# Path-by-path uniqueness for stochastic differential equations under
Krylov–Röckner condition

Lukas Anzeletti, Khoa Lê and Chengcheng Ling  Lukas Anzeletti: Université
Paris-Saclay, CentraleSupélec, MICS and CNRS FR-3487, France
Email<EMAIL_ADDRESS>Khoa Lê: University of Leeds,
School of Mathematics, Leeds LS29JT, U.K.
Email<EMAIL_ADDRESS>Chengcheng Ling: Technische Universität Wien,
Institute of Analysis and Scientific Computing, 1040 Wien, Austria
Email<EMAIL_ADDRESS>
###### Abstract.

We show that any stochastic differential equation (SDE) driven by Brownian
motion with drift satisfying the Krylov–Röckner condition has exactly one
solution in an ordinary sense for almost every trajectory of the Brownian
motion. Additionally, we show that such SDE is strongly complete, i.e. for
almost every trajectory of the Brownian motion, the family of solutions with
different initial data forms a continuous semiflow for all nonnegative times.

AMS 2020 Mathematics Subject Classification: 60H10, 60H50, 60J60.

Keywords: Singular SDEs; path-by-path uniqueness; regularization by noise;
strong completeness.

##  1\. Introduction

For a measurable time-dependent vector field
$b:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$, $d\geqslant 1$, and a
measurable driving signal $\gamma:[0,T]\rightarrow\mathbb{R}^{d}$, we consider
the deterministic ordinary integral equation (ODE)

$y_{t}=y_{0}+\int_{0}^{t}b(r,y_{r})\,\mathrm{d}r+\gamma_{t},\quad{t\in[0,T]}.$
(1.1)

Here, $T>0$ is a finite time horizon which is fixed but arbitrary. Because $b$
is only measurable, a solution $(y_{t})_{t\in[0,T]}$, if it exists, should be
measurable and satisfy the integrability condition

$\displaystyle\int_{0}^{T}|b(r,y_{r})|\,\mathrm{d}r<\infty,$ (1.2)

which ensures that the integral in (1.1) is well-defined in Lebesgue sense.
For such solution, it is immediate that $y-\gamma$ is continuous.

When $\gamma$ is sampled from a stochastic process $W$ on a stochastic basis
$(\Omega,{\mathcal{F}},{\mathbb{P}})$, equation (1.1) transforms into the
stochastic differential equation (SDE)

$\displaystyle
Y_{t}=Y_{0}+\int_{0}^{t}b(s,Y_{s})\,\mathrm{d}s+W_{t},\quad{t\in[0,T]}.$ (1.3)

A solution $Y:\Omega\times[0,T]\to{{\mathbb{R}}^{d}}$ to (1.3) is a stochastic
process such that for a.s. $\omega$, $(Y_{t}(\omega))_{t\in[0,T]}$ is a
solution to (1.1) with $W(\omega)$ in place of $\gamma$.

It is now well-understood that the addition of fast oscillating driving
signals $\gamma,W$ can make both (1.1) and (1.3) well-posed for very irregular
vector fields $b$, which is known as _regularization-by-noise_ phenomenon. For
SDEs driven by Brownian motion with bounded measurable vector fields, [Zvo74,
Ver79] showed that there exists a pathwise unique adapted solution to (1.3).
In [KR05], Krylov and Röckner extended this result to drifts $b$ fulfilling
the condition

$\displaystyle\Big{(}\int_{0}^{T}\big{(}\int_{{\mathbb{R}}^{d}}|b(r,x)|^{p}\,\mathrm{d}x\big{)}^{\frac{q}{p}}\,\mathrm{d}r\Big{)}^{\frac{1}{q}}<\infty,\quad(p,q)\in\mathcal{J}:=\left\\{(p,q)\in(2,\infty)^{2}:\frac{2}{q}+\frac{d}{p}<1\right\\}.$
(1.4)

This condition is also known as the _sub-critical Ladyzhenskaya–Prodi–Serrin_
(LPS) condition in fluid dynamics, see more in [FF13b, BFGM19].

Pathwise uniqueness for SDEs with fractional noise was considered first by
Nualart and Ouknine in [NO02, NO03] and more recently in [BNP19, Lê20, ART21,
GG22a, BLM23b] among others.

Well-posedness for (1.1) is considered in more recent time. Starting from the
seminal works of Davie [Dav07], Catellier–Gubinelli [CG16], there has been an
increasing interest in the problem, [Sha16, HP21, GG22b, GG22a, DG22, CD22]. A
typical result of these articles is that when $\gamma$ is a sample path of an
irregular stochastic process (such as fractional Brownian motion), then for
almost every such sample paths, ODE (1.1) has a unique solution, a property
known by the terminology _path-by-path uniqueness_. The necessary regularity
for the vector field is inversely proportional to the roughness of the
stochastic process from which $\gamma$ is sampled. In a different direction,
when the coefficients are sufficiently regular so that Lyons’ rough path
theory can be applied, then uniqueness of the corresponding rough differential
equation implies path-by-path uniqueness. This is because the stochastic basis
is only used to construct a rough path lift of the driving signal and Lyons’
rough path analysis ([Lyo98]) is applicable for any fixed trajectory of the
lift.

From a practical perspective, path-by-path uniqueness is most relevant in
situations where only a handful samples of the signal can be observed.
Theoretically, “path-by-path uniqueness” is stronger than “pathwise
uniqueness” (which essentially allows to identify two adapted solutions to
(1.3)). Having pathwise uniqueness for (1.3) does not provide uniqueness for
the $W(\omega)$-driven ODE

$\displaystyle
Y_{t}=Y_{0}+\int_{0}^{t}b(r,Y_{r})\,\mathrm{d}r+W_{t}(\omega),\quad
t\in[0,T],$ (1.5)

for a given $\omega$. In fact, examples of SDEs driven by Brownian motion
which have pathwise uniqueness but not path-by-path uniqueness are given in
[SW22, Anz22]. We point to Appendix B for further discussions on the related
concepts.

The main goal of the current article is the following result, which will be
stated more precisely later in Theorems 2.2, 2.4 and 2.5.

Theorem. Assuming that $W$ is a Brownian motion, then path-by-path uniqueness,
path-by-path stability and strong completeness for (1.3) hold under the
Krylov–Röckner condition (1.4).

By path-by-path stability, we mean that for a sequence of functions $(b^{n})$
converging sufficiently fast to $b$ in a suitable topology, there exists an
event of full probability which depends only on $(b^{n})$ and $T$ such that
for every $\omega$ in such event, for every $x\in{{\mathbb{R}}^{d}}$, the
solution $Y^{n}$ to the ODE

$\displaystyle
Y^{n}_{t}=x+\int_{0}^{t}b^{n}(r,Y^{n}_{r})\,\mathrm{d}r+W_{t}(\omega),\quad
t\in[0,T]$

converges uniformly to the unique solution to (1.5) with initial condition
$Y_{0}=x$. This is stronger than standard stability results because the event
of full probability measure only depends on $(b^{n})$. Usually, for each $n$,
we have to choose $Y^{n}$ to be the pathwise unique strong solution and the
set of full measure on which convergence holds depends on the choice of
$Y^{n}$ (respectively the set of $\omega$ for which it fulfills the equation).
Additionally, with this procedure the event of full measure would also depend
on the choice of the initial condition.

Strong completeness refers to the property that for almost all trajectories of
the Brownian motion, the equation can be solved globally regardless of the
initial data. Strong completeness is a rather demanding property and may fail
even for SDEs with bounded smooth (but with unbounded derivatives)
coefficients, with examples given by Li and Scheutzow in [LS11]. Strong
completeness for SDEs driven by Brownian motion with standard regularity
conditions are considered in [Li94, SS17]. Alternatively, when the SDE under
consideration can be lifted to a rough differential equation which has unique
global solution, then it is strongly complete, see [RS17]. The SDE (1.3) under
condition (1.4) falls outside of the standard Itô framework and Lyons’ rough
path theory. Hence, our result provides the first example of strongly complete
singular SDEs driven by Brownian motion.

To the best of our knowledge, path-by-path uniqueness under (1.4) has not been
considered in the literature and path-by-path stability is introduced herein.
[Dav07, Sha16] shows path-by-path uniqueness for SDEs with bounded drifts. For
equations driven general fractional noise with distributional drifts, this
kind of result is shown in [CG16, HP21, GG22a], by reformulating the original
equation into the framework of nonlinear Young differential equations. In this
framework, proofs are considerably simpler, however, the assumed regularity
conditions on the drifts in the aforementioned works exclude the
Krylov–Röckner condition (1.4). Although the solutions of nonlinear Young
differential equations are apparently different from solutions of ordinary
differential equations, it follows from our analysis that solutions of (1.5)
can also be interpreted in nonlinear Young sense, see Remark 4.5.

Our approach is closer to that of [Dav07] taking into account the insights
from [CG16, Sha16, GG22a] and the quantitative John–Nirenberg inequality
recently obtained in [Lê22]. Davie in [Dav07] delivers all arguments from
first principles, which unfortunately increases the level of technicality. His
proof consists of two main parts, one is the regularizing estimates for
averaging operators along Brownian motion ([Dav07, The basic estimate,
Proposition 2.1]) and the other is the uniqueness argument. Catellier and
Gubinelli in [CG16] reformulate and extend Davie’s approach in the framework
of nonlinear Young integration, leading to path-by-path uniqueness for a large
class of fractional SDEs. Although more general, results in [CG16] do not
reproduce Davie’s result due to several technical reasons. Shaposhnikov in
[Sha16] gives a simpler justification for Davie’s uniqueness arguments,
calling out the importance of the flow generated by (1.3). Galeati and
Gerencsér in [GG22a] adopt this argument in combination with estimates for
nonlinear Young integrals. It is known that [Sha16] contains several technical
issues some of which were corrected in [Sha17]. Herein, we discover and
justify another one which was not addressed in [Sha16, Sha17], see Remark 5.3.

As for our strategy, we apply John–Nirenberg inequality to show Davie
estimates under (1.4) (see Proposition 3.1). The John–Nirenberg inequality
effectively reduces estimating all positive moments to estimating only the
second moment. This shortens existing arguments and improves known results
from [Dav07, Rez14]. Building upon Davie estimates, we use nonlinear Young
integration to obtain a key regularizing estimate for averaging operators
along Brownian motion (Lemma 4.3). This allows us to show that equation (1.3)
generates a random continuous semiflow and obtain a priori estimates for any
solution to (1.5) (Propositions 5.1 and 5.2). Applying these priori estimates
and the regularity of the semiflow generated by (1.3), we are able to identify
any solution with the semiflow, see proof of Theorem 2.2 in Section 5.
Although using similar tools as [CG16, Sha16, GG22a], our proofs are logically
different from the aforementioned articles. Additionally, our arguments are
generic and are not strictly tied to Brownian motion, as long as the random
semiflow generated by the SDE (1.3) and the averaging operators along the
driving signal $W$ are sufficiently regular. This leads to a new uniqueness
criterion for ODEs of the form (1.1), which does not require any regularity on
the vector field, but instead relies on the regularizing effect of the driving
signal $\gamma$. These results are summarized in Appendix A.

