ allows us to define another distance on the space $X$, that
with a slight abuse of notation, we denote with $d_{\mathscr{G}}$. For
$u_{1},u_{2}\in X$, we define $d_{\mathscr{G}}:X^{2}\to[0,\infty]$ by

$d_{\mathscr{G}}(u_{1},u_{2}):=\inf\\{|g|:g\in\mathscr{G},u_{1}=\tau_{g}(u_{2})\\}.$

It is easy to check that this is indeed a metric (here (2.1) guarantees that
$d_{\mathscr{G}}(u_{1},u_{2})=0$ implies $u_{1}=u_{2}$). For a function
$\varphi:X\to\mathbb{R}$, we say that $\varphi$ is _$\mathscr{G}$ -Lipschitz_
if there exists a constant $c\geq 0$ such that

$|\varphi(u_{1})-\varphi(u_{2})|\leq cd_{\mathscr{G}}(u_{1},u_{2}),$

and we denote

$\|\varphi\|_{\mathscr{G}-\operatorname{Lip}}:=\sup_{u_{1},u_{2}\in
X,u_{1}\neq
u_{2}}\frac{|\varphi(u_{1})-\varphi(u_{2})|}{d_{\mathscr{G}}(u_{1},u_{2})}.$

We are now ready to introduce the main new properties in this work, the
asymptotic strong Feller and asymptotic coupling properties restricted to the
action of $\mathscr{G}$.

######  Definition 2.2.

We say that $P_{t}$ has the _asymptotic strong Feller_ property _restricted to
the action of $\mathscr{G}$_ on a set $S\subseteq X$, in short
$\mathrm{(rASF)_{S}}$, if there exists a sequence of times $t_{n}\geq 0$ and a
sequence of positive real numbers $\delta_{n}\to 0$ such that for every
$u_{0}\in S$ and every $\varphi\in\mathscr{L}^{\infty}(X)$ with
$\|\varphi\|_{\mathscr{G}-\operatorname{Lip}}<\infty$, we have

$\big{|}P_{t_{n}}\varphi(u_{0})-P_{t_{n}}\varphi(\tau_{g}(u_{0}))\big{|}\leq
2\varepsilon_{0}(u_{0},g)\|\varphi\|_{\infty}+\delta_{n}C(u_{0},g)\|\varphi\|_{\mathscr{G}-\operatorname{Lip}},$
(rASF)

where $C(u_{0},g)<\infty$ for every $u_{0}\in S,g\in\mathscr{G}$, and for
every $u_{0}\in S$,

$\lim_{|g|\to 0}\varepsilon_{0}(u_{0},g)=0.$

######  Definition 2.3.

We say that $P_{t}$ has _asymptotic coupling_ property _restricted to the
action of $\mathscr{G}$_ on a set $S\subseteq X$, in short
$\mathrm{(rAC)_{S}}$, if there exists a sequence of times $t_{n}\geq 0$ and a
sequence of positive real numbers $\delta_{n}\to 0$, such that for every
$u_{0}\in S$, there exist $r=r(u_{0})>0$ so that for every $g\in B_{r(u_{0})}$
and for every $\varphi\in\mathscr{L}^{\infty}(X)$ with
$\|\varphi\|_{\mathscr{G}-\operatorname{Lip}}<\infty$, we have

$\big{|}P_{t_{n}}\varphi(u_{0})-P_{t_{n}}\varphi(\tau_{g}(u_{0}))\big{|}\leq
2\varepsilon_{0}(u_{0},g)\|\varphi\|_{\infty}+\delta_{n}C(u_{0},g)\|\varphi\|_{\mathscr{G}-\operatorname{Lip}},$
(rAC)

where $\varepsilon_{0}(u_{0},g)<1$ and $C(u_{0},g)<\infty$ for every $u_{0}\in
S$ and $g\in B_{r(u_{0})}$.

######  Remark 2.4.

By definition, we have that the property $\mathrm{(rASF)_{S}}$ implies the the
$\mathrm{(rAC)_{S}}$ property with an arbitrarily small constant
$\varepsilon_{0}(u_{0},g)$.

######  Definition 2.5.

We say that a Borel probability measure $\mu$ on $X$ is _invariant_ for
$P_{t}$ if for every $\varphi\in\mathscr{L}^{\infty}(X)$ and for every $t\geq
0$, we have

$\int_{X}P_{t}\varphi(u)d\mu(u)=\int_{X}\varphi(u)d\mu(u).$ (2.3)

###  2.3. Support theorems

We are now able to state our main support theorem, which should be seen as a
generalisation of Proposition 1.2 to our setting.

######  Theorem 2.6 (Support theorem I).

Suppose that $P_{t}$ satisfies $\mathrm{(rAC)_{X}}$, and let $\mu,\nu$ be
_Radon_ probability measures which are invariant for $P_{t}$ and such that
$\mu\perp\nu$. Then there exist two disjoint sets $U_{0},U_{1}\subseteq X$,
both open with respect to the topology induced by $d_{\mathscr{G}}$ and
measurable with respect to the Borel sigma-algebra on $X$, such that

$\mu(U_{0})=1,\nu(U_{0})=0,\hskip 5.0pt\text{and }\mu(U_{1})=0,\nu(U_{1})=1.$

Moreover,

  1. (i)

if $\varepsilon_{0}(u_{0},g)<\frac{1}{2}$ for every $u_{0}\in X,|g|\leq
r(u_{0})$, then the closures of $U_{0},U_{1}$ in the topology induced by
$d_{\mathscr{G}}$ are also disjoint. In particular, this holds if $P_{t}$
satisfies $\mathrm{(rASF)_{X}}$.

  2. (ii)

if $r(u_{0})=\infty$ for every $u_{0}\in X$, we can choose $U_{0}$ to be
$\mathscr{G}$-invariant, i.e. $\tau(\mathscr{G}\times U_{0})=U_{0}$, and
$U_{1}=(U_{0})^{c}$.

In the particular setting of this paper, it is convenient to repackage (ii) of
this support theorem in the following statement, which will have a direct
application in the proof of Theorem 1.1.

######  Theorem 2.7 (Support theorem II).

Let $X/\mathscr{G}$ be the set of the orbits for the action of $\mathscr{G}$,
i.e.

$X/\mathscr{G}=\\{\tau(\mathscr{G}\times\\{x\\}):x\in X\\},$

and let $\pi:X\to X/\mathscr{G}$ be the canonical projection. Let
$\pi_{\sharp}\mathscr{B}$ be the projected sigma algebra over $X/\mathscr{G}$,
i.e.

$E\in\pi_{\sharp}\mathscr{B}\Longleftrightarrow\pi^{-1}(E)\in\mathscr{B}(X).$

Suppose that the semigroup $P_{t}$ satisfies $\mathrm{(rAC)_{X}}$ with
$r(u_{0})=\infty$ for every $u_{0}\in\mathscr{G}$. Then two Radon probability
measures $\mu,\nu$ which are invariant with respect to $P_{t}$ satisfy

$\mu\perp\nu\iff\pi_{\sharp}\mu\perp\pi_{\sharp}\nu,$

and similarly

$\mu\ll\nu\iff\pi_{\sharp}\mu\ll\pi_{\sharp}\nu.$

######  Example 2.8.

Let $X$ be a Polish space, and let $P_{t}$ be a Markov semigroup on
$\mathscr{L}^{\infty}(X)$ such that for an appropriate sequence of times
$t_{n}$, for every $x\in X$ and every $\eta>0$, there exists a radius
$r=r(x,\eta)>0$ and a sequence $\delta_{n}=\delta_{n}(x)\to 0$ such that

$\sup_{y\in
B(x,r)}|P_{t_{n}}\varphi(x)-P_{t_{n}}\varphi(y)|\leq\eta\|\varphi\|_{\infty}+\delta_{n}(x)\|\varphi\|_{\operatorname{Lip}}.$
(2.4)

It is well known that this condition, together with the assumption that
$P_{t}$ is Feller, implies the asymptotic strong Feller property for $P_{t}$
(see for instance [26, Proposition 3.12]). As we saw in the introduction, the
main consequence of the asymptotic strong Feller property is that for every
two invariant measures $\mu,\nu$ with $\mu\perp\nu$, we have that
$\operatorname*{supp}(\mu)\cap\operatorname*{supp}(\nu)=\emptyset$ (as in
Proposition 1.2). We can derive this result as a consequence of Theorem 2.6 as
well. We consider

$\mathscr{G}=S^{\infty}(X):=\\{f:X\to X\text{ bijective }:\sup_{x\in
X}d(x,f(x))\\},$

with

$|f|:=\|f\|_{\infty}=\sup_{x\in X}d(x,f(x)),$

and we define the action of $\mathscr{G}$ over $X$ simply by

$\tau_{f}(x):=f(x).$

It is easy to check that Assumptions 1, 2, 3, 4 hold. Moving to Assumption 5,
we first notice that in general $\tau_{f}$ is not continuous.777When $X$ is
uncountably infinite, we can build $f$ so that $\tau_{f}$ is not even
_measurable_. For instance, for $X=[0,1]\subset\mathbb{R}$, we can take a non-
measurable subset $E\subset[0,1]$ with the cardinality of $\mathbb{R}$, and
consider a bijective map $f$ so that $f(E)=[0,\frac{1}{2})$ and
$f(E^{c})=[\frac{1}{2},1]$. While it is not necessary here, it is fairly easy
to avoid this measurability issue, simply by adding to the definition of
$S^{\infty}(X)$ the requirement $f$, $f^{-1}$ measurable . Indeed, for every
$x_{0},y_{0}\in X$, one can consider the transposition

$f_{(x_{0},y_{0})}(x):=\begin{cases}y_{0}&\text{ if }x=x_{0},\\\ x_{0}&\text{
if }x=y_{0},\\\ x&\text{ otherwise.}\end{cases}$

We have that $f_{(x_{0},y_{0})}\in S^{\infty}(X)$, with
$|f_{(x_{0},y_{0})}|=d(x_{0},y_{0})$. However, unless $x_{0},y_{0}$ are both
isolated points, it is easy to check that $f_{(x_{0},y_{0})}$ is not a
continuous map on $X$. Nevertheless, Assumption 5 still holds. First of all,
we have that

$d(x,\tau_{f}(x))\leq\|f\|_{\infty}=|f|,$

so we have (2.1). Moreover, if $K$ is compact, it is easy to check that

$\tau(B_{r}\times K)=\\{y\in X:\exists x\in K\text{ s.t. }d(x,y)\leq
r\\}=\\{y\in X:d(y,K)\leq r\\},$ (2.5)

which is a closed subset of $X$, hence measurable. Note that in order to show
the first equality, we need to use the fact that for every $x,y$ with
$d(x,y)\leq r$, there exist an element $f\in S^{\infty}$ with $|f|\leq r$ such
that $\tau_{f}(x)=y$. It is not hard to find a Polish space $X$ and two points
$x,y$ so that no homeomorphism satisfies this property.888For instance
$X=[0,1]$, $x=0$, $0<y<1$. This is why it is convenient not to require
continuity of the action $\tau$ on $X$ in Assumption 5.

We have that for $x,y\in X$,

$\displaystyle d_{\mathscr{G}}(x,y)$ $\displaystyle=\inf\\{|f|:f\in
S^{\infty}(X),x=\tau_{f}(y)\\}$ $\displaystyle=\inf\\{\|f\|_{\infty}:f\in
S^{\infty}(X),x=f(y)\\}$ $\displaystyle=d(x,y).$

Notice that the last equality is achieved by taking $f=f_{(x,y)}$. As a
consequence, we have that

$\|\varphi\|_{\mathscr{G}-\operatorname{Lip}}=\sup_{x,y\in X,x\neq
y}\frac{|\varphi(x)-\varphi(y)|}{d(x,y)}=\|\varphi\|_{\operatorname{Lip}}.$

Therefore, by (2.4), we have that $P_{t}$ satisfies the $\mathrm{(rASF)_{X}}$
property, so we can apply the result of Theorem 2.6. Since
$d_{\mathscr{G}}=d$, we have that if $\mu,\nu$ with $\mu\perp\nu$ are
invariant measures for $P_{t}$,999Recall that every Borel probability measure
on a Polish space is a Radon measure. then there exist two open sets
$U_{0},U_{1}\subset X$ with disjoint closures such that

$\mu(U_{0})=1,\nu(U_{0})=0,\hskip 3.0pt\mu(U_{1})=0,\nu(U_{1})=1.$

Therefore,

$\operatorname*{supp}(\mu)\subseteq\overline{U_{0}},\operatorname*{supp}(\nu)\subseteq\overline{U_{1}}\Rightarrow\operatorname*{supp}(\mu)\cap\operatorname*{supp}(\nu)=\emptyset.$

######  Remark 2.9.

Proceeding as in Example 2.8, when $X$ is a Polish space and
$\mathscr{G}=S^{\infty}(X)$, we can relate the various assumptions of this
section with the various results that are exist in the literature, see for
instance [26, 19, 8, 9, 10]. In particular, one can observe that the
definitions of the $\mathrm{(rASF)_{S}}$ and $\mathrm{(rAC)_{S}}$ properties
are strictly more restrictive than what the existing theory for asymptotic
strong Feller and asymptotic coupling properties (respectively) allows. This
is due to the following two requirements in Definition 2.2 and 2.3, which are
both avoidable in the classical case.

  * •

The sequence of times $t_{n}$ is taken to be the same for every $u_{0}\in S$,

  * •

The sequence $\delta_{n}$ is not allowed to depend on $u_{0}\in S$.

In this paper, we ask for these restriction due to the extremely weak
properties of the topology induced by the distance $d_{\mathscr{G}}$. Namely,
the main obstacles to removing the requirements above are the lack of
separability of $(X,d_{\mathscr{G}})$ and the fact that the measure considered
will not be (in general) Radon measures on the space $(X,d_{\mathscr{G}})$.
This means that, in order to develop the theory, some extra uniformity in the
base point $u_{0}$ is required, since it cannot be recovered via
$\sigma$-additivity.

###  2.4. Proof of the support theorems

We start with a couple of preparatory lemmas.