Structure of paper. Main results are stated in Section 2. In Section 3, we
show Davie estimates under (1.4). In Section 4, we show some path-by-path
regularizing estimates, which are consequences of Davie estimates and the
sewing lemma. The proof of the main results are presented in Section 5. In
Appendix A, we summarize a uniqueness criterion for (1.1) which is based on
the regularization effect of the driving signal. Different notions of
solutions and their relations are recalled in Appendix B.

Frequently used notation. Define for a random variable $X$, for any
$m\geqslant 1$, $\|X\|_{L^{m}(\Omega)}:=[{\mathbb{E}}(|X|^{m})]^{1/m}$. For
each $p\in[1,\infty]$, $L^{p}({{\mathbb{R}}^{d}})$ denotes the usual Lebesgue
space on ${\mathbb{R}}^{d}$. For $0\leqslant s\leqslant t$,
${\mathbb{L}}_{p}^{q}([s,t])$ contains all measurable functions
$f:[s,t]\times{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ such that

$\displaystyle\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}:=\Big{(}\int_{s}^{t}\big{(}\int_{{\mathbb{R}}^{d}}|f(r,x)|^{p}\,\mathrm{d}x\big{)}^{\frac{q}{p}}\,\mathrm{d}r\Big{)}^{\frac{1}{q}}<\infty$

for $p,q\in[1,\infty)$ and with obvious modification when $p$ or $q$ is
infinity. We abbreviate
${\mathbb{L}}_{p}^{q}={\mathbb{L}}_{p}^{q}({\mathbb{R}})$. For $0\leqslant
S\leqslant T$, let
$[S,T]^{2}_{\leqslant}:=\left\\{(s,t)\in[S,T]^{2}:S\leqslant s\leqslant
t\leqslant T\right\\}$. We say that a continuous function
$w:[S,T]^{2}_{\leqslant}\to[0,\infty)$ is a control if $w(s,u)+w(u,t)\leqslant
w(s,t)$ whenever $s\leqslant u\leqslant t$. For a function
$(\psi_{t})_{t\in[0,T]}$ of finite variation and each
$(u,v)\in[0,T]_{\leqslant}^{2}$, we define

$\displaystyle[\psi]_{\mathrm{var};[u,v]}:=\sup_{\pi}\sum_{[s,t]\in\pi}|\psi_{t}-\psi_{s}|.$

where the supremum is taken over all partitions $\pi$ of $[u,v]$. It is known
that $(u,v)\mapsto[\psi]_{\mathrm{var};[u,v]}$ is a control. We write
$H\lesssim G$ if $H\leqslant CG$ for some universal finite positive constant
$C$.

##  2\. Main results

Let $\\{{\mathcal{F}}_{t}\\}_{t\geqslant 0}$ be a filtration and assume that
$(\Omega,{\mathcal{F}},\\{{\mathcal{F}}_{t}\\}_{t\geqslant 0},{\mathbb{P}})$
satisfies the usual conditions. For each $s\in[0,T]$, ${\mathbb{E}}_{s}$
denotes the conditional expectation with respect to ${\mathcal{F}}_{s}$, i.e.
${\mathbb{E}}_{s}[\cdot]={\mathbb{E}}_{s}[\,\cdot\,|{\mathcal{F}}_{s}]$. We
assume that $W$ is a standard Brownian motion with respect to
$\\{{\mathcal{F}}_{t}\\}_{t\geqslant 0}$.

A _semiflow_ generated by the ODE (1.1) over the period $[0,T]$ is a
measurable map $(s,t,x)\mapsto\phi^{s,x}_{t}$ defined on
$[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}$ such that for every
$0\leqslant s\leqslant u\leqslant t\leqslant T$ and $x\in{{\mathbb{R}}^{d}}$,

$\displaystyle\int_{s}^{T}|b(r,\phi^{s,x}_{r})|\,\mathrm{d}r<\infty,\quad\phi^{s,x}_{t}=x+\int_{s}^{t}b(r,\phi^{s,x}_{r})\,\mathrm{d}r+\gamma_{t}-\gamma_{s}\quad\text{and}\quad\phi^{s,x}_{t}=\phi^{u,\phi^{s,x}_{u}}_{t}.$

We say that $\phi$ is _locally $\kappa$-Hölder continuous_ if for every
compact set $K\subset{{\mathbb{R}}^{d}}$, there exists a finite constant
$C=C(\kappa,K,T)$ such that for every $x,y\in K$ and
$(s,t)\in[0,T]^{2}_{\leqslant}$,

$\displaystyle|\phi^{s,x}_{t}-\phi^{s,y}_{t}|\leqslant C|x-y|^{\kappa}.$

######  Proposition 2.1 (Random Hölder semiflow).

Under (1.4), there exists an almost surely locally $\kappa$-Hölder continuous
semiflow to (1.3) for every $\kappa\in(0,1)$. In other words, there exists a
jointly measurable map $(s,t,x,\omega)\mapsto X^{s,x}_{t}(\omega)$ defined on
$[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}\times\Omega$ such that for a.s.
$\omega$, $(s,t,x)\mapsto X^{s,x}_{t}(\omega)$ is a locally $\kappa$-Hölder
continuous semiflow to (1.5) for every $\kappa\in(0,1)$.

The semiflow $(X^{s,x}_{t})$ is constructed by taking the continuous extension
of the family of strong solutions over the dyadics. Hence, it inherits other
properties and as such, it is adapted and is a flow of homeomorphisms.
However, we will not use these properties. Proposition 2.1 thus improves upon
[FF13a, Theorem 5.1] by showing joint continuity in all parameters and
allowing the set of full measure on which the flow solves the equation to be
uniform with respect to the initial data. Because $b$ is not continuous,
showing that the continuous extension $X$ forms a family of solutions to (1.3)
is actually a major obstacle. This is overcome thanks to the path-by-path
regularizing estimates obtained by sewing techniques and Davie estimates, see
Lemmas 4.3 and 4.4. These arguments can be applied for other situations and it
is expected that the conclusions of Proposition 2.1 hold for SDEs with bounded
measurable drifts, more details are discussed in Remark 5.3. Furthermore, it
follows from our arguments that a solution to (1.5) is also a nonlinear Young
solution and a regularized solution, see Remark 4.5 for further details.

######  Theorem 2.2 (Path-by-path uniqueness).

There exists an event $\Omega_{b,T}\in{\mathcal{F}}$ which depends only on
$b,T$ and has full probability measure such that for every
$\omega\in\Omega_{b,T}$, any solution $Y$ to (1.5) is identical to
$X^{0,Y_{0}}(\omega)$. As a consequence, path-by-path uniqueness for Equation
(1.3) holds.

######  Remark 2.3.

The dependence on $T$ can easily be removed by considering the event
$\cup_{T=1}^{\infty}\Omega_{b,T}$. The dependence on $b$ seems necessary. When
an initial datum is specified so that $Y_{0}=\xi$ for some
$\xi\in{{\mathbb{R}}^{d}}$, the set $\Omega_{b,T}$ is independent from $\xi$.
In addition, if $\xi$ is a random variable which is not necessary
${\mathcal{F}}_{0}$-measurable, Theorem 2.2 implies uniqueness among non-
adapted solutions.

The event $\Omega_{b,T}$ essentially contains $\omega$ such that

  * (i)

the semiflow $X^{s,x}_{t}(\omega)$ is almost Lipschitz in $x$,

  * (ii)

the averaging operators

$\displaystyle(t,x)\mapsto\int_{0}^{t}b(r,W_{r}(\omega)+x)\,\mathrm{d}r,\,\int_{0}^{t}|b|(r,W_{r}(\omega)+x)\,\mathrm{d}r$

are jointly locally Hölder continuous with exponent $(\alpha,1-\varepsilon)$
for some $\alpha>0$ and some $\varepsilon>0$ sufficiently small.

These properties are rather generic and available not only when $W$ is a
Brownian motion but also for other stochastic processes. The case of
fractional Brownian motion is currently investigated in [BLM23a].

To state the result on path-by-path stability, we introduce some additional
notations. For $\nu\geqslant 0$, $W^{-\nu,p}({\mathbb{R}}^{d})$ is the Bessel
potential space
$W^{-\nu,p}({\mathbb{R}}^{d}):=\\{f:\|f\|_{W^{-\nu,p}({\mathbb{R}}^{d})}:=\|(I-\Delta)^{-\frac{\nu}{2}}f\|_{L^{p}({\mathbb{R}}^{d})}<\infty\\}$.
When $\nu=0$, we have $W^{-\nu,p}({\mathbb{R}}^{d})=L^{p}({\mathbb{R}}^{d})$.
A sequence $(z^{n})$ in a metric space $({\mathcal{Z}},\rho)$ is summable
convergent to an element $z\in{\mathcal{Z}}$ if there is a constant $\eta>0$
such that

$\displaystyle\sum_{n=1}^{\infty}\rho(z^{n},z)^{\eta}<\infty.$

In such case, we write $\operatorname*{{\Sigma-lim}}_{n}z^{n}=z$ in
${\mathcal{Z}}$. It is evident that summable convergence implies convergence
in the usual sense. Conversely, if $\lim_{n}z^{n}=z$, then for every $\eta>0$,
there exists a subsequence $(z^{\ell(n)})$ such that
$\sum_{n=1}^{\infty}\rho(z^{\ell(n)},z)^{\eta}<\infty$, in particular
$\operatorname*{{\Sigma-lim}}_{n}z^{\ell(n)}=z$.

######  Theorem 2.4 (Path-by-path stability).

Let $(p,q)\in{\mathcal{J}}$ and $(b^{n})$ be a sequence of measurable
functions in ${\mathbb{L}}^{q}_{p}$. We assume that $\operatorname*{{\Sigma-
lim}}_{n}b^{n}=b$ in $L^{q}([0,T],W^{-\nu,p}({\mathbb{R}}^{d}))$ where $\nu$
satisfies either

$\displaystyle\nu\in[0,1-\frac{2}{q}),\quad\frac{d}{p}+\frac{2}{q}+\nu<2;$
(2.1) or
$\displaystyle\nu\in[1-\frac{2}{q},1],\quad\frac{d}{p}+\frac{4}{q}<3-2\nu,\quad\frac{d}{p}<\nu.$
(2.2)

Then there exists an event $\Omega^{\prime}_{(b^{n}),b,T}$, which depends on
$(b^{n}),b,T$ and has full probability measure, such that for every
$\omega\in\Omega^{\prime}_{(b^{n}),b,T}$, for every
$(s,x)\in[0,T]\times{\mathbb{R}}^{d}$, whenever $Y^{n}$ are solutions to ODEs

$\displaystyle
Y^{n}_{t}=x+\int_{s}^{t}b^{n}(r,Y^{n}_{r})\,\mathrm{d}r+W_{t}(\omega)-W_{s}(\omega),\quad
t\in[s,T],$

then $\lim_{n}Y^{n}=X^{s,x}(\omega)$ uniformly on $[s,T]$.