######  Lemma 2.10.

Let $K\subseteq X$ be a compact set. Then

$K=\bigcap_{\delta>0}\tau(B_{\delta}\times K).$

###### Proof.

Since $e\in B_{\delta}$ for every $\delta>0$ and $\tau_{e}(K)=K$, we have that
$K\subseteq\bigcap_{\delta>0}\tau(B_{\delta}\times K)$. Therefore, we just
need to show the reverse inclusion $\supseteq$. Let
$u\in\bigcap_{\delta>0}\tau(B_{\delta}\times K)$. Then we have that for every
$\delta>0$, there exist $x_{\delta}\in K$ and $g_{\delta}\in B_{\delta}$ so
that

$x=\tau_{g_{\delta}}(x_{\delta}).$

Since $K$ is compact, there exists $\delta_{n}\to 0$ so that $x_{\delta_{n}}$
has a limit in $K$. Let $x_{0}:=\lim_{n\to\infty}x_{\delta_{n}}$. Since
$x_{0}\in K$, we just need to show that $x=x_{0}$. By (2.1), we have that

$\displaystyle
d(x,x_{0})=\lim_{n\to\infty}d(x,x_{\delta_{n}})=\lim_{n\to\infty}d(\tau_{g_{\delta_{n}}}(x_{\delta_{n}}),x_{\delta_{n}})\leq\lim_{n\to\infty}\sup_{x\in
K,|g|\leq\delta_{n}}d(x,\tau_{g}(x))=0,$

so $x=x_{0}\in K$.

∎

######  Lemma 2.11.

Let $A\subseteq X$, and define

$d_{\mathscr{G}}(x,A):=\min\big{(}\inf\\{|g|:x\in\tau_{g}(A)\\},1).$ (2.6)

Then the function $d_{\mathscr{G}}(\cdot,A):X\to\mathbb{R}$ is
$\mathscr{G}$-Lipschitz with
$\|d_{\mathscr{G}}(\cdot,A)\|_{\mathscr{G}-\operatorname{Lip}}\leq 1$.

###### Proof.

Let $x,y\in X$. If $x,y\not\in\tau(\mathscr{G}\times A)$, then
$d_{\mathscr{G}}(x,A)=d_{\mathscr{G}}(y,A)=1$, and so
$|d_{\mathscr{G}}(x,A)-d_{\mathscr{G}}(y,A)|=0$. If
$x\not\in\tau(\mathscr{G}\times A)$ and $y\in\tau(\mathscr{G}\times A)$, then
$x\not\in\tau(\mathscr{G}\times\\{y\\})$ either, and so
$d_{\mathscr{G}}(x,y)=\infty$. Therefore, we just need to prove that

$d_{\mathscr{G}}(x,A)-d_{\mathscr{G}}(y,A)\leq d_{\mathscr{G}}(x,y)$

under the assumption that $x,y\in\tau(\mathscr{G}\times A)$. Fix
$\varepsilon>0$, and let $g,h\in\mathscr{G}$ be such that $\tau_{g^{-1}}(y)\in
A$, $\tau_{h}(y)=x$, and

$\displaystyle d_{\mathscr{G}}(y,A)\geq\min(|g|,1)-\varepsilon,\hskip
5.0ptd_{\mathscr{G}}(x,y)\leq|h|+\varepsilon.$

Then we have that

$\tau_{g^{-1}h^{-1}}(x)=\tau_{g^{-1}}(\tau_{h^{-1}}(x))=\tau_{g^{-1}}(y)\in
A,$

and so

$\displaystyle d_{\mathscr{G}}(x,A)\leq\min(|hg|,1)\leq|h|+\min(|g|,1)\leq
d_{\mathscr{G}}(x,y)+d_{\mathscr{G}}(y,A)+2\varepsilon.$

We conclude by sending $\varepsilon\to 0$. ∎

We focus on the proof of Theorem 2.6, and Theorem 2.7 will be a proven at the
end of this section as a corollary. We then take $\mu,\nu$, and assume that
$\mu\perp\nu$. Therefore, there exists a Borel set $E\subset X$ such that

$\mu(E)=1,\nu(E)=0.$

Since $\mu$ is a Radon measure, for every $\eta>0$ there exists
$E_{\eta}\subseteq E$ compact such that $\mu(E_{\eta})\geq 1-\eta$. Moreover,
since $\nu(E_{\eta})\leq\nu(E)=0$, by Lemma 2.10 and (2.2), there exists some
$1>\eta^{\prime}=\eta^{\prime}(\eta)>0$ such that
$\nu(\tau(B_{\eta^{\prime}}\times E_{\eta}))\leq\eta$. For $\eta>0$, define
the function

$\psi_{\eta}(u):=\min\Big{(}\frac{1}{\eta^{\prime}}d_{\mathscr{G}}(u,E_{\eta}),1\Big{)}.$
(2.7)

From Lemma 2.11, it follows that

$\|\psi_{\eta}\|_{\mathscr{G}-\operatorname{Lip}}\leq\frac{1}{\eta^{\prime}}.$
(2.8)

Moreover, using Assumption 5, we can check that $\psi_{\eta}$ is a measurable
function. Indeed we have that for $s\in\mathbb{R}$,

$\\{\psi_{\eta}\leq s\\}=\begin{cases}X&\text{ if }s\geq 1,\\\
\tau(B_{\eta^{\prime}s}\times E_{\eta})&\text{ if }0\leq s<1,\\\
\emptyset&\text{ if }s<0,\end{cases}$

and every one of these sets is measurable by (2.2).

Since $P_{t}$ satisfies $\mathrm{(rAC)_{X}}$, for every $\eta>0$, we can find
an index $n_{\eta}\in\mathbb{N}$ such that

$\frac{\delta_{n_{\eta}}}{\eta^{\prime}(\eta)}\leq\eta.$ (2.9)

######  Lemma 2.12.

We have that

$\displaystyle\lim_{\eta\to 0}\|P_{t_{n_{\eta}}}\psi_{\eta}\|_{L^{1}(\mu)}=0,$
(2.10) $\displaystyle\lim_{\eta\to
0}\|\mathbb{1}-P_{t_{n_{\eta}}}\psi_{\eta}\|_{L^{1}(\nu)}=0.$ (2.11)

###### Proof.

Since $P_{t}$ is a Markov semigroup and $\psi_{\eta}\geq 0$, we have that
$P_{t_{n_{\eta}}}\psi_{\eta}\geq 0$ as well. Therefore, by (2.3),

$\displaystyle\|P_{t_{n_{\eta}}}\psi_{\eta}\|_{L^{1}(\mu)}$
$\displaystyle=\int P_{t_{n_{\eta}}}\psi_{\eta}(u)d\mu(u)$
$\displaystyle=\int\psi_{\eta}(u)d\mu(u)$
$\displaystyle\leq\int\mathbb{1}_{E_{\eta}^{c}}(u)d\mu(u)$
$\displaystyle\leq\eta,$

which is converging to $0$ as $\eta\to 0$. This shows (2.10). Similarly, since
$\psi_{\eta}\leq\mathbb{1}$, and $P_{t}$ is Markov, we have that
$\mathbb{1}-P_{t_{n_{\eta}}}\psi_{\eta}\geq 0$. Therefore, again by (2.3),

$\displaystyle\|\mathbb{1}-P_{t_{n_{\eta}}}\psi_{\eta}\|_{L^{1}(\nu)}$
$\displaystyle=1-\int P_{t_{n_{\eta}}}\psi_{\eta}(u)d\nu(u)$
$\displaystyle=\int(1-\psi_{\eta})(u)d\nu(u)$
$\displaystyle\leq\int\mathbb{1}_{\tau(B_{\eta^{\prime}}\times
E_{\eta})}(u)d\nu(u)$ $\displaystyle\leq\eta,$

and this shows (2.11). ∎

In view of Lemma 2.12, recalling that convergence in $L^{1}$ implies a.e.
convergence on a subsequence, we have that on a sequence $\eta_{k}\to 0$,

$\displaystyle P_{t_{n_{\eta_{k}}}}\psi_{\eta_{k}}(u)\to 0\text{ for
}\mu-\text{a.e.\ }u,\hskip 5.0ptP_{t_{n_{\eta_{k}}}}\psi_{\eta_{k}}(u)\to
1\text{ for }\nu-\text{a.e.\ }u.$ (2.12)

For convenience (and abusing slightly of notation), we relabel
$t_{k}:=t_{n_{\eta_{k}}}$, $\delta_{k}:=\delta_{n_{\eta_{k}}}$. We define the
sets

$\displaystyle S_{0}:=\\{u\in
X:\lim_{k\to\infty}P_{t_{k}}\psi_{\eta_{k}}(u)=0\\},$ (2.13) $\displaystyle
S_{1}:=\\{u\in X:\lim_{k\to\infty}P_{t_{k}}\psi_{\eta_{k}}(u)=1\\}.$ (2.14)

Since $P_{t_{k}}\psi_{\eta_{k}}$ are measurable functions for every $k$, we
have that the sets $S_{0}$ and $S_{1}$ are both measurable. Moreover, in view
of (2.12), we have that

$\mu(S_{0})=1,\hskip 10.0pt\nu(S_{1})=1,$ (2.15)

and clearly $S_{0}\cap S_{1}=\emptyset$.

######  Lemma 2.13.

Let $u_{0}\in X$, and let $g\in B_{r(u_{0})}$, where $r(u_{0})$ is as in
(rAC). Then

$\limsup_{k\to\infty}\big{|}P_{t_{k}}\psi_{\eta_{k}}(\tau_{g}(u_{0}))-P_{t_{k}}\psi_{\eta_{k}}(u_{0})\big{|}\leq\varepsilon_{0}(u_{0},g).$
(2.16)

In particular, if $u_{0}\in S_{0}$, then $\tau_{g}(u_{0})\not\in S_{1}$, and
similarly, if $u_{0}\in S_{1}$, then $\tau_{g}(u_{0})\not\in S_{0}$.

###### Proof.

First of all, we notice that for every $\eta\geq 0$,

$\|\psi_{\eta}-\frac{1}{2}\|_{\infty}\leq\frac{1}{2}.$

Therefore, by (rAC), (2.8), and (2.9), we have that

$\displaystyle\big{|}P_{t_{k}}\psi_{\eta_{k}}(\tau_{g}(u_{0}))-P_{t_{k}}\psi_{\eta_{k}}(u_{0})\big{|}$
$\displaystyle=\big{|}P_{t_{k}}\big{(}\psi_{\eta_{k}}-\frac{1}{2}\big{)}(\tau_{g}(u_{0}))-P_{t_{k}}\big{(}\psi_{\eta_{k}}-\frac{1}{2}\big{)}(u_{0})\big{|}$
$\displaystyle\leq
2\varepsilon_{0}(u_{0},g)\big{\|}\psi_{\eta_{k}}-\frac{1}{2}\big{\|}_{\infty}+\delta_{k}C(u_{0},g)\|\psi_{\eta_{k}}\|_{\mathscr{G}-\operatorname{Lip}}$
$\displaystyle\leq\varepsilon_{0}(u_{0},g)+C(u_{0},g)\frac{\delta_{k}}{\eta^{\prime}(\eta_{k})}$
$\displaystyle\leq\varepsilon_{0}(u_{0},g)+C(u_{0},g)\eta_{k}.$

Taking the $\limsup$ as $k\to\infty$, we obtain (2.16).

Therefore, if $u_{0}\in S_{0}$, from (2.13) we have that

$\displaystyle\limsup_{k\to\infty}P_{t_{k}}\psi_{\eta_{k}}(\tau_{g}(u_{0}))$
$\displaystyle=\limsup_{k\to\infty}(P_{t_{k}}\psi_{\eta_{k}}(\tau_{g}(u_{0}))-P_{t_{k}}\psi_{\eta_{k}}(u_{0}))\leq\varepsilon_{0}(u_{0},g)<1,$

so in particular $\tau_{g}(u_{0})\not\in S_{1}$. Similarly, if $u_{0}\in
S_{1}$, from (2.14) we obtain

$\displaystyle\liminf_{k\to\infty}P_{t_{k}}\psi_{\eta_{k}}(\tau_{g}(u_{0}))$
$\displaystyle=1+\liminf_{k\to\infty}(P_{t_{k}}\psi_{\eta_{k}}(\tau_{g}(u_{0}))-P_{t_{k}}\psi_{\eta_{k}}(u_{0}))\geq
1-\varepsilon_{0}(u_{0},g)>0,$

so in particular $\tau_{g}(u_{0})\not\in S_{0}$. ∎

We are finally able to complete the proof of Theorem 2.6 and Theorem 2.7.

###### Proof of Theorem 2.6.

It would be natural to define $U_{0}$ to be the set

$\\{\tau_{g}(u_{0}):u_{0}\in S_{0},g<|B_{r(u_{0})/4}|\\},$

and $U_{1}$ analogously. The problem with this definition is that there is no
guarantee that this set is measurable, hence more work is required.

First of all, since $\mu$ and $\nu$ are Radon measures, we notice that there
exist $\sigma$-compact sets ${\widetilde{S}_{0}}\subset S_{0}$ and
${\widetilde{S}_{1}}\subset S_{1}$ respectively such that

$\mu({\widetilde{S}_{0}})=1,\hskip 5.0pt\nu({\widetilde{S}_{1}})=1.$

We define a function $r_{0}:X\to\mathbb{R}$ by

$r_{0}(u_{0}):=\sup\\{r\geq
0:\tau(B_{r}\times\\{u_{0}\\})\cap{\widetilde{S}_{1}}=\emptyset\\}$ (2.17)

if such an $r\geq 0$ exists, and $r_{0}(u_{0})=0$ otherwise. By Assumptions 3
and 4, we also have that

$r_{0}(u_{0})=\sup\\{r\geq
0:u_{0}\not\in\tau(B_{r}\times{\widetilde{S}_{1}})\\},$ (2.18)

when $r_{0}(u_{0})>0$, and

$r_{0}(u_{0})=0\Leftrightarrow
u_{0}\in\bigcap_{r>0}\tau(B_{r}\times{\widetilde{S}_{1}})=\bigcap_{n\in\mathbb{N}}\tau(B_{n^{-1}}\times{\widetilde{S}_{1}})=:\overline{{\widetilde{S}_{1}}},$
(2.19)

where $\overline{{\widetilde{S}_{1}}}$ corresponds exactly to the closure of
the set ${\widetilde{S}_{1}}$ in the topology induced by $d_{\mathscr{G}}$.
Therefore, by Assumption 5, for every $r>0$, the set

$\displaystyle\\{r_{0}\geq
r\\}=\bigcap_{r^{\prime}<r}\tau(B_{r^{\prime}}\times{\widetilde{S}_{1}})^{c}$

is measurable, and clearly $\\{r_{0}\geq 0\\}=X$, so $r_{0}$ is a measurable
function. Moreover, by Lemma 2.13, we have that for every $u_{0}\in S_{0}$,
$r_{0}(u_{0})\geq r(u_{0})$. Finally, we can check that
$\|r_{0}\|_{\mathscr{G}-\operatorname{Lip}}\leq 1$. Indeed, proceeding as in
the proof of Lemma 2.11, for $x,y\in X$, we can assume that
$d_{\mathscr{G}}(x,y)<\infty$ and $x,y\not\in\overline{{\widetilde{S}_{1}}}$,
and pick $g\in\mathscr{G}$ so that $x=\tau_{g}(y)$,

$\displaystyle d_{\mathscr{G}}(x,y)\geq|g|-\varepsilon.$

We obtain that

$\displaystyle r_{0}(x)$ $\displaystyle=\sup\\{r\geq
0:x\not\in\tau(B_{r}\times{\widetilde{S}_{1}})\\}$ $\displaystyle=\sup\\{r\geq
0:\tau_{g}(y)\not\in\tau(B_{r}\times{\widetilde{S}_{1}})\\}$
$\displaystyle=\sup\\{r\geq
0:y\not\in\tau(\tau_{g^{-1}}(B_{r})\times{\widetilde{S}_{1}})\\}$
$\displaystyle\geq\sup\\{r\geq
0:y\not\in\tau(B_{r+|g|}\times{\widetilde{S}_{1}})\\}$
$\displaystyle=r_{0}(y)-|g|$ $\displaystyle\geq
r_{0}(y)-d_{\mathscr{G}}(x,y)+\varepsilon.$

Therefore, by taking $\varepsilon\to 0$, we obtain that for every $x,y\in X$,

$r_{0}(y)-r_{0}(x)\leq d_{\mathscr{G}}(x,y).$

By swapping the roles of $x$ and $y$, we deduce that
$\|r_{0}\|_{\mathscr{G}-\operatorname{Lip}}\leq 1$.

Proceeding similarly, we can define the map $r_{1}:X\to\mathbb{R}$ by

$r_{1}(u_{0}):=\sup\\{r\geq
0:\tau(B_{r}\times\\{u_{0}\\})\cap{\widetilde{S}_{0}}=\emptyset\\},$ (2.20)

and $r_{1}(u_{0})=0$ if $u_{0}\in\overline{{\widetilde{S}_{0}}}$ (defined
analogously to $\overline{{\widetilde{S}_{1}}}$ in (2.19)). This map will
satisfy the same properties as $r_{0}$, that is, $r_{1}$ is measurable,
$\|r_{1}\|_{\mathscr{G}-\operatorname{Lip}}\leq 1$, and $r_{1}(u_{0})\geq
r(u_{0})$ for every $u_{0}\in S_{1}$. We are finally ready to define the sets
$U_{0}$ and $U_{1}$. Let

$U_{0}:=\\{r_{0}>2r_{1}\\},\hskip 5.0ptU_{1}:=\\{r_{1}>2r_{0}\\}.$

Since $r_{0},r_{1}$ are $\mathscr{G}$-Lipschitz, the sets $U_{0}$ and $U_{1}$
are open with respect to the topology induced by $d_{\mathscr{G}}$. Moreover,

$U_{0}\cap U_{1}\subseteq\\{r_{0}<0\\}\cap\\{r_{1}<0\\}=\emptyset.$

For $u_{0}\in\widetilde{S}_{0}$, by Lemma 2.13 have that

$\displaystyle r_{0}(u_{0})\geq r(u_{0})>0,r_{1}(u_{0})=0,$

so $\widetilde{S}_{0}\subseteq U_{0}$, and similarly
$\widetilde{S}_{1}\subseteq U_{1}$. Therefore,

$\mu(U_{0})=1,\nu(U_{1})=1.$

If moreover $\varepsilon_{0}(u_{0},g)<\frac{1}{2}$ for every $u_{0}\in X$,
$g\in B_{r(u_{0})}$, we consider

$\overline{U_{0}}\cap\overline{U_{1}}\subseteq\\{r_{0}=0\\}\cap\\{r_{1}=0\\}=\overline{{\widetilde{S}_{0}}}\cap\overline{{\widetilde{S}_{1}}}.$

We just need to show that in this case
$\overline{{\widetilde{S}_{0}}}\cap\overline{{\widetilde{S}_{1}}}$ is empty.
By definition,

$\overline{{\widetilde{S}_{0}}}\cap\overline{{\widetilde{S}_{1}}}\subseteq\\{u_{0}:\tau(B_{r(u_{0})}\times\\{u_{0}\\})\cap{\widetilde{S}_{0}}\neq\emptyset,\tau(B_{r(u_{0})}\times\\{u_{0}\\})\cap{\widetilde{S}_{1}}\neq\emptyset\\}.$

Suppose by contradiction that this set is not empty, and let
$u_{0}\in\overline{{\widetilde{S}_{0}}}\cap\overline{{\widetilde{S}_{1}}}$.
Then there exist $g_{0},g_{1}\in B_{r(u_{0})}$ so that $\tau_{g_{0}}(u_{0})\in
S_{0}$, $\tau_{g_{1}}(u_{0})\in S_{1}$. Therefore, by (2.13), (2.14) and
(2.16),

$\displaystyle 1$
$\displaystyle=\lim_{k\to\infty}P_{t_{k}}\psi_{\eta_{k}}(\tau_{g_{1}}(u_{0}))-P_{t_{k}}\psi_{\eta_{k}}(\tau_{g_{0}}(u_{0}))$
$\displaystyle\leq\limsup_{k\to\infty}\big{|}P_{t_{k}}\psi_{\eta_{k}}(\tau_{g_{1}}(u_{0}))-P_{t_{k}}\psi_{\eta_{k}}(u_{0})\big{|}+\limsup_{k\to\infty}\big{|}P_{t_{k}}\psi_{\eta_{k}}(u_{0})-P_{t_{k}}\psi_{\eta_{k}}(\tau_{g_{0}}(u_{0}))\big{|}$
$\displaystyle\leq\varepsilon(u_{0},g_{1})+\varepsilon(u_{0},g_{0})$
$\displaystyle<1,$

contradiction.

We now move to the case where $r(u_{0})=\infty$ for every $U_{0}$. In this
case, we can simply define

$U_{0}:=\tau(\mathscr{G}\times{\widetilde{S}_{0}}),$

and the fact that $U_{0}\cap S_{1}=\emptyset$, hence $\nu(U_{0})=0$, follows
from Lemma 2.13. ∎

###### Proof of Theorem 2.7.

First of all, we notice that if $\pi_{\sharp}\mu\perp\pi_{\sharp}\nu$, then
there exists a set $E\in\pi_{\sharp}\mathscr{B}$ so that
$\pi_{\sharp}\mu(E)=1$, $\pi_{\sharp}\nu(E)=0$, so $\mu(\pi^{-1}(E))=1$ and
$\nu(\pi^{-1}(E))=0$, and we obtain that $\mu\perp\nu$. For the reverse
implication, by Theorem 2.6, there exist a set $U_{0}=\tau(\mathscr{G}\times
U_{0})$ such that $\mu(U_{0})=1$, $\nu(U_{0})=0$. Since

$\displaystyle\pi^{-1}(\pi(U_{0}))$
$\displaystyle=\\{\tau_{g}(u):g\in\mathscr{G},\pi(u)=\pi(u_{0})\text{ for some
}u_{0}\in U_{0}\\}$
$\displaystyle=\\{\tau_{g}(u):g\in\mathscr{G},u=hu_{0}\text{ for some
}h\in\mathscr{G},u_{0}\in U_{0}\\}$
$\displaystyle=\tau(\mathscr{G}\times\tau(\mathscr{G}\times U_{0}))$
$\displaystyle=\tau(\mathscr{G}\times U_{0})$
$\displaystyle=U_{0}\in\mathscr{B}(X),$

we have that $\pi(U_{0})\in\pi_{\sharp}\mathscr{B}$. Therefore, we have that

$\pi_{\sharp}\mu(\pi(U_{0}))=\mu(\pi^{-1}(\pi(U_{0})))=\mu(U_{0})=1,\hskip
5.0pt\pi_{\sharp}\nu(\pi(U_{0}))=\nu(\pi^{-1}(\pi(U_{0})))=\nu(U_{0})=0,$

so $\pi_{\sharp}\mu\perp\pi_{\sharp}\nu$.

We now move to the other equivalence. The implication
$\mu\ll\nu\Rightarrow\pi_{\sharp}\mu\ll\pi_{\sharp}\nu$ follows from the
general property that the push-forward of measures preserves absolute
continuity. Therefore, we focus on the reverse implication. Given $\mu,\nu$
with $\pi_{\sharp}\mu\ll\pi_{\sharp}\nu$, suppose by contradiction that
$\mu\not\ll\nu$. Let then $\mu_{s}$ be the singular part of $\mu$ with respect
to $\nu$, i.e.

$\mu_{s}^{0}:=\frac{d\mu}{d(\mu+\nu)}\mathbb{1}_{\\{\frac{d\nu}{d(\mu+\nu)}=0\\}}(\mu+\nu),\hskip
5.0pt\mu_{s}:=\frac{\mu_{s}^{0}}{\mu_{s}^{0}(X)},$ (2.21)

where $\frac{d\mu}{d(\mu+\nu)},\frac{d\nu}{d(\mu+\nu)}$ denote the Radon-
Nykodim derivatives. It is a standard argument to see that $\mu_{s}$ is a
probability measure invariant for $P_{t}$ (see for instance [16, Lemma 5.11]).
Moreover, we have that $\mu_{s}\perp\nu$. Therefore, by the previous part of
the proof, $\pi_{\sharp}\mu_{s}\perp\pi_{\sharp}\nu$. But $\mu_{s}\ll\mu$, so
$\pi_{\sharp}\mu_{s}\ll\pi_{\sharp}\mu\ll\pi_{\sharp}\nu$, which is a
contradiction. ∎

##  3\. The hyperbolic $P(\Phi)_{2}$-model

###  3.1. On of the $P(\Phi)_{2}$ measure and Wick renormalisation

Let $\mathbb{T}^{2}=\Big{(}\mathbb{R}/[0,2\pi]\Big{)}^{2}$ denote the standard
2-dimensional torus, and let $dx$ be the Lebesgue measure on the torus. We
first start by considering the Gaussian measure $\rho_{0}$ (1.7), formally
given by

$d\rho_{0}(u,u_{t})=\frac{1}{Z}\exp\Big{(}-\frac{1}{2}\int_{\mathbb{T}^{2}}\big{(}|u|^{2}+|\nabla
u|^{2}\big{)}dx-\frac{1}{2}\int_{\mathbb{T}^{2}}|u_{t}|^{2}dx\Big{)}dudu_{t}.$

By expressing the norms above in Fourier series, we get that (formally) the
measure above corresponds to the measure

$d\rho(u,u_{t})=\prod_{n\in\mathbb{Z}^{2}}\frac{1}{Z^{1}_{n}}\exp\Big{(}-\frac{1}{8\pi^{2}}\langle
n\rangle^{2}|\widehat{u}(n)|^{2}\Big{)}d\widehat{u}(n)\prod_{n\in\mathbb{Z}^{2}}\frac{1}{Z^{2}_{n}}\exp\Big{(}-\frac{1}{8\pi^{2}}|\widehat{u_{t}}(n)|^{2}\Big{)}d\widehat{u_{t}}(n)$

restricted to the set
$\\{\widehat{u}(-n)=\overline{\widehat{u}(n)},\widehat{u_{t}}(-n)=\overline{\widehat{u_{t}}(n)}\\}$,
where $\langle n\rangle:=\sqrt{1+|n|^{2}}$. Therefore, we can write

$\rho(u,u_{t})=\operatorname{Law}(\mathbf{U}),$ (3.1)

where $\mathbf{U}=(U,V)$ is given by

$\displaystyle U$
$\displaystyle=\frac{1}{2\pi}\operatorname*{Re}{\Big{(}\sum_{n\in\mathbb{Z}^{2}}\frac{g_{n}}{\langle
n\rangle}e^{in\cdot x}\Big{)}},$ $\displaystyle V$
$\displaystyle=\frac{1}{2\pi}\operatorname*{Re}{\Big{(}\sum_{n\in\mathbb{Z}^{2}}{h_{n}}e^{in\cdot
x}\Big{)}},$

and $\\{g_{n}\\}_{n\in\mathbb{N}},\\{h_{n}\\}_{n\in\mathbb{N}}$ are i.i.d.,
complex-valued normal random variables.

For $\sigma\in\mathbb{R}$, define the Hermite polynomials via their generating
function, i.e.

$H_{n}(x,\sigma^{2}):=\frac{d^{n}}{dt^{n}}\big{(}e^{tx-\frac{1}{2}\sigma^{2}t^{2}}\big{)},$
(3.2)

or equivalently, $H_{n}(x,\sigma^{2})$ are the only functions such that the
equality

$e^{tx-\frac{1}{2}\sigma^{2}t^{2}}=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}H_{n}(x,\sigma^{2})$
(3.3)

holds for every $t\in\mathbb{R}$. For $N\in 2^{\mathbb{\mathbb{N}}}$ dyadic,
define the (sharp) Fourier projector $\pi_{N}$ via the equality

$\widehat{\pi_{N}f}(n)=\widehat{f}(n)\mathbb{1}_{\\{|n|_{\infty}\leq N\\}},$
(3.4)

where for $(n_{1},n_{2})\in\mathbb{Z}^{2}$, we denote
$|(n_{1},n_{2})|_{\infty}=\max\\{|n_{1}|,|n_{2}|\\}$. We then define the
variance

$\sigma_{N}^{2}=\mathbb{E}|{\pi_{N}U}(x)|^{2}=\frac{1}{4\pi}\sum_{|n|_{\infty}\leq
N}\frac{1}{\langle n\rangle^{2}}\sim\log N.$ (3.5)

Finally, for a function $u\in H^{-\varepsilon}$, we denote

${:}\,u^{j}\mspace{2.0mu}{:}=\lim_{N\to\infty}H(\pi_{N}u(\cdot),\sigma_{N}),$
(3.6)

whenever this limit exists in the Sobolev space $H^{-\varepsilon}$. We have
the following properties

######  Proposition 3.1 (Properties of the Wick powers).

Fix $j\in\mathbb{N}$. Then, for every $0\leq\varepsilon\leq\varepsilon_{j}\ll
1$, we have the following.

  1. 1.

Let $u\in H^{-\varepsilon}$ be such that
${:}\,u^{0}\mspace{2.0mu}{:},{:}\,u^{1}\mspace{2.0mu}{:},\dotsc,{:}\,u^{j}\mspace{2.0mu}{:}$
are well-defined and belong to the Sobolev space $W^{-\varepsilon,4}$. Let
$v\in H^{1-\varepsilon}$. Then ${:}\,(u+v)^{j}\mspace{2.0mu}{:}$ is also well
defined, and it satisfies

${:}\,(u+v)^{j}\mspace{2.0mu}{:}=\sum_{h\leq
j}\binom{h}{j}{:}\,u^{h}\mspace{2.0mu}{:}v^{j}\in H^{-\varepsilon}.$

  2. 2.