We say that the SDE (1.3) is _strongly complete_ if for a.s. $\omega$, the ODE
(1.5) generates a continuous semiflow over $[0,\infty)$. This means that there
exists a measurable map

$\displaystyle[0,\infty)^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}\times\Omega$
$\displaystyle\to{{\mathbb{R}}^{d}}$ $\displaystyle(s,t,x,\omega)$
$\displaystyle\mapsto X^{s,x}_{t}(\omega)$

such that for a.s. $\omega$ the following hold:

  1. (i)

$(s,t,x)\mapsto X^{s,x}_{t}(\omega)$ is continuous;

  2. (ii)

for every $0\leqslant s\leqslant u\leqslant t$ and $x\in{{\mathbb{R}}^{d}}$

$\displaystyle\int_{s}^{t}|b(r,X^{s,x}_{r}(\omega))|\,\mathrm{d}r<\infty,\quad
X^{s,x}_{t}(\omega)=x+\int_{s}^{t}b(r,X^{s,r}_{r}(\omega))\,\mathrm{d}r+W_{t}(\omega)-W_{s}(\omega)$

and

$\displaystyle X^{s,x}_{t}(\omega)=X^{u,X^{s,x}_{u}(\omega)}_{t}(\omega).$

Our definition given above is different from the literatures’ in that ii
demands that the event on which the semiflow solves the equation is
independent from initial data.noinline, color=green!20!whitenoinline,
color=green!20!whitetodo: noinline, color=green!20!whiteLukas: not sure what
we mean here (I guess it is the following…). To me pathwise stochastic sense
is ordinary+adapted so it is stronger. Instead the difference is that the full
measure set on which it solves the equation is independent of $x$ right? which
does not have to be the case in [LS11] I think Maybe just add at the end:
(i.e. it has to solve the equation on a set of full measure that is
independent of $x$) Heuristically, strong completeness requires that the
random semiflow is defined globally over all nonnegative times $[0,\infty)$
and continuously in $(s,t,x)$ over
$[0,\infty)^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}$. That the semiflow is
generated by the equation means that for almost every $\omega$ and every
initial data $(s,x)$, the map $t\mapsto X^{s,x}_{t}(\omega)$ is a global
solution which does not blow up in finite time. The later property can be
thought as “path-by-path non-explosion”. SDEs which have global pathwise
solutions without explosion may fail to have path-by-path non-explosion with
examples given in [LS11].

Our next result provides a general principle which asserts that if over any
finite period of time, an SDE generates a random continuous semiflow and has
path-by-path uniqueness, then the SDE is strongly complete. Although this fact
has been observed in [RS17] in the context of rough differential equations,
singular SDEs is another distinct class of equations. It is therefore
necessary to draw the connections.

######  Theorem 2.5 (Strong completeness).

Suppose that for every $T>0$, (1.3) generates a random continuous semiflow
over $[0,T]$ and that path-by-path uniqueness holds over $[0,T]$. Then (1.3)
is strongly complete.

###### Proof.

Let $(X^{s,x}_{t})$ and $(\bar{X}^{s,x}_{t})$ be random continuous semiflows
respectively over $[0,T]$ and $[0,\bar{T}]$ for some $T<\bar{T}$. Then by
path-by-path uniqueness
$\bar{X}|_{[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}}=X$. This implies the
existence (and uniqueness) of a random continuous semiflow over all
nonnegative time. ∎

The proofs of Propositions 2.1, 2.2 and 2.4 are presented in Section 5.

##  3\. Davie estimates

We show a variant of _Davie estimates_ in [Dav07, Propositions 2.1, 2.2].

######  Proposition 3.1.

Let $f$ be a Schwartz function in ${\mathbb{L}}^{q}_{p}$ for some
$(p,q)\in{\mathcal{J}}$. Then there is a constant $C=C(d,p,q,T)$ such that for
every $(s,t)\in[0,T]^{2}_{\leqslant}$ and $m\geqslant 1$,

$\displaystyle\left\|\int_{s}^{t}\nabla
f(r,W_{r})\,\mathrm{d}r\right\|_{L^{m}(\Omega)}\leqslant
C\Gamma\left(m(\frac{1}{2}+\frac{d}{2p})+1\right)\|f\|_{{\mathbb{L}}^{q}_{p}([s,t])}(t-s)^{\frac{1}{2}-\frac{1}{q}-\frac{d}{2p}},$
(3.1)

where $\Gamma(r)=\int_{0}^{\infty}u^{r-1}e^{-u}\,\mathrm{d}u$ is the Gamma
function. Consequently, for every bounded continuous function $f$ in
${\mathbb{L}}^{q}_{p}([0,T])$, every $x,y\in{\mathbb{R}}^{d}$, every
$(s,t)\in[0,T]^{2}_{\leqslant}$ and $m\geqslant 1$, we have

$\left\|\int_{s}^{t}[f(r,W_{r}+x)-f(r,W_{r}+y)]\,\mathrm{d}r\right\|_{L^{m}(\Omega)}\\\
\leqslant
C\Gamma\left(m(\frac{1}{2}+\frac{d}{2p})+1\right)\|f\|_{{\mathbb{L}}^{q}_{p}([s,t])}(t-s)^{\frac{1}{2}-\frac{1}{q}-\frac{d}{2p}}|x-y|.$
(3.2)

The precise growth constant $\Gamma\left(m(\frac{1}{2}+\frac{d}{2p})+1\right)$
is not essential to our current purposes. Nevertheless, it improves upon
previous known estimates [Rez14]. To prove this result, we follow a recent
approach from [Lê22] based on the quantitative John–Nirenberg inequality. We
prepare two auxiliary lemmas, putting

$\displaystyle p_{t}(x):={\mathbf{1}}_{(t>0)}(2\pi
t)^{-\frac{d}{2}}e^{-\frac{|x|^{2}}{2t}}.$

######  Lemma 3.2.

The operator $f\mapsto D(f)$ defined by

$\displaystyle D(f)(t,x):=\int_{{\mathbb{R}}\times{{\mathbb{R}}^{d}}}\nabla
p_{s}(y)\cdot\nabla f(t-s,x-y)\,\mathrm{d}(s,y),$

is bounded on ${\mathbb{L}}^{q}_{p}$ for every $(p,q)\in(1,\infty)^{2}$.

###### Proof.

The operator $D$ is associated to the Fourier multiplier

$\displaystyle\Phi(x_{0},x)=c\frac{x^{2}}{x^{2}+\mathrm{i}x_{0}},\quad
x^{2}=x_{1}^{2}+...+x_{d}^{2},$

for some absolute constant $c$, where $\mathrm{i}$ is the imaginary unit. A
result from [Liz70, Corollary 1, pg. 234] asserts that $D$ is a bounded
operator on ${\mathbb{L}}^{q}_{p}$ provided that $\Phi$ admits continuous (for
$x_{0},...,x_{d}\neq 0$) purely mixed derivatives of orders $k\leqslant d+1$
such that

$\displaystyle\left|x_{j_{1}}...x_{j_{k}}\partial^{k}_{x_{j_{1}}...x_{j_{k}}}\Phi\right|\leqslant
M$

for all distinct $j_{1},...,j_{k}$ and for some finite constant $M$. Such
condition can be verified directly. ∎

######  Lemma 3.3 (Quantitative John–Nirenberg inequality).

Let $V$ be an adapted continuous process, $w$ be a deterministic control and
$\alpha\in(0,1)$. Assume that

$\displaystyle\|{\mathbb{E}}_{s}|V_{t}-V_{s}|\|_{L^{\infty}(\Omega)}\leqslant
w(s,t)^{\alpha}\quad\forall(s,t)\in[0,T]^{2}_{\leqslant}.$

Then, there exists a finite constant $C_{\alpha}>0$ such that for every
$(s,t)\in[0,T]^{2}_{\leqslant}$ and every $m\geqslant 1$,

$\displaystyle\|\sup_{u\in[s,t]}|V_{u}-V_{s}|\|_{L^{m}(\Omega)}\leqslant
C_{\alpha}\Gamma(m(1-\alpha)+1)w(s,t)^{\alpha}.$

###### Proof.

Without the growth constant $\Gamma(m(1-\alpha)+1)$, this result is a direct
consequence of [SV06, Excercise A.3.2]. To obtain the precise growth constant,
we follow [Lê22] closely. Continuity and the assumption imply that

$\displaystyle\|{\mathbb{E}}_{\sigma}|V_{\tau}-V_{\sigma}|\|_{L^{\infty}(\Omega)}\leqslant
w(s,t)^{\alpha}$

for every $(s,t)\in[0,T]^{2}_{\leqslant}$ and all stopping times $\sigma,\tau$
satisfying $s\leqslant\sigma\leqslant\tau\leqslant t$. The class of all
processes with such property is denoted by
$\mathrm{VMO}^{(1/\alpha)-\mathrm{var}}$ in [Lê22, Section 3]. We then apply
[Lê22, Corollary 3.5] to obtain the desired estimate. ∎

###### Proof of Proposition 3.1.

In view of Lemma 3.3, it suffices to estimate the conditional second moment.
Define for each $(s,t)\in[0,T]_{\leqslant}^{2}$,

$\displaystyle V_{t}:=\int_{0}^{t}\nabla
f(r,W_{r})\,\mathrm{d}r,\quad\text{and}\quad
I_{s,t}:=\frac{1}{2}{\mathbb{E}}_{s}\big{|}V_{t}-V_{s}\big{|}^{2}.$ (3.3)

Using integration by parts, tower property of conditional expectation and
Fubini’s Theorem we have, almost surely,

$\displaystyle I_{s,t}$
$\displaystyle={\mathbb{E}}_{s}\int_{s}^{t}\int^{t}_{r_{2}}{\mathbb{E}}_{r_{2}}[\nabla
f(r_{1},W_{r_{1}})]\cdot\nabla
f(r_{2},W_{r_{2}})\,\mathrm{d}r_{1}\,\mathrm{d}r_{2}$
$\displaystyle=-{\mathbb{E}}_{s}\int_{s}^{t}\int^{t}_{r_{2}}\big{[}\int_{{\mathbb{R}}^{d}}f(r_{1},y+W_{r_{2}})\nabla
p_{r_{1}-r_{2}}(y)\,\mathrm{d}y\big{]}\cdot\nabla
f(r_{2},W_{r_{2}})\,\mathrm{d}r_{1}\,\mathrm{d}r_{2}$
$\displaystyle=-\int_{s}^{t}\int^{t}_{r_{2}}\int_{{\mathbb{R}}^{d}}\big{[}\int_{{\mathbb{R}}^{d}}f(r_{1},y+z+W_{s})\nabla
p_{r_{1}-r_{2}}(y)\,\mathrm{d}y\big{]}\cdot\nabla
f(r_{2},z+W_{s})p_{r_{2}-s}(z)\,\mathrm{d}z\,\mathrm{d}r_{1}\,\mathrm{d}r_{2}$
$\displaystyle=:I_{1}+I_{2}$

where

$\displaystyle I_{1}$
$\displaystyle=\int_{s}^{t}\int^{t}_{r_{2}}\int_{{\mathbb{R}}^{d}}\big{[}\int_{{\mathbb{R}}^{d}}f(r_{1},y+z+W_{s})\nabla
p_{r_{1}-r_{2}}(y)\,\mathrm{d}y\big{]}\cdot f(r_{2},z+W_{s})\nabla
p_{r_{2}-s}(z)\,\mathrm{d}z\,\mathrm{d}r_{1}\,\mathrm{d}r_{2},$ $\displaystyle
I_{2}$
$\displaystyle=\int_{s}^{t}\int^{t}_{r_{2}}\int_{{\mathbb{R}}^{d}}\big{[}\int_{{\mathbb{R}}^{d}}\nabla
f(r_{1},y+z+W_{s})\cdot\nabla
p_{r_{1}-r_{2}}(y)\,\mathrm{d}y\big{]}f(r_{2},z+W_{s})p_{r_{2}-s}(z)\,\mathrm{d}z\,\mathrm{d}r_{1}\,\mathrm{d}r_{2}.$