Let $p(x)=a_{j}x^{j}+\dotsb+a_{0}$ be a polynomial of degree $j$, and let
$\\{p_{h}\\}_{1\leq h\leq j}$ be the unique polynomials such that

$p(x+y)-p(y)=\sum_{h=1}^{j}p_{h}(x)y^{h}.$

Suppose that $u\in H^{-\varepsilon}$ is such that
${:}\,u^{0}\mspace{2.0mu}{:},{:}\,u^{1}\mspace{2.0mu}{:},\dotsc,{:}\,u^{j}\mspace{2.0mu}{:}$
are well-defined in $W^{-\varepsilon,4}$, and let $v\in H^{1-\varepsilon}$.
Then

${:}\,p(u+v)\mspace{2.0mu}{:}-{:}\,p(u)\mspace{2.0mu}{:}=\sum_{h=1}^{j}{:}\,p_{h}(u)\mspace{2.0mu}{:}v^{h}.$

  3. 3.

For $\rho_{0}$-a.e. $u$, we have that ${:}\,u^{j}\mspace{2.0mu}{:}$ is well
defined and belongs to the Sobolev space $W^{-\varepsilon,\infty}$. Moreover,
for every $p<\infty$, we have that

$\int\|{:}\,u^{j}\mspace{2.0mu}{:}\|_{W^{-\varepsilon,\infty}}^{p}d\rho_{0}(u,u_{t})<\infty.$

###### Proof.

This is essentially a collection of well-known results, so we keep the proof
short. From (3.3), we see that we must have

$\frac{d}{dx}H_{n}(x,\sigma^{2})=nH_{n-1}(x,\sigma^{2}),$

and so by Taylor series expansion, we obtain

$H_{j}(x+y,\sigma^{2})=\sum_{h=0}^{j}\binom{n}{j}H_{h}(x,\sigma^{2})y^{h}.$

By taking limits and exploiting the continuity of the map
$W^{-\varepsilon,4}\times H^{1-\varepsilon}\to H^{-\varepsilon}$ given by
$(u,v)\mapsto uv$ (when $\varepsilon$ is small enough), we obtain 1. Notice
that when $p(x)=x^{j}$, then the statement of 2. coincides with the statement
of 1. Therefore, 2. follows from 1. by linearity in the coefficients of $p$.

Finally, 3. can be found (for instance) in [22, Lemma 2.3]. ∎

We conclude this subsection with the following result, which provides a
rigorous definition of the measure $\rho$ in (1.5).

######  Proposition 3.2.

Let $P=a_{2k}x^{2k}+\dotsb+a_{0}$ be a polynomial of even degree, and consider
the functional

$F_{P}(u)=\exp\Big{(}-\int_{\mathbb{T}^{2}}{:}\,P(u)\mspace{2.0mu}{:}dx\Big{)}.$

By Proposition 3.1, the functional $F_{P}(u)$ is well-defined for
$\rho_{0}$-almost every $u,u_{t}$ (as a function of the first variable).
Moreover, we have that for every $p<\infty$,

$F_{p}\in L^{p}(\rho).$

Therefore, we define the measure $\rho$ to be

$\rho(u,u_{t})=\frac{F_{P}(u)}{\int
F_{P}(u)d\rho_{0}(u,u_{t})}\rho_{0}(u,u_{t})$

###### Proof.

This is an immediate corollary of [38, Theorem V.7]. See also [22, Lemma 2.3].
∎

###  3.2. Local and global theory for (SDNLW)

In this subsection, we quickly recap the existing local and global well-
posedness theory for (SDNLW),

$\displaystyle\partial_{t}^{2}u+\partial_{t}u+(1-\Delta)u+{:}\,p(u)\mspace{2.0mu}{:}=\sqrt{2}\xi.$

For the purpose of this section, it is convenient to write the equation in
vectorial form in the variable $\mathbf{u}=\begin{pmatrix}u\\\
u_{t}\end{pmatrix}$,

$\partial_{t}\begin{pmatrix}u\\\ u_{t}\end{pmatrix}=\begin{pmatrix}0&1\\\
-(1-\Delta)&-1\end{pmatrix}\begin{pmatrix}u\\\
u_{t}\end{pmatrix}-\begin{pmatrix}0\\\
{:}\,p(u)\mspace{2.0mu}{:}\end{pmatrix}+\begin{pmatrix}0\\\
\sqrt{2}\xi\end{pmatrix}.$ (3.7)

We define the linear propagator for this equation to be

$\displaystyle S(t)$ $\displaystyle=\exp\bigg{(}t\begin{pmatrix}0&1\\\
-(1-\Delta)&-1\end{pmatrix}\bigg{)}$ (3.8)
$\displaystyle=e^{-\frac{t}{2}}\begin{pmatrix}\cos\Big{(}t\sqrt{\frac{3}{4}-\Delta}\Big{)}+\frac{1}{2}\frac{\sin\Big{(}t\sqrt{\frac{3}{4}-\Delta}\Big{)}}{\sqrt{\frac{3}{4}-\Delta}}&\frac{\sin\Big{(}t\sqrt{\frac{3}{4}-\Delta}\Big{)}}{\sqrt{\frac{3}{4}-\Delta}}\\\
-\Big{(}\sqrt{\frac{3}{4}-\Delta}+\frac{1}{4\sqrt{\frac{3}{4}-\Delta}}\Big{)}\sin\Big{(}t\sqrt{\frac{3}{4}-\Delta}\Big{)}&\cos\Big{(}t\sqrt{\frac{3}{4}-\Delta}\Big{)}-\frac{1}{2}\frac{\sin\Big{(}t\sqrt{\frac{3}{4}-\Delta}\Big{)}}{\sqrt{\frac{3}{4}-\Delta}}\end{pmatrix}.$

Immediately from this definition, we obtain

$\|S(t)\mathbf{u}_{0}\|_{\mathcal{H}^{s}}\lesssim
e^{-\frac{t}{2}}\|\mathbf{u}_{0}\|_{\mathcal{H}^{s}}.$ (3.9)

for every $s\in\mathbb{R}$ (recall the definition
$\mathcal{H}^{s}:=H^{s}\times H^{s-1}$). As mentioned in Section 1.2, we want
to express the solution of (3.7) as linear solution + nonlinear remainder. To
this scope, we define

$\vec{\psi}[\xi](t):=\int_{0}^{t}S(t-t^{\prime})\begin{pmatrix}0\\\
\sqrt{2}\xi\end{pmatrix}dt^{\prime},$ (3.10)

and we call the components of $\vec{\psi}[\xi](t)$

$\vec{\psi}[\xi](t)=\begin{pmatrix}\psi[\xi](t)\\\
\psi_{t}[\xi](t)\end{pmatrix}.$ (3.11)

Notice that this way, $\vec{\psi}[\xi]$ is the solution of the linear equation

$\begin{cases}\psi_{tt}+\psi_{t}+(1-\Delta)\psi=\sqrt{2}\xi,\\\
\vec{\psi}[\xi](0)=(0,0).\end{cases}$

We point out here that $\vec{\psi}$ is a low-regularity object. Namely, for
every $t>0$, $\vec{\psi}\in\mathcal{H}^{-\varepsilon}$ for every
$\varepsilon>0$, but $\vec{\psi}\not\in\mathcal{H}^{0}$.101010This can easily
seen (for instance) from the fact that
$\int|\pi_{N}\psi|^{2}-\mathbb{E}\int|\pi_{N}\psi|^{2}$ converges almost
surely (to $\int{:}\,\psi^{2}\mspace{2.0mu}{:}$), but
$\mathbb{E}\int|\pi_{N}\psi|^{2}\to\infty$ as $N\to\infty$, which together
imply that $\psi\not\in L^{2}$. Both the almost sure convergence and the
divergence of $\mathbb{E}\int|\pi_{N}\psi|^{2}$ follow from arguments similar
to the ones in Section 3.1. Finally, for an initial data $\mathbf{u}_{0}$, we
define $\mathbf{v}$ by
$\mathbf{u}(t):=S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t)+\mathbf{v}(t)$, where
$\mathbf{u}$ (formally) solves (3.7). This way,
$\mathbf{v}(t)=\begin{pmatrix}v\\\ v_{t}\end{pmatrix}$ will solve the equation

$\begin{cases}\partial_{t}\begin{pmatrix}v\\\
v_{t}\end{pmatrix}=\begin{pmatrix}0&1\\\
-(1-\Delta)&-1\end{pmatrix}\begin{pmatrix}v\\\
v_{t}\end{pmatrix}-\begin{pmatrix}0\\\
{:}\,p(S(t)\mathbf{u}_{0}+\psi[\xi](t)+\mathbf{v}(t))\mspace{2.0mu}{:}\end{pmatrix},\vspace{5pt}\\\
\begin{pmatrix}v(0)\\\ v_{t}(0)\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix}.\end{cases}$ (3.12)

We obtain the following local well-posedness statement.

######  Theorem 3.3 ([20, Theorem 1.1],[22, Proposition 4.1]).

Define

$\Xi(\mathbf{u}_{0},\xi)(t):=\big{(}S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t),{:}\,(S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t))^{2}\mspace{2.0mu}{:},\dotsc,{:}\,(S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t))^{2k-1}\mspace{2.0mu}{:}\big{)}.$

For $0<\varepsilon\leq\varepsilon_{k}\ll 1$, let
$\mathbf{u}_{0}\in\mathcal{H}^{-\varepsilon}$ be such that
$\Xi(\mathbf{u}_{0},\xi)\in(L^{2}_{\mathrm{loc}}(\mathbb{R}^{+};W^{-\varepsilon,4}))^{2k-1}\text{
a.s. }$ Then the equation (SDNLW) is almost surely locally well-posed. More
precisely, there exists a random time
$T^{*}=T^{*}(\|\Xi\|_{(L^{2}([0,1];W^{-\varepsilon,4}))^{2k-1}})>0$ such that
the equation (3.12) has a unique solution $\mathbf{v}\in
C([0,T^{*}],\mathcal{H}^{1-\varepsilon})$. For $t<T^{*}$, we denote

$\Phi_{t}(\mathbf{u}_{0},\xi):=S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t)+\mathbf{v}(t).$

Moreover, the map $\Xi\mapsto\mathbf{v}$ is continuous as a map from
$(L^{1}([0,1];W^{-\varepsilon,4}))^{2k-1}$ to
$C([0,T^{*}],\mathcal{H}^{1-\varepsilon})$.

We remark here that while in the papers cited, Theorem 3.3 was shown only in
the case $p(x)=x^{2k-1}$. However, the result for a general polynomial $p$ is
a straightforward and easy modification of the arguments in [22]. The same
remark will hold for Theorem 3.5 below.

On the basis of Theorem 3.3, we define the set of “good” initial data to be

$\Gamma:=\\{\mathbf{u}_{0}\in\mathcal{H}^{-\varepsilon}:\Xi(\mathbf{u}_{0},\xi)\in\bigotimes_{j=1}^{2k-1}W^{-\varepsilon,\frac{4(2k-1)}{j}}\text{
a.s.}\\}.$ (3.13)

######  Lemma 3.4.

Let $\mathbf{u}_{0}\in\Gamma$, and suppose that $\mathbf{v}(t)\in
C([0,T),\mathcal{H}^{1-\varepsilon})$ solves (3.12) on the interval $[0,T)$.
Then
$\mathbf{u}(t)=S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t)+\mathbf{v}(t)\in\Gamma$
for every $0\leq t<T$. Moreover, let $t_{0}<T$, and suppose that
$\mathbf{v}_{t_{0}}\in C([0,T_{1}),\mathcal{H}^{1-\varepsilon})$ solves (3.12)
on some interval $[0,T_{1})$ with initial data $\mathbf{u}(t_{0})$ and noise
$\tilde{\xi}:=\xi(t-t_{0},\cdot)$. Let

$\mathbf{u}_{t_{0}}(t)=S(t)\mathbf{u}(t_{0})+\vec{\psi}[\xi(t-t_{0},\cdot)]+\mathbf{v}_{t_{0}}(t).$

Then

  1. (i)

For $t_{0}\leq t\leq\min(T,t_{0}+T_{1})$,

$\mathbf{u}(t)=\mathbf{u}_{t_{0}}(t-t_{0}),$

  2. (ii)

For $t_{0}\leq t\leq t_{0}+T_{1}$, let
$\widetilde{\mathbf{v}}(t):=\mathbf{u}_{t_{0}}(t-t_{0})-S(t)\mathbf{u}_{0}-\psi[\xi](t)$.
Then $\widetilde{\mathbf{v}}$ satisfies $\widetilde{\mathbf{v}}\in
C([t_{0},t_{0}+T_{1}),\mathcal{H}^{1-\varepsilon}),$ and
$\widetilde{\mathbf{v}}$ solves (3.12) on the interval $[t_{0},t_{0}+T_{1})$
with initial data $\mathbf{u}_{0}$ and noise $\xi$.

###### Proof.

For $t\in\mathbb{R}$, define $\xi_{t}(s):=\xi(t+s)$. Since the law of white
noise is independent of time, we have that
$\operatorname{Law}(\xi_{t})=\operatorname{Law}(\xi)$ for every
$t\in\mathbb{R}$. Notice that, by (3.10) and the semigroup property of $S(t)$,
$S(t_{1}+t_{2})=S(t_{1})S(t_{2})$, we have that
$\Xi(S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t),\xi_{t})(s)=\Xi(\mathbf{u}_{0},\xi)(t+s).$
Therefore, we have that if $\mathbf{u}_{0}\in\Gamma$, then
$S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t)\in\Gamma$ as well. The fact that we
also have $S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t)+\mathbf{v}(t)\in\Gamma$ for
$\varepsilon\leq\varepsilon_{k}$ small enough is a direct consequence of
Proposition 3.1, 1. and standard product estimates.