To estimate $I_{1}$, we apply Hölder’s inequality (below $p^{\prime}$
satisfies $\frac{1}{p}+\frac{1}{p^{\prime}}=1$) to see that for any
$v\in{\mathbb{R}}^{d}$,

$\displaystyle\int_{s}^{t}\int^{t}_{r_{2}}\int_{{\mathbb{R}}^{d}}\big{[}\int_{{\mathbb{R}}^{d}}f(r_{1},y+z+v)\nabla
p_{r_{1}-r_{2}}(y)\,\mathrm{d}y\big{]}f(r_{2},z+v)\nabla
p_{r_{2}-s}(z)\,\mathrm{d}z\,\mathrm{d}r_{1}\,\mathrm{d}r_{2}$
$\displaystyle\leqslant\int_{s}^{t}\int^{t}_{r_{2}}\|f(r_{1},z+v+\cdot)\|_{L^{p}({\mathbb{R}}^{d})}\|\nabla
p_{r_{1}-r_{2}}\|_{L^{p^{\prime}}({\mathbb{R}}^{d})}\|f(r_{2},v+\cdot)\|_{L^{p}({\mathbb{R}}^{d})}\|\nabla
p_{r_{2}-s}\|_{L^{p^{\prime}}({\mathbb{R}}^{d})}\,\mathrm{d}r_{1}\,\mathrm{d}r_{2}$
$\displaystyle\lesssim\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}\int_{s}^{t}\big{(}\int_{r_{2}}^{t}(r_{1}-r_{2})^{(-\frac{1}{2}-\frac{d}{2p})\frac{q}{q-1}}\,\mathrm{d}r_{1}\big{)}^{1-\frac{1}{q}}\|f(r_{2})\|_{L^{p}({\mathbb{R}}^{d})}(r_{2}-s)^{-\frac{1}{2}-\frac{d}{2p}}\,\mathrm{d}r_{2}$
$\displaystyle\lesssim\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}\int_{s}^{t}(t-r_{2})^{\frac{1}{2}-\frac{d}{2p}-\frac{1}{q}}\|f(r_{2})\|_{L^{p}({\mathbb{R}}^{d})}(r_{2}-s)^{-\frac{1}{2}-\frac{d}{2p}}\,\mathrm{d}r_{2}$
$\displaystyle\lesssim\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}(t-s)^{\frac{1}{2}-\frac{d}{2p}-\frac{1}{q}}\int_{s}^{t}\|f(r_{2})\|_{L^{p}({\mathbb{R}}^{d})}(r_{2}-s)^{-\frac{1}{2}-\frac{d}{2p}}\,\mathrm{d}r_{2}$
$\displaystyle\lesssim\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}(t-s)^{\frac{1}{2}-\frac{d}{2p}-\frac{1}{q}}\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}(t-s)^{\frac{1}{2}-\frac{d}{2p}-\frac{1}{q}}\lesssim\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}^{2}(t-s)^{1-\frac{2}{q}-\frac{d}{p}}.$

This yields that, almost surely,

$\displaystyle|I_{1}|\lesssim\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}^{2}(t-s)^{1-\frac{2}{q}-\frac{d}{p}}.$

To estimate $I_{2}$, we apply Hölder inequality and Lemma 3.2 to see that

$\displaystyle|I_{2}|\lesssim\|f\|_{{\mathbb{L}}^{q}_{p}([s,t])}\|{\mathbf{1}}_{[s,t]}f^{W_{s}}p^{s}\|_{{\mathbb{L}}^{q^{\prime}}_{p^{\prime}}}$

where

$\displaystyle f^{W_{s}}(r,z)=f(r,z+W_{s}),\quad p^{s}_{r}(z)=p_{r-s}(z),$

and $p^{\prime},q^{\prime}$ denote the Hölder conjugates of $p,q$
respectively. Applying Hölder inequality again, we have

$\displaystyle\|{\mathbf{1}}_{[s,t]}f^{W_{s}}p^{s}\|_{{\mathbb{L}}^{q^{\prime}}_{p^{\prime}}}\leqslant\|{\mathbf{1}}_{[s,t]}p^{s}\|_{{\mathbb{L}}^{q/(q-2)}_{p/(p-2)}}\|f\|_{{\mathbb{L}}^{q}_{p}([s,t])}.$

Using the fact that $(p,q)\in{\mathcal{J}}$ and some elementary calculations,
we have

$\displaystyle\|{\mathbf{1}}_{[s,t]}p^{s}\|_{{\mathbb{L}}^{q/(q-2)}_{p/(p-2)}}=c(t-s)^{1-\frac{d}{p}-\frac{2}{q}}$

for some constant $c$ depending on $p,q$. Combining with the previous
estimates, we obtain that

$\displaystyle|I_{s,t}|\lesssim\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}^{2}(t-s)^{1-\frac{2}{q}-\frac{d}{p}}\quad
a.s.$

Consequently,

$\displaystyle\|{\mathbb{E}}_{s}|V_{t}-V_{s}|\|_{L^{\infty}(\Omega)}\lesssim\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}(t-s)^{\frac{1}{2}-\frac{1}{q}-\frac{d}{2p}}.$

It is evident that $V$ is continuous. Applying Lemma 3.3, we obtain (3.1).

Next we show (3.2). By approximations, we can assume that $f$ has bounded
continuous first derivatives. We observe that

$\displaystyle\int_{s}^{t}f(r,W_{r}+x)$
$\displaystyle-f(r,W_{r}+y)\,\mathrm{d}r=(x-y)\cdot\int_{0}^{1}\int_{s}^{t}\nabla
f\big{(}r,W_{r}+\theta x+(1-\theta)y\big{)}\,\mathrm{d}r\,\mathrm{d}\theta.$

Then it follows from (3.1) that

$\displaystyle\big{\|}\int_{s}^{t}f(r,W_{r}+x)-f(r,W_{r}+y)\,\mathrm{d}r\big{\|}_{L^{m}(\Omega)}$
$\displaystyle\leqslant\sup_{\theta\in[0,1]}\big{\|}\int_{s}^{t}\nabla
f\big{(}r,W_{r}+\theta
x+(1-\theta)y\big{)}\,\mathrm{d}r\big{\|}_{L^{m}(\Omega)}|x-y|$
$\displaystyle\leqslant|x-y|\sup_{z\in{\mathbb{R}}^{d}}\big{\|}\int_{s}^{t}\nabla
f(r,W_{r}+z)\,\mathrm{d}r\big{\|}_{L^{m}(\Omega)}$
$\displaystyle\lesssim\Gamma\left(m(\frac{1}{2}+\frac{d}{2p})+1\right)\|f\|_{{\mathbb{L}}^{q}_{p}([s,t])}(t-s)^{\frac{1}{2}-\frac{1}{q}-\frac{d}{2p}}|x-y|.$

The proof is completed. ∎

##  4\. Path-by-path estimates

Let $f$ be a measurable function on ${\mathbb{R}}\times{{\mathbb{R}}^{d}}$. If
$\omega$ is such that $\int_{0}^{T}|f(r,W_{r}(\omega))|\,\mathrm{d}r<\infty$
then the function $t\mapsto\int_{0}^{t}f(r,W_{r}(\omega))\,\mathrm{d}r$ is
continuous. The following result extends this argument for the function
$(t,x)\mapsto\int_{0}^{t}f(r,W_{r}(\omega)+x)\,\mathrm{d}r$. The integrability
condition $\int_{0}^{T}|f(r,W_{r}(\omega)+x)|\,\mathrm{d}r<\infty$ is no
longer sufficient and one has to replace it with the quantity
$\Xi_{T,K}(f)(\omega)$ defined below.

######  Lemma 4.1.

Let $K\subset{{\mathbb{R}}^{d}}$ be a compact set containing $0$,
$\alpha\in(0,\frac{1}{2}-\frac{d}{2p}-\frac{1}{q})$ for some
$(p,q)\in{\mathcal{J}}$ and $\varepsilon\in(0,1)$. There exists a function
$\Xi_{T,K,\alpha,\varepsilon}=\Xi_{T,K}:{\mathbb{L}}^{q}_{p}\to L^{1}(\Omega)$
such that

  1. (i)

$\Xi_{T,K}$ satisfies the triangle inequality, i.e.
$\Xi_{T,K}(f+g)\leqslant\Xi_{T,K}(f)+\Xi_{T,K}(g)$;

  2. (ii)

for every $f\in{\mathbb{L}}^{q}_{p}$,

$\displaystyle{\mathbb{E}}[\Xi_{T,K}(f)]\lesssim\|f\|_{{\mathbb{L}}^{q}_{p}([0,T])};$
(4.1)

  3. (iii)

for every bounded measurable function $f$ in ${\mathbb{L}}^{q}_{p}$, there
exists an event $\Omega_{f,T,K}$ of full measure such that for every
$\omega\in\Omega_{f,T,K}$, every $(s,t)\in[0,T]^{2}_{\leqslant}$ and every
$x,y\in K$,

$\displaystyle\Big{|}\int_{s}^{t}[f(r,W_{r}(\omega)+x)-f(r,W_{r}(\omega)+y)]\,\mathrm{d}r\Big{|}$
$\displaystyle\leqslant\Xi_{T,K}(f)(\omega)|x-y|^{1-\varepsilon}(t-s)^{\alpha},$
(4.2)
$\displaystyle\Big{|}\int_{s}^{t}f(r,W_{r}(\omega)+x)\,\mathrm{d}r\Big{|}$
$\displaystyle\leqslant\Xi_{T,K}(f)(\omega)(t-s)^{\alpha}.$ (4.3)

###### Proof.

For each function $f\in{\mathbb{L}}^{q}_{p}$, we define

$\Xi_{T,K}(f)=C_{T,K}\Big{(}\iint_{[0,T]^{2}}\Big{|}\frac{\int_{s}^{t}f(r,W_{r})\,\mathrm{d}r}{|t-s|^{\gamma_{1}}}\Big{|}^{m}\,\mathrm{d}s\,\mathrm{d}t\\\
+\iint_{[0,T]^{2}\times
K^{2}}\Big{|}\frac{\int_{s}^{t}[f(r,W_{r}+x)-f(r,W_{r}+y)]\,\mathrm{d}r}{|t-s|^{\gamma_{1}}|x-y|^{\gamma_{2}}}\Big{|}^{m}\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}x\,\mathrm{d}y\Big{)}^{\frac{1}{m}}$

for some absolute constants $C_{T,K}>0$, $m\geqslant 2$ sufficiently large and
$\gamma_{1}\in(\frac{2}{m},\frac{1}{m}+(\frac{1}{2}-\frac{d}{2p}-\frac{1}{q}))$,
$\gamma_{2}\in(\frac{2}{m},1+\frac{1}{m})$ fixed. The constants $C_{T,K}$ and
$m$ will be determined at a later step. It is obvious that i holds. Using
(3.2) and the following estimate from [LL21, Lemma 4.5]

$\displaystyle{\mathbb{E}}\Big{(}\int_{s}^{t}f(r,W_{r})\,\mathrm{d}r\Big{)}^{m}\lesssim\|f\|_{{\mathbb{L}}_{p}^{q}([s,t])}^{m}(t-s)^{m(1-\frac{d}{p}-\frac{2}{q})},$
(4.4)

we see that ii holds.