We now move to showing (i). From the uniqueness in Theorem 3.3 (see also [22,
Proposition 4.1]), it is enough to check that

$\overline{v}(t-t_{0}):=S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t)+\mathbf{v}(t)-S(t-t_{0})\big{(}S(t_{0})\mathbf{u}_{0}+\vec{\psi}[\xi](t_{0})+\mathbf{v}(t_{0})\big{)}-\vec{\psi}[\xi_{t_{0}}](t-t_{0})$

solves (3.12) with initial data $\mathbf{u}(t_{0})$ and noise $\xi_{t_{0}}$.
This is a straightforward (but tedious) computation. The proof of (ii) is
completely analogous. ∎

The previous lemma allows us to define a stochastic flow on the set

$\widetilde{X}:=\mathcal{H}^{-\varepsilon}\cup\\{\infty\\}.$

Here $\infty$ denotes a “cemetery state”, that we use to keep track of when
the flow blows up (or it is not well defined). Indeed, we declare that

$\Phi_{t}(\mathbf{u}_{0},\xi)=\begin{cases}S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t)+\mathbf{v}(t)&\begin{aligned}
&\text{if }\mathbf{u}_{0}\in\Gamma\text{ and }\mathbf{v}\in
C([0,t],\mathcal{H}^{1-\varepsilon})\text{ solves \eqref{SNLWv},}\\\ &\text{if
such a solution exists},\end{aligned}\\\ \infty&\text{ otherwise}.\end{cases}$
(3.14)

In view of Lemma 3.4, the flow as defined will satisfy the semigroup property

$\Phi_{t+s}(\mathbf{u}_{0},\xi)=\Phi_{t}(\Phi_{s}(\mathbf{u}_{0},\xi),\xi(t-t_{0},\cdot)).$
(3.15)

Moreover, we have the following global well-posedeness statement.

######  Theorem 3.5 (Theorem 1.7, [22]).

The renormalised SdNLW (SDNLW) is almost surely globally well-posed with
initial data distributed according to the renormalised Gibbs measure $\rho$ in
(1.5). Furthermore, the renormalised Gibbs measure $\rho$ is invariant under
the dynamics.

More precisely, for $\rho$-almost every $\mathbf{u}_{0}$, we have that
$\Phi_{t}(\mathbf{u}_{0},\xi)\neq\infty$ a.s., and for every bounded, Borel
measurable functional $F:\mathcal{H}^{-\varepsilon}\to\mathbb{R}$,

$\int\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0},\xi))]d\rho(\mathbf{u}_{0})=\int
F(\mathbf{u}_{0})d\rho(u_{0}).$ (3.16)

######  Remark 3.6.

Since the map $\mathbf{u}_{0}\mapsto\Phi_{t}(\mathbf{u}_{0},\xi)$ is not
continuous in the topology of $\mathcal{H}^{-\varepsilon}$, one might wonder
if it is actually possible to define the semigroup

$P_{t}\varphi(\mathbf{u}_{0}):=\mathbb{E}[\Phi_{t}(\mathbf{u}_{0},\xi)]$

over the space of bounded, Borel measurable functions over $\widetilde{X}$.
Namely, it is not a priori clear whether the map

$\mathbf{u}_{0}\mapsto\Phi_{t}(\mathbf{u}_{0},\xi)$

is Borel measurable. However, by (3.6), one obtains that the set

$\Gamma=\\{\mathbf{u}_{0}:\Xi(\mathbf{u}_{0},\xi)\in\bigotimes_{j=1}^{2k-1}W^{-\varepsilon,\frac{4(2k-1)}{j}}\text{
a.s.}\\}$

is measurable, since the map
$(\mathbf{u}_{0},\xi)\mapsto\Xi(\mathbf{u}_{0},\xi)$ is measurable on the set
where it is well defined, and the set where this map is not well defined can
be expressed as the limsup of measurable sets (hence it is measurable). Since
furthermore the maps $\Xi\mapsto S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t)$,
$\Xi\mapsto\mathbf{v}(t)$ are continuous, we also obtain that the map
$\Gamma\ni\mathbf{u}_{0}\mapsto\Phi_{t}(\mathbf{u}_{0},\xi)$ is measurable.
Finally, one has that $\Phi_{t}(\mathbf{u}_{0},\xi)=\infty$ for
$\mathbf{u}_{0}\not\in\Gamma$ and $t>0$. We obtain that the map
$\widetilde{X}\ni\mathbf{u}_{0}\mapsto\Phi_{t}(\mathbf{u}_{0},\xi)$ is also
measurable. At this point, the semigroup property $P_{t+s}=P_{t}P_{s}$ follows
from (3.15).

###  3.3. Construction of the Girsanov shift

Define the set

$\displaystyle Y=\Big{\\{}$
$\displaystyle\mathbf{u}_{0}\in\mathcal{H}^{-\varepsilon}:\Phi_{t}(\mathbf{u}_{0},\xi)\neq\infty\text{
for every }t\geq 0,$ (3.17) $\displaystyle\ \text{and for every }p\geq
1,\text{ there exists }C=C(\mathbf{u}_{0},p)\text{ s.t. }$ $\displaystyle\
\mathbb{E}\int_{0}^{\infty}\sup_{1\leq h\leq
2k-1}\Big{\|}e^{-\frac{t}{32}}{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\Big{\|}_{W^{-\frac{\varepsilon}{2},4}}^{2}dt\leq
C.\Big{\\}}$

######  Lemma 3.7.

$Y\in\mathscr{B}(\mathcal{H}^{-\varepsilon})$. Moreover, $\rho(Y)=1$.

###### Proof.

The fact that $Y$ is Borel measurable follows from the same arguments as in
Remark 3.6. In order to show that $\rho(Y)=1$, it is enough to show that

$\int\Big{(}\mathbb{E}\int_{0}^{\infty}\sup_{1\leq h\leq
2k-1}\Big{\|}e^{-\frac{t}{32}}{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\Big{\|}_{W^{-\frac{\varepsilon}{2},4}}^{2}dt\Big{)}d\rho(\mathbf{u}_{0})<\infty.$

By Tonelli’s theorem, by invariance of $\rho$, and Proposition 3.1, 3.
together with 3.2, we have that

$\displaystyle\int\Big{(}\mathbb{E}\int_{0}^{\infty}\sup_{1\leq h\leq
2k-1}\Big{\|}e^{-\frac{t}{32}}{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\Big{\|}_{W^{-\frac{\varepsilon}{2},4}}^{2}dt\Big{)}d\rho(\mathbf{u}_{0})$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int\mathbb{E}\Big{[}\sup_{1\leq h\leq
2k-1}\Big{\|}e^{-\frac{t}{32}}{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\Big{\|}_{W^{-\frac{\varepsilon}{2},4}}^{2}\Big{]}d\rho(\mathbf{u}_{0})dt$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\int\sup_{1\leq h\leq
2k-1}\Big{\|}e^{-\frac{t}{32}}{:}\,p_{h}(\mathbf{u}_{0})\mspace{2.0mu}{:}\Big{\|}_{W^{-\frac{\varepsilon}{2},4}}^{2}d\rho(\mathbf{u}_{0})dt$
$\displaystyle\lesssim$ $\displaystyle\int\sup_{1\leq h\leq
2k-1}\Big{\|}{:}\,p_{h}(\mathbf{u}_{0})\mspace{2.0mu}{:}\Big{\|}_{W^{-\frac{\varepsilon}{2},4}}^{2}d\rho(\mathbf{u}_{0})$
$\displaystyle<$ $\displaystyle\ \infty.$

∎

The goal of this subsection is to prove the following.

######  Proposition 3.8.

For every $\mathbf{u}_{0}\in Y$ and
$\mathbf{v}_{0}\in\mathcal{H}^{1-\varepsilon}$, there exist
$\varepsilon_{0}(\mathbf{u}_{0},\mathbf{v}_{0})>0$ so that if
$F:\mathcal{H}^{-\varepsilon}\to\mathbb{R}$ is a Borel function, Lipschitz
with respect to the distance

$d_{\mathcal{H}^{1-\varepsilon}}(\mathbf{u}_{0},\mathbf{u}_{1}):=\min(\|\mathbf{u}_{0}-\mathbf{u}_{1}\|_{\mathcal{H}^{1-\varepsilon}},1),$

we have that

$\displaystyle|\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi))-F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{1};\xi))]|$
(3.18) $\displaystyle\leq
2\varepsilon_{0}(\mathbf{u}_{0},\mathbf{v}_{0},\mathbf{v}_{1})\|F\|_{\infty}+e^{-\frac{t}{4}}(\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}+\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}})\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}},$

with $\varepsilon_{0}(\mathbf{u}_{0},\mathbf{v}_{0},\mathbf{v}_{1})<1$.

Moreover, in the special case $\mathbf{v}_{1}=0$, we can choose
$\varepsilon_{0}$ so that

$\lim_{\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}\to
0}{\varepsilon_{0}(\mathbf{u}_{0},\mathbf{v}_{0},0)}=0.$ (3.19)

In order to show this, we need two preparatory lemmas.

######  Lemma 3.9.

Let $\varepsilon>0$ small enough, and let $\mathbf{u}_{0}\in Y$,
$\mathbf{v}_{0}\in\mathcal{H}^{1-\varepsilon}$. Then there exists $h\in
L^{2}(\mathbb{R}_{+},\mathbb{T}^{2})$, adapted with respect to the natural
filtration induced by $\xi$, such that

  1. (i)

For every $t\geq 0$,

$\|\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi+h)-\Phi_{t}(\mathbf{u}_{0};\xi)\|_{\mathcal{H}^{1-\varepsilon}}\leq\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}e^{-\frac{t}{4}},$

  2. (ii)

There exists an almost surely finite constant $K=K(\mathbf{u}_{0},\xi)$ such
that

$\int_{0}^{\infty}\|h(t)\|_{L^{2}(\mathbb{T}^{2})}^{2}\leq
K(1+\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}^{2k-2})\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}},$

and $\mathbb{E}|K|<+\infty$.

###### Proof.

The statement is trivially true for $\mathbf{v}_{0}=0$, by choosing $h=0$.
Therefore, we can safely assume $\mathbf{v}_{0}\neq 0$. By Proposition 3.1,
2., we have that

${:}\,p(\Phi_{t}(\mathbf{u}_{0};\xi)+w)\mspace{2.0mu}{:}-{:}\,p(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}=\sum_{h=1}^{2k}{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}w^{h},$
(3.20)

where $\\{p_{h}\\}_{1\leq h\leq 2k-1}$ are appropriate polynomials of degree
$h$. For a given function $\delta(t):\mathbb{R}^{+}\to\mathbb{R}^{+}$ that we
will determine later, let $\mathbf{w}=\begin{pmatrix}w\\\ w_{t}\end{pmatrix}$
be the solution of the equation

$w_{tt}+w_{t}+(1-\Delta)w=-\sum_{h=1}^{2k-1}(1-e^{\delta(t)\Delta}){:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}w^{h}$
(3.21)

with initial data $\mathbf{w}_{0}=\mathbf{v}_{0}$. Therefore, defining
$\mathcal{T}_{\delta}z:=z-e^{\delta\Delta}z$, (3.21) can be rewritten as

$w_{tt}+w_{t}+(1-\Delta)w=-\sum_{h=1}^{2k-1}\mathcal{T}_{\delta(t)}({:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:})w^{h}.$
(3.22)

Since
$\|\mathcal{T}_{\delta}z\|_{W^{-\varepsilon,4}}\lesssim\delta^{\frac{\varepsilon}{4}}\|z\|_{W^{-\frac{\varepsilon}{2},4}}$,
by choosing

$\delta(t)=\big{(}A\sup_{h}\|{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\|_{W^{-\frac{\varepsilon}{2},4}}\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}^{h-1}\big{)}^{-\frac{4}{\varepsilon}},$
(3.23)

for $\varepsilon$ small enough and $A$ big enough, we have

$\displaystyle\left\|-\sum_{h=1}^{2k-1}\mathcal{T}_{\delta(t)}({:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:})w^{h}\right\|_{H^{-\varepsilon}}\leq\frac{1}{4}\Big{[}\sup_{1\leq
h\leq
2k-1}\Big{(}\frac{\|w\|_{H^{1-\varepsilon}}}{\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}}\Big{)}^{h-1}\Big{]}\|w\|_{H^{1-\varepsilon}}.$

Therefore, by (3.9),

$\displaystyle~{}\|\mathbf{w}(t)\|_{\mathcal{H}^{1-\varepsilon}}$
$\displaystyle\leq$
$\displaystyle~{}\|S(t)\mathbf{w}(0)\|_{\mathcal{H}^{1-\varepsilon}}+\left\|\int_{0}^{t}S(t-t^{\prime})\Big{(}-\sum_{h=1}^{2k-1}\mathcal{T}_{\delta(t)}({:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:})w^{h}\Big{)}\mathrm{d}t^{\prime}\right\|_{\mathcal{H}^{1-\varepsilon}}$
$\displaystyle\leq$
$\displaystyle~{}e^{-\frac{t}{2}}\|\mathbf{w}(0)\|_{\mathcal{H}^{1-\varepsilon}}+\frac{1}{4}\int_{0}^{t}e^{-\frac{t-t^{\prime}}{2}}\left\|{-\sum_{h=1}^{2k-1}\mathcal{T}_{\delta(t)}({:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:})w^{h}}\right\|_{H^{-\varepsilon}}\mathrm{d}t^{\prime}$
$\displaystyle\leq$ $\displaystyle\
e^{-\frac{t}{2}}\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}+\frac{1}{4}\int_{0}^{t}e^{-\frac{t-t^{\prime}}{2}}\Big{[}\sup_{1\leq
h\leq
2k-1}\Big{(}\frac{\|w(t^{\prime})\|_{H^{1-\varepsilon}}}{\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}}\Big{)}^{h-1}\Big{]}\|w(t^{\prime})\|_{H^{1-\varepsilon}}\mathrm{d}t^{\prime}.$

From this, by an easy Gronwall argument, we obtain

$\|\mathbf{w}\|_{H^{1-\varepsilon}}\leq
e^{-\frac{t}{4}}\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}.$ (3.24)

Moreover, if we define

$h(t):=\frac{1}{\sqrt{2}}\sum_{j=1}^{2k-1}(1-\mathcal{T}_{\delta(t)})({:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:})w^{j},$
(3.25)

by (3.20) it is easy to see that
$\tilde{\mathbf{u}}(t):=\Phi_{t}(\mathbf{u}_{0};\xi)+\mathbf{w}(t)$ solves the
equation (3.7) with forcing $\xi+h$, and that $h$ is adapted. Therefore,

$\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi+h)=\Phi_{t}(\mathbf{u}_{0};\xi)+\mathbf{w}(t),$

so ($\mathrm{i}$) is proven for this particular choice of $h$. Moreover, for
$\varepsilon$ small enough,

$\displaystyle\phantom{\lesssim}\|h(t)\|_{L^{2}}$
$\displaystyle\lesssim\sum_{j=1}^{2k-1}\delta^{-\frac{\varepsilon}{4}}\|{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\|_{W^{-\frac{\varepsilon}{2},4}}\|w\|_{H^{1-\varepsilon}}^{j}$
$\displaystyle\lesssim
A\big{(}\sup_{j}\|{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\|_{W^{-\frac{\varepsilon}{2},4}}\big{)}^{2}\big{(}1+\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}^{2k-2}\big{)}e^{-\frac{t}{4}}\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}$
$\displaystyle\leq
K(t)\big{(}1+\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}^{2k-2}\big{)}\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}e^{-\frac{t}{8}},$

for

$K(t)\lesssim\big{(}e^{-\frac{t}{16}}\sup_{h}\|{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\|_{W^{-\frac{\varepsilon}{2},4}}\big{)}^{2}.$

Therefore, we have ($\mathrm{ii}$) with

$K\lesssim\int_{0}^{\infty}\big{(}e^{-\frac{t}{16}}\sup_{h}\|{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\|_{W^{-\frac{\varepsilon}{2},4}}\big{)}^{2}dt$

Furthermore, by definition (3.17) of $Y$,

$\displaystyle\mathbb{E}|K|$
$\displaystyle\lesssim\mathbb{E}\Big{|}\int_{0}^{\infty}e^{-\frac{t}{16}}\big{(}e^{-\frac{t}{32}}\sup_{h}\|{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\|_{W^{-\frac{\varepsilon}{2},4}}\big{)}^{2}dt\Big{|}$
$\displaystyle\lesssim\mathbb{E}\int_{0}^{\infty}\big{|}e^{-\frac{t}{32}}\sup_{h}\|{:}\,p_{h}(\Phi_{t}(\mathbf{u}_{0};\xi))\mspace{2.0mu}{:}\|_{W^{-\frac{\varepsilon}{2},4}}\big{|}^{2}dt$
$\displaystyle<\infty.$

∎

######  Lemma 3.10.

Let $0\leq f_{1},f_{2}$ with $\mathbb{E}[f_{1}],\mathbb{E}[f_{2}]<\infty$, and
let $\eta>0$. We have that

$\mathbb{E}[|f_{1}-f_{2}|]\leq\mathbb{E}[f_{1}]+\mathbb{E}[f_{2}]-\eta\big{(}\mathbb{P}(\\{f_{1}\geq\eta\\})+\mathbb{P}(\\{f_{2}\geq\eta\\})-1\big{)}.$
(3.26)

###### Proof.

We have that

$\displaystyle\mathbb{E}[|f_{1}-f_{2}|]$
$\displaystyle=\mathbb{E}[(f_{1}-f_{2})\mathbb{1}_{\\{f_{1}\geq
f_{2}\\}}]+\mathbb{E}[(f_{2}-f_{1})\mathbb{1}_{\\{f_{2}\geq f_{1}\\}}]$
$\displaystyle=\mathbb{E}[(f_{1}-f_{2})\mathbb{1}_{\\{f_{1}\geq
f_{2}\geq\eta\\}}]+\mathbb{E}[(f_{2}-f_{1})\mathbb{1}_{\\{f_{2}\geq
f_{1}\geq\eta\\}}]$
$\displaystyle\phantom{=}+\mathbb{E}[(f_{1}-f_{2})\mathbb{1}_{\\{f_{1}\geq\eta>f_{2}\\}}]+\mathbb{E}[(f_{2}-f_{1})\mathbb{1}_{\\{f_{2}\geq\eta>f_{1}\\}}]$
$\displaystyle\phantom{=}+\mathbb{E}[(f_{1}-f_{2})\mathbb{1}_{\\{\eta>f_{1}\geq
f_{2}\\}}]+\mathbb{E}[(f_{2}-f_{1})\mathbb{1}_{\\{\eta>f_{2}\geq f_{1}\\}}]$
$\displaystyle\leq\mathbb{E}[(f_{1}-\eta)\mathbb{1}_{\\{f_{1}\geq
f_{2}\geq\eta\\}}]+\mathbb{E}[(f_{2}-\eta)\mathbb{1}_{\\{f_{2}\geq
f_{1}\geq\eta\\}}]$
$\displaystyle\phantom{=}+\mathbb{E}[f_{1}\mathbb{1}_{\\{f_{1}\geq\eta>f_{2}\\}}]+\mathbb{E}[f_{2}\mathbb{1}_{\\{f_{2}\geq\eta>f_{1}\\}}]$
$\displaystyle\phantom{=}+\mathbb{E}[f_{1}\mathbb{1}_{\\{\eta>f_{1}\geq
f_{2}\\}}]+\mathbb{E}[f_{2}\mathbb{1}_{\\{\eta>f_{2}\geq f_{1}\\}}]$
$\displaystyle=\mathbb{E}[f_{1}\mathbb{1}_{\\{f_{1}\geq
f_{2}\\}}]+\mathbb{E}[f_{2}\mathbb{1}_{\\{f_{2}\geq
f_{1}\\}}]-\eta(\mathbb{P}(\\{f_{1}\geq
f_{2}\geq\eta\\})+\mathbb{P}(\\{f_{2}\geq f_{1}\geq\eta\\}))$
$\displaystyle\leq\mathbb{E}[f_{1}]+\mathbb{E}[f_{2}]-\eta\mathbb{P}(\\{\min(f_{1},f_{2})\geq\eta\\}).$

At this point, (3.26) follows from

$\displaystyle\mathbb{P}(\\{\min(f_{1},f_{2})\geq\eta\\})$
$\displaystyle=1-\mathbb{P}(\\{f_{1}<\eta\\}\cup\\{f_{2}<\eta\\})$
$\displaystyle\geq
1-\mathbb{P}(\\{f_{1}<\eta\\})-\mathbb{P}(\\{f_{2}<\eta\\})$
$\displaystyle\geq\mathbb{P}(\\{f_{1}\geq\eta\\})+\mathbb{P}(\\{f_{2}\geq\eta\\}-1.$

∎

###### Proof of Proposition 3.8.

Let $\mathbf{v}_{0},\mathbf{v}_{1}\in\mathcal{H}^{1-\varepsilon}$, and let
$h_{\mathbf{v}_{j}}$ be as in Lemma 3.9. For $M>0$, let
$\tau_{\mathbf{v}_{j}}^{M}$ be the first time such that
$\|h_{\mathbf{v}_{j}}(t)\|_{L^{2}([0,\tau_{\mathbf{v}_{j}}^{M}]\times\mathbb{T}^{2})}=M$
(with $\tau_{\mathbf{v}_{j}}^{M}=+\infty$ if $\|h_{L^{2}_{t,x}}\|<M$ for every
$t$), and let
$h_{\mathbf{v}_{j}}^{M}(t):=h(\min(t,\tau^{M}_{\mathbf{v}_{j}}))$. Let

$\mathcal{E}(h)(t):=\exp\Big{(}-\frac{1}{2}\int_{0}^{t}\|h(t^{\prime})\|_{L^{2}}^{2}+\int_{0}^{t}\left\langle
h(t^{\prime}),\xi\right\rangle_{L^{2}}\Big{)}.$

We first observe that in order to prove (3.18), by exchanging the roles of
$\mathbf{v}_{0}$ and $\mathbf{v}_{1}$, it is enough to prove the estimate
without the absolute value. By Girsanov, we have that

$\displaystyle\phantom{\lesssim}\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi))-F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{1};\xi))]$
$\displaystyle=\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi+h_{\mathbf{v}_{0}}^{M}))\mathcal{E}(h_{\mathbf{v}_{0}}^{M})-F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{1};\xi+h_{\mathbf{v}_{1}}^{M}))\mathcal{E}(h_{\mathbf{v}_{1}}^{M})]$
$\displaystyle=\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi+h_{\mathbf{v}_{0}}^{M}))\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}<t\\}}]$
(I)
$\displaystyle\phantom{=}+\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi+h_{\mathbf{v}_{0}}^{M}))\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}-F(\Phi_{t}(\mathbf{u}_{0};\xi))\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}]$ (II)
$\displaystyle\phantom{=}+\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0};\xi))\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}-F(\Phi_{t}(\mathbf{u}_{0};\xi))\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{1}}\geq
t\\}}]$ (III)
$\displaystyle\phantom{=}-\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{1};\xi+h_{\mathbf{v}_{1}}^{M}))\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{1}}\geq
t\\}}-F(\Phi_{t}(\mathbf{u}_{0};\xi))\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{1}}\geq
t\\}}]$ (IV)
$\displaystyle\phantom{=}-\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{1};\xi+h_{\mathbf{v}_{1}}^{M}))\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{1}}<t\\}}].$
(V)

We have that

$|\eqref{2di}|\leq\|F\|_{\infty}\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}<t\\}}],\hskip
15.0pt|\eqref{2dv}|\leq\|F\|_{\infty}\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{1}}<t\\}}],$
(3.27)

by Lemma 3.9, (i), and recalling that
$\mathbb{E}[\mathcal{E}(h^{M}_{\mathbf{v}_{j}})]=1$,

$\displaystyle|\eqref{2dii}|$
$\displaystyle=\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi+h_{\mathbf{v}_{0}}))\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}-F(\Phi_{t}(\mathbf{u}_{0};\xi))\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}]$ (3.28)
$\displaystyle=\mathbb{E}\big{[}\big{(}F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi+h_{\mathbf{v}_{0}}))-F(\Phi_{t}(\mathbf{u}_{0};\xi))\big{)}\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}\big{]}$ $\displaystyle\leq
e^{-\frac{t}{4}}\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}},$

and similarly

$|\eqref{2div}|\leq
e^{-\frac{t}{4}}\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\|\mathbf{v}_{1}\|_{\mathcal{H}^{1-\varepsilon}}.$
(3.29)

Finally, by (3.26), for every $\eta>0$,

$\displaystyle|\eqref{2diii}|$
$\displaystyle\leq\|F\|_{\infty}\mathbb{E}|\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}-\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{1}}\geq
t\\}}|$ (3.30)
$\displaystyle\leq\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}]+\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{1}}\geq
t\\}}]$
$\displaystyle\phantom{\leq}-\eta(\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\geq\eta\\}\cap\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\})+\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\geq\eta\\}\cap\\{\tau^{M}_{\mathbf{v}_{1}}\geq
t\\})-1).$

Putting (3.27), (3.28), (3.29), and (3.30) together, and by symmetry between
$\mathbf{v}_{0}$ and $\mathbf{v}_{1}$, we obtain

$\displaystyle\phantom{\lesssim}\mathbb{E}|F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi))-F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{1};\xi))|$
(3.31)
$\displaystyle\leq\|F\|_{\infty}\big{(}\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}<t\\}}]+\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{1}}<t\\}}]+\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}]+\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}]$
$\displaystyle\phantom{\leq\|F\|_{\infty}\big{(})}-\eta(\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\geq\eta\\}\cap\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\})+\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\geq\eta\\}\cap\\{\tau^{M}_{\mathbf{v}_{1}}\geq
t\\})-1)\big{)}$
$\displaystyle\phantom{\leq}+\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}e^{-\frac{t}{4}}(\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}+\|\mathbf{v}_{1}\|_{\mathcal{H}^{1-\varepsilon}})$
$\displaystyle\leq\|F\|_{\infty}(2-\eta(\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\geq\eta\\}\cap\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\})+\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\geq\eta\\}\cap\\{\tau^{M}_{\mathbf{v}_{1}}\geq
t\\})-1)$
$\displaystyle\phantom{\leq}+\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}e^{-\frac{t}{4}}(\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}+\|\mathbf{v}_{1}\|_{\mathcal{H}^{1-\varepsilon}}).$

Therefore, by taking limits for $M\to\infty$, we obtain (3.18) with

$\displaystyle\varepsilon_{0}=1-\frac{1}{2}\sup_{\eta>0}\eta\inf_{t>0}\limsup_{M\to\infty}(\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\geq\eta\\})+\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{1}}^{M})\geq\eta\\})-1).$
(3.32)

By Chebishev, for $0<\eta<1$ we have that

$\displaystyle\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{j}}^{M})<\eta\\})$
$\displaystyle=\mathbb{P}\Big{(}\Big{\\{}\frac{1}{2}\int_{0}^{t}\|h(t^{\prime})\|_{L^{2}}^{2}-\int_{0}^{t}\left\langle
h(t^{\prime}),\xi\right\rangle_{L^{2}}>\log\eta^{-1}\Big{\\}}\Big{)}$
$\displaystyle\lesssim\frac{\mathbb{E}\Big{[}\int_{0}^{t}\|h_{\mathbf{v}_{j}}^{M}(t^{\prime})\|_{L^{2}}^{2}\Big{]}}{\log\eta^{-1}}+\frac{\mathbb{E}\Big{|}\int_{0}^{t}\left\langle
h_{\mathbf{v}_{j}}^{M}(t^{\prime}),\xi\right\rangle_{L^{2}}\Big{|}^{2}}{(\log\eta^{-1})^{2}}$
$\displaystyle\lesssim\frac{\mathbb{E}\|h_{\mathbf{v}_{j}}\|_{L^{2}}^{2}}{\log\eta^{-1}}$

Therefore, by Lemma 3.9, (ii), choosing

$\eta=\exp\Big{(}-C(\mathbf{u}_{0})(1+\|v_{0}\|_{\mathcal{H}^{1-\varepsilon}}^{2k-2}+\|v_{1}\|_{\mathcal{H}^{1-\varepsilon}}^{2k-2})^{2}(\|v_{0}\|_{\mathcal{H}^{1-\varepsilon}}+\|v_{1}\|_{\mathcal{H}^{1-\varepsilon}})^{2}\Big{)}$

for some $C(\mathbf{u}_{0})\gg\mathbb{E}|K(\mathbf{u}_{0},\xi)|$, we obtain
that
$\mathbb{P}(\\{\mathcal{E}(h_{\mathbf{v}_{j}}^{M})\geq\eta\\})\geq\frac{3}{4}$,
and so $\varepsilon_{0}(\mathbf{u}_{0},\mathbf{v}_{0},\mathbf{v}_{1})<1$.