It remains to show that iii holds. Assume first that $f$ is bounded continuous
so that the map $(t,x)\mapsto\int_{0}^{t}f(r,W_{r}(\omega)+x)\,\mathrm{d}r$ is
continuous for a.s. $\omega$. From (3.2) and (4.4), applying the
multiparameter Garsia–Rodemich–Rumsey inequality ([HL13, Theorem 3.1]), there
exist constants $C_{T,K}$ and $m$ such that (4.2) and (4.3) hold.

Below, we remove the continuity restriction.

Step 1. We show iii for $f={\mathbf{1}}_{U}$ where $U$ is an open set of
finite measure in ${\mathbb{R}}\times{{\mathbb{R}}^{d}}$. By Urysohn lemma,
there exists a sequence of increasing continuous functions $f^{n}$ converging
pointwise to $f$. From i and ii, we can choose a further subsequence, still
denoted by $(f^{n})$ such that $\lim_{n}\Xi_{T,K}(f^{n})=\Xi_{T,K}(f)$ a.s. We
then apply (4.2) and (4.3) for $f^{n}$ and take limit in $n$ to obtain iii.

Step 2. We show iii for a general bounded measurable $f$ in
${\mathbb{L}}^{q}_{p}$. Let $M$ be a constant such that $|f|\leqslant M$. By
Lusin theorem, for any $n\in\mathbb{N}$, there exists a continuous function
$f^{n}$ and an open set $U^{n}$ such that $|U^{n}|\leqslant 2^{-n}$ and
$|f-f^{n}|\leqslant 2M{\mathbf{1}}_{U^{n}}$. This implies that
$\lim_{n}f^{n}=f$ in ${\mathbb{L}}^{q}_{p}$. Using ii, we can choose
subsequences, still denoted by $n$ such that a.s.

$\displaystyle\lim_{n}\Xi_{T,K}({\mathbf{1}}_{U^{n}})=0\quad\text{and}\quad\lim_{n}\Xi_{T,K}(f^{n})=\Xi_{T,K}(f).$
(4.5)

Define

$\displaystyle
A[f](\omega)=|\int_{s}^{t}[f(r,W_{r}(\omega)+x)-f(r,W_{r}(\omega)+y)]\,\mathrm{d}r|.$

Then

$\displaystyle A[f]\leqslant A[f^{n}]+A[f-f^{n}].$ (4.6)

Applying (4.2) for the continuous function $f^{n}$, we have a.s.

$\displaystyle
A[f^{n}]\leqslant\Xi_{T,K}(f^{n})|x-y|^{1-\varepsilon}(t-s)^{\alpha}\quad\forall(s,t,x,y)\in[0,T]^{2}_{\leqslant}\times
K^{2}.$

Applying (4.2) and (4.3) for ${\mathbf{1}}_{U^{n}}$, we have a.s.

$\displaystyle A[f-f^{n}]$ $\displaystyle\leqslant
2M\int_{s}^{t}[{\mathbf{1}}_{U^{n}}(r,W_{r}+x)+{\mathbf{1}}_{U^{n}}(r,W_{r}+y)]\,\mathrm{d}r$
$\displaystyle\leqslant
4M\Xi_{T,K}({\mathbf{1}}_{U^{n}})(t-s)^{\alpha}\quad\forall(s,t,x,y)\in[0,T]^{2}_{\leqslant}\times
K^{2}.$

Hence, from (4.6), taking limit in $n$ and using (4.5), we see that (4.2)
holds. The estimate (4.3) is obtained in a similar way. ∎

######  Remark 4.2.

Via a truncation procedure as is done in the last step in the proof of Lemma
4.3 below, one can remove the assumption of boundedness in Lemma 4.1iii. The
current formulation is sufficient for our purpose.

The following result provides an alternative perspective to [Dav07, Lemmas 3.3
and 3.4] and [Sha16, Lemmas 3.3 and 3.4].

######  Lemma 4.3.

Let $f:[0,T]\times{{\mathbb{R}}^{d}}\to\mathbb{R}$ be a function in
${\mathbb{L}}^{q}_{p}$ for some $(p,q)\in{\mathcal{J}}$,
$K\subset{{\mathbb{R}}^{d}}$ be a compact set containing $0$, and
$\alpha\in(0,\frac{1}{2}-\frac{d}{2p}-\frac{1}{q})$. Then there exists an
event $\Omega_{f,T,K}$ with full probability such that for every
$\omega\in\Omega_{f,T,K}$, every $\varepsilon\in(0,\alpha)$, there is a
deterministic constant $C=C(\varepsilon,\alpha)$ such that for every
$(s,t)\in[0,T]^{2}_{\leqslant}$, every function $\psi:[0,T]\to K$ of finite
variation, we have

$\displaystyle\Big{|}\int_{s}^{t}f(r,W_{r}(\omega)+\psi_{r})\,\mathrm{d}r-\int_{s}^{t}f(r,W_{r}(\omega)+\psi_{s})\,\mathrm{d}r\Big{|}\leqslant
C\Xi_{T,K}(f)(\omega)[\psi]_{\mathrm{var};[s,t]}^{1-\varepsilon}(t-s)^{\alpha}$
(4.7) and
$\displaystyle\Big{|}\int_{s}^{t}f(r,W_{r}(\omega)+\psi_{r})\,\mathrm{d}r\Big{|}\leqslant
C\Xi_{T,K}(f)(\omega)(1+[\psi]_{\mathrm{var};[s,t]}^{1-\varepsilon})(t-s)^{\alpha}.$
(4.8)

###### Proof.

Since (4.8) is a consequence of (4.7) and (4.3), we only focus on (4.7).

Step 1. We first show the result assuming that $f$ is a bounded continuous
function. For $(s,t)\in[0,T]^{2}_{\leqslant}$ define
$A_{s,t}=\int_{s}^{t}f(r,W_{r}+\psi_{s})\,\mathrm{d}r$ so that, for
$u\in[s,t]$,

$\displaystyle\delta
A_{s,u,t}:=A_{s,t}-A_{s,u}-A_{u,t}=\int_{u}^{t}[f(r,W_{r}+\psi_{s})-f(r,W_{r}+\psi_{u})]\,\mathrm{d}r.$

Applying (4.2), we have

$\displaystyle|\delta
A_{s,u,t}|\leqslant\Xi_{T,K}(f)|\psi_{u}-\psi_{s}|^{1-\varepsilon}(t-s)^{\alpha}\leqslant\Xi_{T,K}(f)[\psi]_{\mathrm{var};[s,t]}^{1-\varepsilon}(t-s)^{\alpha}.$

That $\varepsilon<\alpha$ ensures that $1-\varepsilon+\alpha>1$. Since $f$ is
continuous, we have for every $t\in[0,T]$,

$\displaystyle\int_{0}^{t}f(r,W_{r}+\psi_{r})\,\mathrm{d}r=\lim_{|\pi|\downarrow
0}\sum_{[a,b]\in\pi}A_{a,b}$

where $\pi$ is any partition of $[0,t]$. We apply the sewing lemma ([FH14]) to
obtain (4.7).

Step 2. To remove the continuity assumption and show the result for a bounded
measurable function $f$, we use similar arguments as in Lemma 4.1.

First, using Urysohn lemma, one can show that (4.7) holds for
$f={\mathbf{1}}_{U}$ for any open set $U$ of finite measure. In particular, we
have a.s.

$\displaystyle|\int_{s}^{t}{\mathbf{1}}_{U}(r,W_{r}+\psi_{r})\,\mathrm{d}r|\lesssim\Xi_{T,K}({\mathbf{1}}_{U})(1+[\psi]_{\mathrm{var};[s,t]}^{1-\varepsilon})(t-s)^{\alpha}$
(4.9)

for every $(s,t)\in[0,T]^{2}_{\leqslant}$ and every $\psi$.

Next, applying Lusin theorem, we can find for each integer $n\geqslant 1$ a
continuous bounded function $f^{n}$ and an open set $U^{n}$ such that
$|U^{n}|\leqslant 2^{-n}$ and $|f-f^{n}|\leqslant 2M{\mathbf{1}}_{U^{n}}$,
where $M$ is a constant such that $|f|\leqslant M$. This implies that
$\lim_{n}f^{n}=f$ in ${\mathbb{L}}^{q}_{p}$. For each $s\leqslant t$, we put

$\displaystyle
J[f]=\int_{s}^{t}f(r,W_{r}+\psi_{r})\,\mathrm{d}r-\int_{s}^{t}f(r,W_{r}+\psi_{s})\,\mathrm{d}r$

so that $J[f]=J[f^{n}]+J[f-f^{n}]$. By the previous step,

$\displaystyle|J[f^{n}]|\leqslant
C\Xi_{T,K}(f^{n})[\psi]_{\mathrm{var};[s,t]}^{1-\varepsilon}(t-s)^{\alpha}.$

Using (4.3) and (4.9),

$\displaystyle|J[f-f^{n}]|$ $\displaystyle\lesssim
M|\int_{s}^{t}{\mathbf{1}}_{U^{n}}(r,W_{r}+\psi_{r})\,\mathrm{d}r|+\sup_{x\in
K}|\int_{s}^{t}{\mathbf{1}}_{U^{n}}(r,W_{r}+x)\,\mathrm{d}r|$
$\displaystyle\lesssim
MT^{\alpha}(1+[\psi]^{1-\varepsilon}_{\mathrm{var};[s,t]})\Xi_{T,K}({\mathbf{1}}_{U^{n}}).$

By Lemma 4.1iii, we can further choose a subsequence, still denoted by $n$,
such that

$\displaystyle\lim_{n}\Xi_{T,K}(f^{n})=\Xi_{T,K}(f)\quad\text{and}\quad\lim_{n}\Xi_{T,K}({\mathbf{1}}_{U^{n}})=0\quad\text{a.s.}$

We emphasize that the null events only depend on $f,T,K$ and are independent
from $s,t,\psi$. Then, by passing through the limit in $n$, we obtain that

$\displaystyle|J[f]|\leqslant
C\Xi_{T,K}(f)[\psi]_{\mathrm{var};[s,t]}^{1-\varepsilon}(t-s)^{\alpha}.$

Step 3. Consider now the case $f\in{\mathbb{L}}^{q}_{p}$. For each integer
$M\geqslant 1$, define $f^{M}=f{\mathbf{1}}_{(|f|\leqslant M)}$. Note that
$(f^{M})$ (respectively $(|f^{M}|)$) is a sequence of bounded functions
converging to $f$ (respectively $|f|$) in ${\mathbb{L}}^{q}_{p}$. We can
choose a sequence $(M_{n})$ such that
$(\Xi_{T,K}(f^{M_{n}}),\Xi_{T,K}(|f^{M_{n}}|))$ converges to
$(\Xi_{T,K}(f),\Xi_{T,K}(|f|))$ a.s. Using this, (4.8) (with $|f^{M_{n}}|$)
and monotone convergence we then have a.s. $\omega$

$\displaystyle\int_{s}^{t}|f(r,W_{r}(\omega)+\psi_{r})|dr$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{s}^{t}|f^{M_{n}}|(r,W_{r}(\omega)+\psi_{r})dr$
$\displaystyle\leqslant
C\Xi_{T,K}(|f|)(\omega)(1+[\psi]^{1-\varepsilon}_{\mathrm{var};[s,t]})(t-s)^{\alpha}<\infty.$

From the above estimate and (4.3), applying the Lebesgue dominated convergence
theorem, we have $\lim_{n}J[f-f^{M_{n}}]=0$. As in the previous step, we have

$\displaystyle|J[f]|$ $\displaystyle\leqslant|J[f^{M_{n}}]|+|J[f-f^{M_{n}}]|$
$\displaystyle\lesssim\Xi_{T,K}(f^{M_{n}})[\psi]_{\mathrm{var};[s,t]}^{1-\varepsilon}(t-s)^{\alpha}+|J[f-f^{M_{n}}]|.$

We take the limit in $n$ to obtain (4.7).

This concludes the proof. ∎

######  Corollary 4.4.

Let $f$ be a function in ${\mathbb{L}}^{q}_{p}$ for some
$(p,q)\in{\mathcal{J}}$. Let $({\mathcal{Z}},\rho)$ be a metric space and
$(t,z)\mapsto\psi^{z}_{t}$ be a bounded function from
$[0,T]\times{\mathcal{Z}}$ to $\mathbb{R}$ such that

  1. (i)

for each $z\in{\mathcal{Z}}$, $t\mapsto\psi^{z}_{t}$ has finite variation and
$\sup_{z\in{\mathcal{Z}}}[\psi^{z}]_{\mathrm{var};[0,T]}<\infty$;

  2. (ii)

the family $\\{z\mapsto\psi^{z}_{t}\\}_{t\in[0,T]}$ is uniformly
equicontinuous, i.e.

$\displaystyle\lim_{h\downarrow
0}\sup_{t\in[0,T]}\sup_{\rho(z,\bar{z})\leqslant
h}|\psi^{z}_{t}-\psi^{\bar{z}}_{t}|=0.$

Then there exist an event $\Omega_{f,T}$ of full measure and a sequence of
bounded continuous functions $(f^{n})$ which are independent from $\psi$ and
$({\mathcal{Z}},\rho)$, such that $\lim_{n}f^{n}=f$ in ${\mathbb{L}}^{q}_{p}$
and for every $\omega\in\Omega_{f,T}$,

$\displaystyle\lim_{n}\int_{0}^{t}f^{n}(r,W_{r}(\omega)+\psi^{z}_{r})\,\mathrm{d}r=\int_{0}^{t}f(r,W_{r}(\omega)+\psi^{z}_{r})\,\mathrm{d}r\text{
uniformly in }(t,z)\in[0,T]\times{\mathcal{Z}}.$

Consequently, the map

$\displaystyle(t,z)\mapsto\int_{0}^{t}f(r,W_{r}(\omega)+\psi^{z}_{r})\,\mathrm{d}r$

is continuous on $[0,T]\times{\mathcal{Z}}$.

###### Proof.

Let $(K^{i})$ be an increasing sequence of compacts such that
$\bigcup_{i}K^{i}=\mathbb{R}$. Let $\Omega_{f,T,K^{i}}$ be as in Lemma 4.3. In
view of (4.1), we can choose a sequence of bounded continuous functions
$(f^{n})$ such that $\lim_{n}\Xi_{T,K^{i}}(f^{n}-f)=0$ a.s. for all
$i\in\mathbb{N}$. Because $\psi$ is bounded, there exists $j\in\mathbb{N}$
such that $\psi^{z}_{t}\in K^{j}$ for all $(t,z)\in[0,T]\times{\mathcal{Z}}$.
Applying (4.8) and i, we have for any
$\omega\in\Omega_{f,T}:=\bigcap_{i}\Omega_{f,T,K^{i}}$,

$\displaystyle\lim_{n}\int_{0}^{t}f^{n}(r,W_{r}(\omega)+\psi^{z}_{r})\,\mathrm{d}r=\int_{0}^{t}f(r,W_{r}(\omega)+\psi^{z}_{r})\,\mathrm{d}r\text{
uniformly in }(t,z)\in[0,T]\times{\mathcal{Z}}.$

Due to ii, the function
$(t,z)\mapsto\int_{0}^{t}f^{n}(r,W_{r}(\omega)+\psi^{z}_{r})\,\mathrm{d}r$ is
continuous for each $n$. This shows the claim. ∎

######  Remark 4.5.

We record some consequential observations which may be useful for other
purposes.

(i) From Lemma 4.3, it follows that for a.s. $\omega$, a solution of (1.5) is
also a solution in the framework of nonlinear Young integrals. More precisely,
for a.s. $\omega$, if $Y$ is a solution to (1.5), then there is a control $w$
and a number $\beta>1$ such that for $\psi=Y-W$,

$\displaystyle|\psi_{t}-\psi_{s}-\int_{s}^{t}b(r,W_{r}(\omega)+\psi_{s})\,\mathrm{d}r|\leqslant
w(s,t)^{\beta}\quad\forall(s,t)\in[0,T]^{2}_{\leqslant}.$

Via the sewing lemma, this means that $\psi$ is a solution to the nonlinear
Young integral equation ([HL17])

$\displaystyle\psi_{t}=\psi_{s}+\int_{s}^{t}b^{W}(\,\mathrm{d}r,\psi_{r})\quad\forall(s,t)\in[0,T]^{2}_{\leqslant},$

where $\int_{s}^{t}b^{W}(\,\mathrm{d}r,\psi_{r})$ is the nonlinear Young
integral defined as the limit of the Riemann sums

$\displaystyle\sum_{[u,v]}\int_{u}^{v}b(r,W_{r}+\psi_{u})\,\mathrm{d}r.$

While nonlinear Young integral equations have been a central theme in previous
works [CG16, GG22b, HP21, GG22a, ART21], it was not known if (1.5) under (1.4)
can be formulated in this framework.

(ii) From Corollary 4.4, it follows that for a.s. $\omega$, a solution of
(1.5) is also a _regularized solution_ in the following sense: For a.s.
$\omega$, if $Y$ is a solution to (1.5) then there exists a sequence of
bounded continuous function $(b^{n})$ and a continuous function
$V:[0,T]\to{{\mathbb{R}}^{d}}$ such that
$V_{t}=\lim_{n}\int_{0}^{t}b^{n}(r,Y_{r})\,\mathrm{d}r$ uniformly on $[0,T]$
and $Y_{t}=Y_{0}+V_{t}+W_{t}(\omega)$ for all $t\in[0,T]$. Regularized
solutions of stochastic differential equations appear in [BC01, ABLM20,
BLM23b, ART21].

##  5\. Proof of main results

For a given $\omega$ and a solution $Y$ to (1.5), we obtain in Proposition 5.1
a priori estimates on the variations of $Y-W$ on arbitrary intervals. This
allows us to show the existence of random Hölder continuous semiflow
$(X^{s,x}_{t})_{s,t,x}$ , i.e. Proposition 2.1. We then show in Proposition
5.2 another set of a priori estimates on $Y_{t}-X^{s,Y_{s}}_{t}(\omega)$ for
any given $(s,t)\in[0,T]^{2}_{\leqslant}$. Having these properties at our
disposal, we proceed to prove path-by-path uniqueness and stability, i.e.
Theorems 2.2 and 2.4.

######  Proposition 5.1.

Let $K\subset{{\mathbb{R}}^{d}}$ be a compact set containing $0$,
$(u,v)\in[0,T]^{2}_{\leqslant}$ and
$\alpha\in(0,\frac{1}{2}-\frac{d}{2p}-\frac{1}{q})$. Let $\omega$ be such that
(4.7) and (4.8) with $f=|b|$ hold. Let $Y:[u,v]\to{{\mathbb{R}}^{d}}$ be a
solution to (1.5) on $[u,v]$, i.e.
$\int_{u}^{v}|b(r,Y_{r})|\,\mathrm{d}r<\infty$ and

$\displaystyle
Y_{t}=Y_{s}+\int_{s}^{t}b(r,Y_{r})\,\mathrm{d}r+W_{t}(\omega)-W_{s}(\omega)\quad\forall(s,t)\in[u,v]^{2}_{\leqslant}.$

Assume that $Y_{t}-W_{t}(\omega)\in K$ for all $t\in[u,v]$. Then there exists
a finite constant $C=C(\varepsilon,\alpha)$ such that

$\displaystyle\int_{u}^{v}|b|(r,Y_{r})\,\mathrm{d}r\leqslant\Xi_{T,K}(|b|)(\omega)(v-u)^{\alpha}+C\big{(}\Xi_{T,K}(|b|)(\omega)(v-u)^{\alpha}\big{)}^{\frac{1}{\varepsilon}}.$
(5.1)

###### Proof.

We omit the dependence on $\omega$ in this proof. Define $\psi=Y-W$ and note
that

$\displaystyle[\psi]_{\mathrm{var};[u,v]}\leqslant\int_{u}^{v}|b|(r,Y_{r})\,\mathrm{d}r.$

Applying Lemma 4.3, we have

$\displaystyle\int_{u}^{v}|b|(r,Y_{r})\,\mathrm{d}r\leqslant\Xi_{T,K}(|b|)(v-u)^{\alpha}+C\Xi_{T,K}(|b|)\left(\int_{u}^{v}|b(r,Y_{r})|\,\mathrm{d}r\right)^{1-\varepsilon}(v-u)^{\alpha}.$

Applying Young’s inequality, we have for every $\varepsilon^{\prime}>0$

$\displaystyle\Xi_{T,K}(|b|)$
$\displaystyle\Big{|}\int_{u}^{v}|b|(r,Y_{r})\,\mathrm{d}r\Big{|}^{1-\varepsilon}(v-u)^{\alpha}\leqslant\varepsilon^{\prime}\int_{u}^{v}|b|(r,Y_{r})\,\mathrm{d}r+C_{\varepsilon^{\prime}}\big{(}\Xi_{T,K}(|b|)(v-u)^{\alpha}\big{)}^{\frac{1}{\varepsilon}}.$