We now move to the case $\mathbf{v}_{0}\to 0$, $\mathbf{v}_{1}=0$. From
(3.27), (3.28), (3.29), and (3.30), we obtain that

$\displaystyle\mathbb{E}[F(\Phi_{t}(\mathbf{u}_{0}+\mathbf{v}_{0};\xi))-F(\Phi_{t}(\mathbf{u}_{0};\xi))]$
$\displaystyle\leq\|F\|_{\infty}\big{(}\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}<t\\}}]+\mathbb{E}|\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}-1|\big{)}$
$\displaystyle\phantom{\leq}+\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}e^{-\frac{t}{4}}\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}.$

Therefore, we can choose

$\varepsilon_{0}(\mathbf{u}_{0},\mathbf{v}_{0},0)=\frac{1}{2}\sup_{t>0}\inf_{M>0}\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}<t\\}}]+\mathbb{E}|\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}-1|$ (3.33)

Fixing $M>0$, by Lemma 3.9, (ii), and by the fact that
$\mathbb{E}|[\mathcal{E}(h_{\mathbf{v}_{0}}^{M})|^{2}\leq C(M)$, we obtain
that

$\lim_{\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}\to
0}\mathbb{E}|\mathcal{E}(h_{\mathbf{v}_{0}}^{M})-1|=0.$

Therefore, by (3.33), by definition of $\tau_{\mathbf{v}_{0}}^{M}$ and by
Lemma 3.9, (ii) again,

$\displaystyle\limsup_{\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}\to
0}\varepsilon_{0}(\mathbf{u}_{0},\mathbf{v}_{0},0)$
$\displaystyle\leq\frac{1}{2}\limsup_{\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}\to
0}\sup_{t>0}\mathbb{E}[\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}<t\\}}]+\mathbb{E}|\mathcal{E}(h_{\mathbf{v}_{0}}^{M})\mathbb{1}_{\\{\tau^{M}_{\mathbf{v}_{0}}\geq
t\\}}-1|$
$\displaystyle\leq\limsup_{\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}\to
0}C(M)^{\frac{1}{2}}\mathbb{P}(\\{\tau_{\mathbf{v}_{0}}^{M}<\infty\\})^{\frac{1}{2}}$
$\displaystyle\leq\limsup_{\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}\to
0}C(M)^{\frac{1}{2}}\mathbb{P}(\|h_{\mathbf{v}_{0}}\|_{L^{2}_{t,x}}^{2}\geq
M\\})^{\frac{1}{2}}$ $\displaystyle=0.$

∎

###  3.4. Restricted asymptotic strong Feller and restricted coupling
properties

In view of Proposition 3.8, we consider the space

$X:=Y+\mathcal{H}^{1-\varepsilon}\cup\\{\infty\\},$ (3.34)

with the distance

$d_{X}(\mathbf{u}_{0},\mathbf{u}_{1})=\begin{cases}\min(\|\mathbf{u}_{0}-\mathbf{u}_{1}\|_{\mathcal{H}^{-\varepsilon}},1)&\text{
if }\mathbf{u}_{0},\mathbf{u}_{1}\neq\infty,\\\ 1&\text{ if
}\mathbf{u}_{0}=\infty,\mathbf{u}_{1}\neq\infty\text{ or
}\mathbf{u}_{0}\neq\infty,\mathbf{u}_{1}=\infty,\\\ 0&\text{ if
}\mathbf{u}_{0}=\mathbf{u}_{1}=\infty.\end{cases}$ (3.35)

In view of (3.17), (3.13), and Proposition 3.1, 3., we have that
$X\subseteq\Gamma\cup\\{\infty\\}$. Therefore, the flow $\Phi_{t}$ is well
defined for every $\mathbf{u}_{0}\in X$. Moreover, by the decomposition

$\Phi_{t}(\mathbf{u}_{0},\xi)=S(t)\mathbf{u}_{0}+\vec{\psi}[\xi](t)+\mathbf{v}(t),$

and the fact that $S(t)$ is bounded on $\mathcal{H}^{1-\varepsilon}$, we have
that $\Phi_{t}(X,\xi)\subseteq X$ for every $t\geq 0$ a.s. We want to use the
flow to define a Markov semigroup $P_{t}$ on $X$ so that Assumptions 1,2 are
satisfied. Notice that, because of the implicit definition of the space $X$
(or more precisely, of the space $Y$ in (3.17)), we have no guarantee a priori
that the set $X\subset H^{-\varepsilon}$ is Borel measurable.111111More
precisely, while it is possible to show that the set $Y$ is Borel-measurable
by adapting the arguments of Remark 3.6, it is unclear if the set
$Y+\mathcal{H}^{1-\varepsilon}$ remains measurable. However, if
$\varphi:X\to\mathbb{R}$ is a Borel function, there exists
$\widetilde{\varphi}:\Gamma\cup\\{\infty\\}\to\mathbb{R}$ Borel so that
$\widetilde{\varphi}|_{X}=\varphi$.121212For indicator functions of open and
closed sets, this follows from the fact that closed and open sets in $X$ are
the intersection of a (respectively) closed or open set in $\Gamma\cup\infty$
with $X$. For indicator functions of Borel sets, we obtain this from the
definition of the Borel $\sigma$-algebra as the smallest $\sigma$-algebra that
contains open sets, together with the previous step and the fact that the
intersection of a $\sigma$-algebra with a set is a $\sigma$-algebra on that
set. Finally, for a general Borel function, we obtain the result by writing it
as the limit of simple functions, and extending every single function to
$\Gamma\cup\\{\infty\\}$. Notice that the set where the sequence of simple
functions converges is measurable, and will contain $X$ by construction. We
will use this fact (and more generally, the consequences of this construction)
liberally throughout the rest of the paper. Moreover, by invariance of $X$, we
have that for every $\mathbf{u}_{0}\in X$,

$\mathbb{E}[\varphi(\Phi_{t}(\mathbf{u}_{0},\xi))]=\mathbb{E}[\widetilde{\varphi}(\Phi_{t}(\mathbf{u}_{0},\xi))].$

Therefore, for $s\in\mathbb{R}$,

$\\{u_{0}\in X:\mathbb{E}[\varphi(\Phi_{t}(\mathbf{u}_{0},\xi))]\leq
s\\}=\\{\mathbf{u}_{0}\in\Gamma\cup\\{\infty\\}:\mathbb{E}[\widetilde{\varphi}(\Phi_{t}(\mathbf{u}_{0},\xi))]\leq
s\\}\cap X,$

which is the intersection of a Borel set in $\Gamma\cup\\{\infty\\}$ (see
Remark 3.6) and $X$, so it is Borel in $X$. Therefore, we can define

$P_{t}\varphi(\mathbf{u}_{0}):=\mathbb{E}[\varphi(\Phi_{t}(\mathbf{u}_{0},\xi))].$
(3.36)

In view of Lemma 3.4, we have that $P_{t}$ is a Markov semigroup on
$\mathscr{L}^{\infty}(X)$, thus the space $X$ together with the semigroup
$P_{t}$ satisfies Assumptions 1, 2. In order to put ourselves in the framework
of Section 2, we define

$\mathscr{G}=\mathcal{H}^{1-\varepsilon},$

with

$|\mathbf{v}_{0}|:=\min(\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}},1),$
(3.37)

and for $\mathbf{u}_{0}\in X$,
$\mathbf{v}_{0}\in\mathscr{G}=\mathcal{H}^{1-\varepsilon}$,

$\tau_{\mathbf{v}_{0}}(\mathbf{u}_{0}):=\begin{cases}\mathbf{u}_{0}+\mathbf{v}_{0}&\text{
if }\mathbf{u}_{0}\neq\infty\\\ \infty&\text{ if
}\mathbf{u}_{0}=\infty.\end{cases}$ (3.38)

It easy to check that Assumptions 1–5 hold, with the possible exception of the
measurability condition (2.2). In order to show that $\tau(B_{r}\times K)$ is
measurable, we will need the following lemma.

######  Lemma 3.11.

Let $C\subseteq\mathcal{H}^{-\varepsilon}$ be a closed set in
$\mathcal{H}^{-\varepsilon}$. Then for every $r\geq 0$,

$C+\\{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}:\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}\leq
r\\}$ (3.39)

is closed in $\mathcal{H}^{-\varepsilon}$.

###### Proof.

Since $\mathcal{H}^{-\varepsilon}$ is a metric space, it is enough to show
that if $(x_{n})_{n\in\mathbb{N}}$ is a sequence in $C$,
$\|\mathbf{v}_{n}\|_{\mathcal{H}^{1-\varepsilon}}\leq r$, and

$\lim_{n\to\infty}x_{n}+\mathbf{v}_{n}=x,$

then $x\in
C+\\{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}:\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}\leq
r\\}.$ By compactness of Sobolev embeddings, the ball
$\\{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}:\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}\leq
r\\}$ is compact in $\mathcal{H}^{-\varepsilon}$, so there exists a
subsequence $\mathbf{v}_{n_{k}}$ such that for some
$\mathbf{v}\in\mathcal{H}^{-\varepsilon}$,

$\lim_{k\to\infty}\mathbf{v}_{n_{k}}=\mathbf{v},$

and $\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}\leq r$. Therefore,

$\lim_{n\to\infty}x_{n_{k}}=x-\mathbf{v}.$

Since $C$ is closed, we have that $x-\mathbf{v}\in C$. Therefore,

$x=\overbrace{x-\mathbf{v}}^{\in C}+\hskip 5.0pt\mathbf{v}\hskip 5.0pt\in
C+\\{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}:\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}\leq
r\\}.$

∎

For a compact set $K$, we have that

$\tau(B_{r}\times
K)=\begin{cases}K+\\{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}:\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}\leq
r\\}&\text{ if }r<1,\\\
K+\mathcal{H}^{1-\varepsilon}=\bigcup_{n\in\mathbb{N}}K+\\{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}:\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}\leq
n\\}&\text{ if }r\geq 1.\end{cases}$

Since $K$ is compact as a subset of $X$, then it is also compact (hence
closed) as a subset of $\mathcal{H}^{-\varepsilon}$. Therefore, by Lemma 3.11,
$\tau(B_{r}\times K)$ is a closed set (for $r<1$) or a countable union of
closed sets (for $r\geq 1$) in $\mathcal{H}^{-\varepsilon}$ hence it is Borel.
Since $\tau(B_{r}\times K)\subseteq X$, then $\tau(B_{r}\times K)$ is Borel
measurable as a subset of $X$.

In conclusion, we have that $X$ defined in (3.34) with the distance (3.35),
the semigroup $P_{t}$ defined in (3.36) and the group
$\mathscr{G}=\mathcal{H}^{1-\varepsilon}$ with the absolute value (3.37) and
the action (3.38) satisfy Assumptions 1–5. Moreover, we have the following.

######  Proposition 3.12.

The semigroup $P_{t}$ on $\mathscr{L}^{\infty}(X)$ has the following
properties.

  1. (i)

The semigroup $P_{t}$ has the asymptotic strong Feller property restricted to
the action of $\mathcal{H}^{1-\varepsilon}$ on the set $Y$.

  2. (ii)

The semigroup $P_{t}$ has the asymptotic coupling property restricted to the
action of $\mathcal{H}^{1-\varepsilon}$ on the set $X$, with
$r(\mathbf{u}_{0})=\infty$ for every $\mathbf{u}_{0}\in X$.

###### Proof.

Both statements follow from the estimates (3.18) and (3.19) and the respective
definitions, except for the technical issue that Proposition 3.8 holds for
functions $\widetilde{F}$ which are Borel-measurable in
$\Gamma\cup\\{\infty\\}$, while we need the estimates to hold true for
functions $F$ which are Borel-measurable in $X$.

Therefore, the proposition is proven if we show that for every
$F:X\to\mathbb{R}$ Borel measurable with $\|F\|_{\infty}<\infty$,
$\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}<\infty$, there exists
$\widetilde{F}:\Gamma\cup\\{\infty\\}\to\mathbb{R}$ Borel measurable such that
$\|\widetilde{F}\|_{\infty}\leq\|F\|_{\infty}$,
$\|\widetilde{F}\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\leq\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}$,
and that satisfies

$\widetilde{F}(\mathbf{u}_{0})=F(\mathbf{u}_{0})\text{ for every
}\mathbf{u}_{0}\in X.$ (3.40)

Since $F$ is a measurable function, there exists an extension
$\overline{F}:\Gamma\cup\\{\infty\\}\to\mathbb{R}$ which is measurable,
$\|\overline{F}\|_{\infty}=\|F\|_{\infty}$, and satisfies
$\overline{F}(\mathbf{u}_{0})=F(\mathbf{u}_{0})$ for every $\mathbf{u}_{0}\in
X$. Define

$\widetilde{F}(\mathbf{u}_{0}):=\inf_{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}}\overline{F}(\mathbf{u}_{0}+\mathbf{v})+\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}.$
(3.41)

It is easy to check that
$-\|\overline{F}\|_{\infty}\leq\widetilde{F}(\mathbf{u}_{0})\leq\overline{F}(\mathbf{u}_{0})$,
so $\|\widetilde{F}\|_{\infty}\leq\|\overline{F}\|_{\infty}=\|F\|_{\infty}$,
and that
$\|\widetilde{F}\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\leq\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}$.
Therefore, if $\mathbf{v}_{n}$ is a countable dense subset of
$\mathcal{H}^{1-\varepsilon}$, we also have that

$\widetilde{F}(\mathbf{u}_{0}):=\inf_{n\in\mathbb{N}}\overline{F}(\mathbf{u}_{0}+\mathbf{v}_{n})+\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\|\mathbf{v}_{n}\|_{\mathcal{H}^{1-\varepsilon}},$

so $\widetilde{F}$ is also measurable. Finally, for $\mathbf{u}_{0}\in X$,
since $\mathbf{u}_{0}+\mathbf{v}\in X$ as well for every
$\mathbf{v}\in\mathcal{H}^{1-\varepsilon}$, we have that

$\displaystyle\overline{F}(\mathbf{u}_{0})=F(\mathbf{u}_{0})$
$\displaystyle\geq\widetilde{F}(\mathbf{u}_{0})$
$\displaystyle=\inf_{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}}F(\mathbf{u}_{0}+\mathbf{v})+\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}$
$\displaystyle\geq
F(\mathbf{u}_{0})+\inf_{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}}-|F(\mathbf{u}_{0}+\mathbf{v})-F(\mathbf{u}_{0})|+\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}$
$\displaystyle\geq
F(\mathbf{u}_{0})+\inf_{\mathbf{v}\in\mathcal{H}^{1-\varepsilon}}-\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}+\|F\|_{\mathcal{H}^{1-\varepsilon}-\operatorname{Lip}}\|\mathbf{v}\|_{\mathcal{H}^{1-\varepsilon}}$
$\displaystyle=F(\mathbf{u}_{0}),$

so we obtain (3.40). ∎

######  Remark 3.13.

In what follows, we are never going to use the fact that the semigroup has the
$\mathrm{(rASF)_{Y}}$ property, and we will only focus on the consequences of
the $\mathrm{(rAC)_{X}}$ property. The reason is self-evident from the support
statements of Theorem 2.6 and 2.7, that require the relevant property to hold
on the whole space.