We choose $\varepsilon^{\prime}$ sufficiently small to get (5.1). ∎

###### Proof of Proposition 2.1.

For each $s,x$, let $(\tilde{X}^{s,x}_{t})_{t\in[s,T]}$ be the unique strong
solution to (1.3) started from $x$ at time $s$. By definition of a solution,
for every $\omega$ in a set of full measure that depends on $s$ and x, we have
$\int_{s}^{T}|b(r,\tilde{X}^{s,x}_{r}(\omega))|\,\mathrm{d}r<\infty$ and

$\displaystyle\tilde{X}^{s,x}_{t}(\omega)=x+\int_{s}^{t}b(r,\tilde{X}^{s,x}_{r}(\omega))\,\mathrm{d}r+W_{t}(\omega)-W_{s}(\omega)\quad\forall
t\in[s,T].$ (5.2)

Let $m\geqslant 2$ be a fixed number. By [GL23, Theorem 1.2 (1.2)]

$\displaystyle\|\tilde{X}^{s,x}_{t}-\tilde{X}^{s,y}_{t}\|_{L^{m}(\Omega)}\lesssim\|x-y\|_{L^{m}(\Omega)}$
(5.3)

for any initial random points $x,y$ which are ${\mathcal{F}}_{s}$-measurable.
Applying [LL21, Lemma 4.5], we get

$\displaystyle\|\tilde{X}^{s,x}_{t}-\tilde{X}^{s,x}_{t^{\prime}}\|_{L^{m}(\Omega)}\lesssim\big{\|}\int_{t^{\prime}}^{t}b(r,\tilde{X}_{r}^{s,x})\,\mathrm{d}r\big{\|}_{L^{m}(\Omega)}\lesssim|t-t^{\prime}|^{\alpha}$

for some $\alpha\in[\frac{1}{2},1)$. Using (5.3) and pathwise uniqueness, we
have for $s^{\prime}<s$,

$\displaystyle\|\tilde{X}^{s,x}_{t}-\tilde{X}^{s^{\prime},x}_{t}\|_{L^{m}(\Omega)}=\|\tilde{X}^{s,x}_{t}-\tilde{X}^{s,\tilde{X}^{s^{\prime},x}_{s}}_{t}\|_{L^{m}(\Omega)}\lesssim\|x-\tilde{X}^{s^{\prime},x}_{s}\|_{L^{m}(\Omega)}\lesssim|s-s^{\prime}|^{\frac{1}{2}}.$

It follows, that

$\displaystyle\|\tilde{X}^{s,x}_{t}-\tilde{X}^{s^{\prime},y}_{t^{\prime}}\|_{L^{m}(\Omega)}\lesssim|x-y|+|t-t^{\prime}|^{\alpha}+|s-s^{\prime}|^{\frac{1}{2}}.$
(5.4)

Since $m$ can be arbitrarily large, applying Kolmogorov continuity criterion
([RY99, Theorem (2.1) Chapter I]), we see that for a.s. $\omega$, the map
$(s,t,x)\mapsto\tilde{X}^{s,x}_{t}(\omega)$ is locally Hölder continuous on

$\\{(s,t,x)\in[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}:s,t,x\text{ are
dyadic}\\}$

with exponents
$(\alpha^{\prime},\beta,\kappa)\in(0,\alpha)\times(0,1/2)\times(0,1)$. Because
for each $s,x$, $(\tilde{X}^{s,x}_{t}(\omega))_{t\in[s,T]}$ is continuous, the
map $(s,t,x)\mapsto\tilde{X}^{s,x}_{t}(\omega)$ is locally Hölder continuous
on

$G:=\\{(s,t,x)\in[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}:s,x\text{ are
dyadic}\\}$

with the same exponents.

Let $\Omega^{\prime}$ be the event of full measure on which (5.2) holds
whenever $s,x$ are dyadic, and $\tilde{X}$ is locally Hölder continuous on
$G$. For each $\omega\in\Omega^{\prime}$, let $X(\omega)$ be the (unique)
continuous extension of $\tilde{X}(\omega)\big{|}_{G}$ to
$[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}$. We show that
$(X^{s,x}_{t})_{s,t,x}$ is the desired semiflow. Hölder regularity is clear
and hence, we focus on showing the semiflow properties. Since $b$ is not
continuous, it is non-trivial that for almost every $\omega$ and for each
$(s,x)$, the map $t\mapsto X^{s,x}_{t}(\omega)$ satisfies the equation (1.5).
The main difficulty is to show that the continuous extension of the map

$\displaystyle(s,t,x)\mapsto\int_{s}^{t}b(r,\tilde{X}^{s,x}_{r}(\omega))\,\mathrm{d}r=\int_{s}^{t}b(r,X^{s,x}_{r}(\omega))\,\mathrm{d}r$

which is defined on $G$, is identical to the map

$\displaystyle(s,t,x)\mapsto\int_{s}^{t}b(r,X^{s,x}_{r}(\omega))\,\mathrm{d}r,$

which is defined on $[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}$.

Let $H\subset{{\mathbb{R}}^{d}}$ be a compact set and put
${\mathcal{Z}}=[0,T]\times H$. For each $\omega\in\Omega^{\prime}$, we verify
the conditions i and ii of Corollary 4.4. For each
$(t,s,x)\in[0,T]\times{\mathcal{Z}}$, define
$\psi^{s,x}_{t}(\omega):=X^{s,x}_{{\mathord{{\rm
min}}}(s,t)}(\omega)-W_{{\mathord{{\rm min}}}(s,t)}(\omega)$, and if $s,x$ are
dyadic, define
$\tilde{\psi}^{s,x}_{t}(\omega):=\tilde{X}^{s,x}_{{\mathord{{\rm
min}}}(s,t)}(\omega)-W_{{\mathord{{\rm min}}}(s,t)}(\omega)$. Note that
$\psi(\omega)$ is the continuous extension of $\tilde{\psi}(\omega)$. From
(5.2), applying Proposition 5.1, we can find a constant $C_{T}(\omega)$ such
that whenever $s,x$ are dyadic,

$\displaystyle[\psi^{s,x}(\omega)]_{\mathrm{var};[0,T]}=[\tilde{\psi}^{s,x}(\omega)]_{\mathrm{var};[0,T]}\leqslant\int_{s}^{T}|b|(r,\tilde{X}^{s,x}_{r}(\omega))\,\mathrm{d}r\leqslant
C_{T}(\omega).$

This means that for any partition $\pi$ of $[0,T]$ and $(s,x)$ dyadic in
${\mathcal{Z}}$

$\displaystyle\sum_{[u,v]\in\pi}|\psi^{s,x}_{v}(\omega)-\psi^{s,x}_{u}(\omega)|\leqslant
C_{T}(\omega).$

By continuity, the above estimate also holds for every
$(s,x)\in{\mathcal{Z}}$. This implies that

$\displaystyle\sup_{(s,x)\in{\mathcal{Z}}}[\psi^{s,x}(\omega)]_{\mathrm{var};[0,T]}\leqslant
C_{T}(\omega),$

verifying i. Uniform equicontinuity condition ii is satisfied because
$X(\omega)$ is locally Hölder continuous on
$[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}$. Let $\Omega_{b,T}$ be the
event in Corollary 4.4. Let $\omega\in\Omega^{\prime}\cap\Omega_{b,T}$.
Applying Corollary 4.4, we see that the map

$\displaystyle(t,s,x)\mapsto\int_{0}^{t}b(r,W_{r}(\omega)+\psi^{s,x}_{r}(\omega))\,\mathrm{d}r=\int_{0}^{t}b(r,X^{s,x}_{r}(\omega))\,\mathrm{d}r$

is continuous on $[0,T]\times{\mathcal{Z}}$.

It is now clear from (5.2) that for every
$\omega\in\Omega^{\prime}\cap\Omega_{b,T}$, for every
$(s,t,x)\in[0,T]^{2}_{\leqslant}\times H$, we have

$\displaystyle
X^{s,x}_{t}(\omega)=x+\int_{s}^{t}b(r,X^{s,x}_{r}(\omega))\,\mathrm{d}r+W_{t}(\omega)-W_{s}(\omega).$

By exhausting ${{\mathbb{R}}^{d}}$ with a sequence of increasing compact sets
$(H^{i})$, we see that the above equation holds for every
$(s,t,x)\in[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}$. By pathwise
uniqueness, we have for every $s\leqslant u\leqslant t$ and every
$x\in{{\mathbb{R}}^{d}}$

$\displaystyle{X}^{s,x}_{t}={X}^{u,{X}^{s,x}_{u}}_{t}\quad\text{a.s.}$ (5.5)

Note that the exceptional null event (5.5) depends on $s,u,t,x$. However,
because all processes in (5.5) are continuous, one can deduce that for a.s.
$\omega$, ${X}^{s,x}_{t}={X}^{u,{X}^{s,x}_{u}}_{t}$ for any $s,u,t,x$, which
means that $(X^{s,x}_{t}(\omega))_{s,t,x}$ is a semiflow. ∎

The next result, together with Proposition 5.1, provide _a priori_ estimates
for any solution.

######  Proposition 5.2.

Let $K\subset{{\mathbb{R}}^{d}}$ be a compact set which contains $0$. Let
$\Omega_{b,T,K}$ be the event with full probability on which $(X^{s,x}_{t})$
is a $\kappa$-Hölder continuous semiflow for every $\kappa\in(0,1)$ and (4.7)
with $f\in\\{b,|b|\\}$ holds. Let $\omega\in\Omega_{b,T,K}$ and $Y$ be a
solution to (1.5). Suppose that $X^{s,Y_{s}}_{t}(\omega)-W_{t}(\omega)$
belongs to $K$ for every $(s,t)\in[0,T]^{2}_{\leqslant}$ with $s\in Q_{Y}$.
Then, there exist

  * (i)

a control $w$ which depends only on
$Y,\Xi_{T,K}(b)(\omega),\Xi_{T,K}(|b|)(\omega),d,p,q$;

  * (ii)

a constant $\beta>1$ which depends only on $d,p,q$

such that

$\displaystyle|Y_{t}-X^{s,Y_{s}}_{t}(\omega)|\leqslant
w(s,t)^{\beta}\quad\forall(s,t)\in[0,T]^{2}_{\leqslant}.$ (5.6)

###### Proof.

We omit the dependence of $\omega$ in the argument below. Define $\psi=Y-W$
and $\xi^{s,x}_{\cdot}=X^{s,x}_{{\mathord{{\rm
min}}}(s,\cdot)}-W_{{\mathord{{\rm min}}}(s,\cdot)}$. The proof consists of
two steps.