If on top of the $\mathrm{(rAC)_{X}}$ property, we also had the
$\mathrm{(rASF)_{X}}$ property, it would actually be possible to show that the
flow is globally well posed on $X\setminus\\{\infty\\}$, in the sense that for
every $\mathbf{u}_{0}\in X$, $\mathbf{u}_{0}\neq\infty$, then for every $t\geq
0$, that $\Phi_{t}(\mathbf{u}_{0},\xi)\neq\infty$ a.s. This would provide a
global well posedness statement for every initial data of the form
$\mathbf{u}_{0}\in Y+\mathcal{H}^{1-\varepsilon}$. Unfortunately, the
techniques of this paper are not strong enough to provide such a result (and
it is unclear if it is true to begin with). The technical reason is that, in
the construction of the Girsanov shift of Lemma 3.9, we have no way to control
exponential moments of $h$ when
$\|\mathbf{v}_{0}\|_{\mathcal{H}^{1-\varepsilon}}\gg 1$. Indeed, it is in
principle possible that for initial data not on the set $Y$, the solution
blows up with positive probability. What the Girsanov shift guarantees, that
is encoded in the $\mathrm{(rAC)_{X}}$ property, is that the solution starting
from $\mathbf{u}_{0}+\mathbf{v}_{0}$ will follow (in law) the trajectory of
the flow starting from $\mathbf{u}_{0}$ with strictly positive probability.

Nevertheless, we decided to include the definition of $\mathrm{(rASF)_{S}}$
property (and to show it on the set $Y$) in order to make it is easier to draw
comparisons with the existing theory.

###  3.5. Ergodicity of the $P(\Phi)_{2}$ measure

We now move to proving the ergodicity of the measure $\rho$. In order to be
able to apply the theory described in Section 2, we need to a procedure to
associate to a measure $\mu$ defined on $\mathcal{H}^{-\varepsilon}$ a measure
$\iota^{\ast}\mu$ defined on $X$. We recall that from the definition (3.34),
it is not clear if the space $X$ is measurable, so we cannot simply define
$\iota^{\ast}\mu$ as the restriction of $\mu$ to $X$. However, we have the
following.

######  Lemma 3.14.

Let $\mu$ be a finite, nonnegative measure on
$\mathscr{B}(\mathcal{H}^{-\varepsilon})$. Then there exists a unique Radon
measure $\iota^{\ast}\mu$ such that for every compact set $K\subseteq X$,

$\iota^{\ast}\mu(K)=\mu(K).$ (3.42)

Moreover, if $\mu$ is concentrated in $\widetilde{X}=\Gamma\cup\\{\infty\\}$
and it is an invariant probability measure in the sense that (3.16) holds, and
$\iota^{\ast}\mu(X)=1$, then $\iota^{\ast}\mu$ is an invariant probability
measure for $P_{t}$ defined in (3.36).

###### Proof.

Since Radon measures are uniquely determined by their values on compact sets,
uniqueness follows from (3.42), so we just need to show existence. Let

$\lambda:=\sup_{K\subseteq X\text{
compact}}\mu(K)\leq\mu(\mathcal{H}^{-\varepsilon})<\infty,$

and let $K_{n}\subseteq X$ be an increasing sequence of compact sets so that
$\lim_{n\to\infty}\mu(K_{n})=\lambda$. For $E\subseteq X$ Borel, we define

$\iota^{\ast}\mu(E):=\lim_{n\to\infty}\mu(E\cap K_{n}).$

Notice that since $E$ is a Borel set in $X$, there exists a Borel set
$E^{\prime}\subseteq\mathcal{H}^{-\varepsilon}$ so that

$E=E^{\prime}\cap X,$

so $E\cap K_{n}=E^{\prime}\cap K_{n}$ is a Borel set in
$\mathcal{H}^{-\varepsilon}$, and $\iota^{\ast}\mu(E)$ is well defined. By
monotone convergence, $\iota^{\ast}\mu$ is $\sigma$-additive, so
$\iota^{\ast}\mu$ actually defines a measure. We now check that it is indeed a
Radon measure. Recalling that finite measures on a compact metric space are
Radon, it is enough to show tightness, or more specifically that

$\lim_{n\to\infty}\iota^{\ast}\mu(X\setminus K_{n})=0.$

Noticing that by definition $\iota^{\ast}\mu(K_{n})=\mu(K_{n})$, we have that

$\displaystyle\lim_{n\to\infty}\iota^{\ast}\mu(X\setminus K_{n})$
$\displaystyle=\lim_{n\to\infty}\big{(}\iota^{\ast}\mu(X)-\iota^{\ast}\mu(K_{n})\big{)}=\lim_{m\to\infty}\mu(K_{m})-\lim_{n\to\infty}\mu(K_{n})=0,$

so $\iota^{\ast}\mu$ is a Radon measure. Finally, we show (3.42). For
$K\subseteq X$ compact, recalling the definition of $\lambda$, we have that

$\displaystyle 0\leq\lim_{n\to\infty}\mu(K\setminus
K_{n})=\lim_{n\to\infty}\mu(K\cup
K_{n})-\mu(K_{n})\leq\lambda-\lim_{n\to\infty}\mu(K_{n})=0.$

Therefore,

$\displaystyle\iota^{\ast}\mu(K)=\lim_{n\to\infty}\mu(K\cap
K_{n})=\lim_{n\to\infty}\mu(K)-\mu(K\setminus K_{n})=\mu(K).$

We now assume (3.16) and that $\iota^{\ast}\mu(X)=1$, and show that
$\iota^{\ast}\mu$ is invariant for $P_{t}$. We first show that for every
function $f\in L^{\infty}(\mathcal{H}^{-\varepsilon})$,

$\int f(\mathbf{u}_{0})d\iota^{\ast}\mu(\mathbf{u}_{0})=\int
f(\mathbf{u}_{0})d\mu(\mathbf{u}_{0}).$ (3.43)

Since simple functions are dense in
$\mathscr{L}^{\infty}(\mathcal{H}^{-\varepsilon})$, it is enough to show that
this holds for $f=\mathbb{1}_{E^{\prime}}$, where $E^{\prime}$ is a Borel
subset of $\mathcal{H}^{-\varepsilon}$. In this setting, (3.43) reduces to
showing that

$\iota^{\ast}\mu(E\cap X)=\mu(E).$

By definition of $\iota^{\ast}\mu$, we have that $\iota^{\ast}\mu(E\cap
X)=\lim_{n}\mu(E\cap K_{n})$. Moreover, since $\iota^{\ast}\mu(X)=1$, we have
that $\lim_{n\to\infty}\mu(K_{n}^{c})=0$. Therefore,

$\displaystyle\mu(E)$ $\displaystyle=\lim_{n\to\infty}\mu(E\cap
K_{n})+\mu(E\setminus K_{n})$ $\displaystyle=\lim_{n\to\infty}\mu(E\cap
K_{n})$ $\displaystyle=\iota^{\ast}\mu(E\cap X),$

so we have 3.43. In order to prove invariance, fix
$\varphi\in\mathscr{L}^{\infty}(X)$, and let
$\widetilde{\varphi}\in\mathscr{L}^{\infty}(\mathcal{H}^{-\varepsilon})$ be
such that $\widetilde{\varphi}|_{X}=\varphi$. By (3.36), (3.43), and (3.16),
we have that

$\displaystyle\int
P_{t}\varphi(\mathbf{u}_{0})d\iota^{\ast}\mu(\mathbf{u}_{0})$
$\displaystyle=\int\mathbb{E}[\widetilde{\varphi}(\Phi_{t}(\mathbf{u}_{0},\xi))]d\iota^{\ast}\mu(\mathbf{u}_{0})$
$\displaystyle=\int\mathbb{E}[\widetilde{\varphi}(\Phi_{t}(\mathbf{u}_{0},\xi))]d\mu(\mathbf{u}_{0})$
$\displaystyle=\int\widetilde{\varphi}(\mathbf{u}_{0})d\mu(\mathbf{u}_{0})$
$\displaystyle=\int\varphi(\mathbf{u}_{0})d\iota^{\ast}\mu(\mathbf{u}_{0}).$

∎

The next lemma, even if technically fairly simple, is one of the central
pieces of the proof strategy for Theorem 1.1. It essentially states that the
support Theorem 2.7 is enough to obtain a contradiction further down the line.
In our setting, the statement should be interpreted as a kind of
“irreducibility under the action of $\mathcal{H}^{1-\varepsilon}$” for the
Gaussian measure $\rho_{0}$ (and as a consequence, for the invariant measure
$\rho$).

######  Lemma 3.15.

Let $\pi:X\to X/\mathcal{H}^{1-\varepsilon}$ be the canonical projection, and
consider the measure $\pi_{\sharp}\iota^{\ast}\rho_{0}$, where $\rho_{0}$ is
the gaussian measure (1.7). For every measurable set
$E\in\pi_{\sharp}\mathscr{B}(X)$, we have that

$\pi_{\sharp}\iota^{\ast}\rho_{0}(E)=0\hskip 10.0pt\text{ or }\hskip
10.0pt\pi_{\sharp}\iota^{\ast}\rho_{0}(E)=1.$

###### Proof.

We start by showing this property for the measure
$\pi^{\prime}_{\sharp}\rho_{0}$ instead of $\pi_{\sharp}\iota^{\ast}\rho_{0}$,
where
$\pi^{\prime}:\mathcal{H}^{-\varepsilon}\to\mathcal{H}^{-\varepsilon}/\mathcal{H}^{1-\varepsilon}$
denotes the canonical projection. By (3.1), we can see $\mu$ as the law of the
random variable $\mathbf{U}=(U,V)$, with

$\displaystyle U$
$\displaystyle=\frac{1}{2\pi}\operatorname*{Re}{\Big{(}\sum_{n\in\mathbb{Z}^{2}}\frac{g_{n}}{\langle
n\rangle}e^{in\cdot x}\Big{)}},$ $\displaystyle V$
$\displaystyle=\frac{1}{2\pi}\operatorname*{Re}{\Big{(}\sum_{n\in\mathbb{Z}^{2}}{h_{n}}e^{in\cdot
x}\Big{)}},$

where $g_{n}$, $h_{n}$, are i.i.d., centred, complex valued gaussian random
variables, with $\mathbb{E}g_{n}^{2}=\mathbb{E}h_{n}^{2}=0$,
$\mathbb{E}|g_{n}|^{2}=\mathbb{E}|h_{n}|^{2}=1$. Call
$\mathbf{U}_{>N}=(U_{>N},V_{>N})$, with

$\displaystyle U_{>N}$
$\displaystyle=\frac{1}{2\pi}\operatorname*{Re}{\Big{(}\sum_{n\in\mathbb{Z}^{2},|n|_{\infty}>N}\frac{g_{n}}{\langle
n\rangle}e^{in\cdot x}\Big{)}},$ $\displaystyle V_{>N}$
$\displaystyle=\frac{1}{2\pi}\operatorname*{Re}{\Big{(}\sum_{n\in\mathbb{Z}^{2},|n|_{\infty}>N}{h_{n}}e^{in\cdot
x}\Big{)}},$

and let $\mathbf{U}_{\leq N}=\pi_{N}\mathbf{U}=\mathbf{U}-\mathbf{U}_{>N}$.
Since $e^{in\cdot x}\in\mathcal{H}^{1-\varepsilon}$ for every
$n\in\mathbb{Z}^{2}$, we have that $\mathbf{U}_{\leq
N}\in\mathcal{H}^{1-\varepsilon}$. By definition, for a set
$E\in\pi^{\prime}_{\sharp}\mathscr{B}(\mathcal{H}^{-\varepsilon})$, we have

$\pi^{\prime}_{\sharp}\rho_{0}(E)=\mathbb{P}(\mathbf{U}\in(\pi^{\prime})^{-1}(E)).$

Moreover, since
$(\pi^{\prime})^{-1}(E)=(\pi^{\prime})^{-1}(E)+\mathcal{H}^{1-\varepsilon}$,
we have the equivalence

$\mathbf{U}\in(\pi^{\prime})^{-1}(E)\iff\mathbf{U}_{>N}\in(\pi^{\prime})^{-1}(E).$

Therefore,

$\\{\mathbf{U}\in(\pi^{\prime})^{-1}(E)\\}=\\{\mathbf{U}_{>N}\in(\pi^{\prime})^{-1}(E)\\}.$

This shows that for every $n\in\mathbb{N}$, the event
$\\{\mathbf{U}\in(\pi^{\prime})^{-1}(E)\\}$ belongs to the $\sigma$-algebra
generated by $\\{g_{n},h_{n}:n>|n|_{\infty}\\}$. By Kolmogorov’s zero-one law,
this implies that $\mathbb{P}(\\{\mathbf{U}\in(\pi^{\prime})^{-1}(E)\\})=0$ or
$\mathbb{P}(\\{\mathbf{U}\in(\pi^{\prime})^{-1}(E)\\})=1$. Let now
$F\in\pi_{\sharp}\mathscr{B}(X/\mathcal{H}^{-\varepsilon})$. By definition,
this means that there exists a set
$\widetilde{F}\in\mathscr{B}(\mathcal{H}^{-\varepsilon})$ such that

$\pi^{-1}(F)=\widetilde{F}\cap X,\quad\pi^{-1}(F^{c})=\widetilde{F}^{c}\cap
X.$ (3.44)

By Lemma 3.7, $\rho_{0}(Y)=1$. Therefore, there exists a $\sigma$-compact set
$\widetilde{Y}\subseteq Y$ such that $\rho_{0}(\widetilde{Y})=1$ as well. In
particular, by Lemma 3.11,
$\widetilde{Y}+\mathcal{H}^{1-\varepsilon}\in\mathscr{B}(\mathcal{H}^{-\varepsilon})$,
and clearly $\rho_{0}(\widetilde{Y}+\mathcal{H}^{1-\varepsilon})=1$ as well.
Therefore, by (3.43),

$\displaystyle\iota^{\ast}\mu(\pi^{-1}(F))$
$\displaystyle=\iota^{\ast}\mu(\widetilde{F}\cap
X)=\mu(\widetilde{F})=\mu(\widetilde{F}\cap(\widetilde{Y}+\mathcal{H}^{1-\varepsilon})).$
(3.45)

Morever, by (3.44), since $\widetilde{Y}+\mathcal{H}^{1-\varepsilon}\subseteq
Y+\mathcal{H}^{1-\varepsilon}\subseteq X$,

$\displaystyle\widetilde{F}\cap(\widetilde{Y}+\mathcal{H}^{1-\varepsilon})=(\widetilde{F}\cap
X)\cap(\widetilde{Y}+\mathcal{H}^{1-\varepsilon})=\pi^{-1}(F)\cap(\widetilde{Y}+\mathcal{H}^{1-\varepsilon})=\pi^{-1}(F\cap\pi(\widetilde{Y})).$

Notice that, since $X\setminus\\{\infty\\}$ is invariant under the action of
$\mathcal{H}^{1-\varepsilon}$ over $\mathcal{H}^{-\varepsilon}$, for every set