Step 1. We show that for $\varepsilon\in(0,\alpha)$, there exist a constant
$C=C(\Xi_{T,K}(|b|),\varepsilon)$ such that

$\Big{|}\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,Y_{r})]\,\mathrm{d}r\Big{|}\leqslant
C\left(\int_{s}^{t}|b|(r,Y_{r})\,\mathrm{d}r\right)^{1-\varepsilon}(t-s)^{\alpha}\\\
+C(t-s)^{\frac{\alpha}{\varepsilon}}\quad\forall(s,t)\in[0,T]^{2}_{\leqslant}.$
(5.7)

As $\psi_{s}=\xi_{s}^{s,Y_{s}}$, we obtain that

$\displaystyle\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,Y_{r})]\,\mathrm{d}r$
$\displaystyle=\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,W_{r}+\xi^{s,Y_{s}}_{s})]\,\mathrm{d}r-\int_{s}^{t}[|b|(r,Y_{r})-|b|(r,W_{r}+\psi_{s})]\,\mathrm{d}r.$

We note that $\xi^{s,Y_{s}}$ and $\psi$ are contained in $K$ and have finite
variation such that

$\displaystyle[\xi^{s,Y_{s}}]_{\mathrm{var};[s,t]}\leqslant\int_{s}^{t}|b|(r,X^{s,Y_{s}}_{r})\,\mathrm{d}r\quad\text{and}\quad[\psi]_{\mathrm{var};[s,t]}\leqslant\int_{s}^{t}|b|(r,Y_{r})\,\mathrm{d}r.$

Applying Lemma 4.3, we have for every $\varepsilon\in(0,\alpha)$ and every
$(s,t)\in[0,T]^{2}_{\leqslant}$,

$\displaystyle\Big{|}\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,W_{r}+\xi^{s,Y_{s}}_{s})]\,\mathrm{d}r\Big{|}\lesssim\Xi_{T,K}(|b|)\left(\int_{s}^{t}|b|(r,X^{s,Y_{s}}_{r})\,\mathrm{d}r\right)^{1-\varepsilon}(t-s)^{\alpha},$
$\displaystyle\Big{|}\int_{s}^{t}[|b|(r,Y_{r})-|b|(r,W_{r}+\psi_{s})]\,\mathrm{d}r\Big{|}\lesssim\Xi_{T,K}(|b|)\left(\int_{s}^{t}|b|(r,Y_{r})\,\mathrm{d}r\right)^{1-\varepsilon}(t-s)^{\alpha}.$

We also have

$\displaystyle\Big{|}\int_{s}^{t}|b|(r,X^{s,Y_{s}}_{r})\,\mathrm{d}r\Big{|}^{1-\varepsilon}\leqslant\Big{|}\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,Y_{r})]\,\mathrm{d}r\Big{|}^{1-\varepsilon}+\Big{|}\int_{s}^{t}|b|(r,Y_{r})\,\mathrm{d}r\Big{|}^{1-\varepsilon}.$
(5.8)

It follows that

$\Big{|}\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,Y_{r})]\,\mathrm{d}r\Big{|}\lesssim\Xi_{T,K}(|b|)\Big{|}\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,Y_{r})]\,\mathrm{d}r\Big{|}^{1-\varepsilon}(t-s)^{\alpha}\\\
+{\Xi_{T,K}(|b|)}\left(\int_{s}^{t}|b|(r,Y_{r})\,\mathrm{d}r\right)^{1-\varepsilon}(t-s)^{\alpha}.$

Applying Young’s inequality, we have for every $\varepsilon^{\prime}>0$

$\displaystyle\Xi_{T,K}(|b|)$
$\displaystyle\Big{|}\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,Y_{r})]\,\mathrm{d}r\Big{|}^{1-\varepsilon}(t-s)^{\alpha}$
$\displaystyle\leqslant\varepsilon^{\prime}\Big{|}\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,Y_{r})]\,\mathrm{d}r\Big{|}+C_{\varepsilon^{\prime}}\big{(}\Xi_{T,K}(|b|)(t-s)^{\alpha}\big{)}^{\frac{1}{\varepsilon}}.$
(5.9)

Combining the previous two inequality and choosing $\varepsilon^{\prime}$
sufficiently small yields (5.7).

Step 2. We show (5.6). Let $(s,t)\in[0,T]^{2}_{\leqslant}$ be fixed. Using the
equations and the identity $\psi_{s}=\xi_{s}^{s,Y_{s}}$, we obtain that

$\displaystyle|X^{s,Y_{s}}_{t}-Y_{t}|$
$\displaystyle=\left|\int_{s}^{t}[b(r,X^{s,Y_{s}}_{r})-b(r,Y_{r})]\,\mathrm{d}r\right|$
$\displaystyle\leqslant\left|\int_{s}^{t}[b(r,X^{s,Y_{s}}_{r})-b(r,W_{r}+\xi^{s,Y_{s}}_{s})]\,\mathrm{d}r\right|+\left|\int_{s}^{t}[b(r,Y_{r})-b(r,W_{r}+\psi_{s})]\,\mathrm{d}r\right|.$

Applying Lemma 4.3, we have for every $\varepsilon\in(0,\alpha)$,

$\displaystyle\Big{|}\int_{s}^{t}[b(r,X^{s,Y_{s}}_{r})-b(r,W_{r}+\xi^{s,Y_{s}}_{s})]\,\mathrm{d}r\Big{|}\lesssim\Xi_{T,K}(b)\left(\int_{s}^{t}|b|(r,X^{s,Y_{s}}_{r})\,\mathrm{d}r\right)^{1-\varepsilon}(t-s)^{\alpha},$
$\displaystyle\Big{|}\int_{s}^{t}[b(r,Y_{r})-b(r,W_{r}+\psi_{s})]\,\mathrm{d}r\Big{|}\lesssim\Xi_{T,K}(b)\left(\int_{s}^{t}|b|(r,Y_{r})\,\mathrm{d}r\right)^{1-\varepsilon}(t-s)^{\alpha}.$

Combining with (5.8), we have

$\displaystyle|X^{s,Y_{s}}_{t}-Y_{t}|\lesssim\Xi_{T,K}(b)$
$\displaystyle\Big{|}\int_{s}^{t}[|b|(r,X^{s,Y_{s}}_{r})-|b|(r,Y_{r})]\,\mathrm{d}r\Big{|}^{1-\varepsilon}(t-s)^{\alpha}$
$\displaystyle+\Xi_{T,K}(b)\left(\int_{s}^{t}|b|(r,Y_{r})\,\mathrm{d}r\right)^{1-\varepsilon}(t-s)^{\alpha}.$

We apply (5) and (5.7) to get

$\displaystyle|X_{t}^{s,Y_{s}}-Y_{t}|\leqslant
C(\varepsilon,\Xi_{T,K}(|b|),\Xi_{T,K}(b))\left(\left(\int_{s}^{t}|b|(r,Y_{r})\,\mathrm{d}r\right)^{1-\varepsilon}(t-s)^{\alpha}+(t-s)^{\alpha/\varepsilon}\right).$

Choosing $\varepsilon$ small enough such that $\alpha/\varepsilon>1$ yields
(5.6). ∎

###### Proof of Theorem 2.2.

Let $K_{n}=\\{x\in{{\mathbb{R}}^{d}}:|x|\leqslant n\\}$ ($n\in{\mathbb{N}}$)
and $\Omega_{b,T}=\cup_{k=1}^{\infty}\cap_{n=k}^{\infty}\Omega_{b,T,K_{n}}$
for $\Omega_{b,T,K_{N}}$ as in Proposition 5.2. Note that $\Omega_{b,T}$ has
full probability. Let $\omega\in\cap_{n=k}^{\infty}\Omega_{b,T,K_{n}}$ for
some $k$ and let $Y$ be a solution to (1.5). By continuity, we can choose
$N\geqslant k$ such that $K_{N}$ contains $Y_{t}$ and
$X^{s,Y_{s}}_{t}(\omega)-W_{t}(\omega)$ for every
$(s,t)\in[0,T]^{2}_{\leqslant}$. Let $w$ be the control and $\beta$ be the
constant found in Proposition 5.2. We note that $w$ depends on $N$. Let
$\tau\in(0,T]$ be a fixed but arbitrary number and define

$\displaystyle F(t)=X^{t,Y_{t}}_{\tau},\quad t\in[0,\tau].$

For $(s,t)\in[0,\tau]^{2}_{\leqslant}$, we obtain by the semiflow property
that

$\displaystyle
F(t)-F(s)=X^{t,Y_{t}}_{\tau}-X^{s,Y_{s}}_{\tau}=X^{t,Y_{t}}_{\tau}-X^{t,X^{s,Y_{s}}_{t}}_{\tau}.$

Then, using Hölder continuity of the semiflow (Proposition 2.1), we have

$\displaystyle|F(t)-F(s)|\lesssim|Y_{t}-X^{s,Y_{s}}_{t}|^{\kappa},$

for some $\kappa\in(0,1)$ which can be chosen to be arbitrarily close to $1$.
Applying (5.6), we have

$\displaystyle|F(t)-F(s)|\lesssim w(s,t)^{\kappa\beta}.$

Choosing $\kappa$ so that $\kappa\beta>1$, for any sequence of partitions
$\Pi_{n}=\\{t_{i}\\}_{i=0}^{N_{n}}$ of $[0,\tau]$ with mesh converging to $0$,
we get that

$\displaystyle|F(\tau)-F(0)|\leqslant\sum_{i=1}^{N_{n}}|F(t_{i})-F(t_{i-1})|\lesssim|\sup_{i\leqslant
N_{n}}w(t_{i-1},t_{i})|^{\kappa\beta-1}w(0,\tau)\overset{n\rightarrow\infty}{\longrightarrow}0.$

Hence $F(\tau)=F(0)$. Since $\tau$ was arbitrarily chosen in $(0,T]$, this
means that $Y$ and $X^{0,Y_{0}}$ are identical. ∎

###### Proof of Theorem 2.4.

In the proof, we write $\|b^{n}-b\|$ for
$\|b^{n}-b\|_{L^{q}([0,T],W^{-\nu,p}({\mathbb{R}}^{d}))}$, where $p,q$ and
$\nu$ satisfy either (2.1) or (2.2) as in the statement. Denote by
$(X^{s,x,n}_{t})_{s,t,x}$ the Hölder continuous semiflow of solutions to

$\displaystyle X_{t}=X_{0}+\int_{0}^{t}b^{n}(r,X_{r})\,\mathrm{d}r+W_{t},\quad
t\in[0,T].$ (5.10)

In view of Theorem 2.2, it suffices to show that there exists an event
$\Omega^{\prime}_{(b^{n}),b,T}$ such that for every
$\omega\in\Omega^{\prime}_{(b^{n}),b,T}$,

$\displaystyle\lim_{n\rightarrow\infty}X^{n}(\omega)=X(\omega)\text{ uniformly
over compact sets of }[0,T]^{2}_{\leqslant}\times{{\mathbb{R}}^{d}}.$ (5.11)

From (5.4) and [GL23, Theorem 1.2, Remark 2.13], putting
$V_{s,t,x}=X^{s,x,n}_{t}-X^{s,x}_{t}$, we can find an
$\alpha\in[\frac{1}{2},1)$ such that for every $m\geqslant 1$,
$t,t^{\prime}\in[0,T]$ and $x,x^{\prime}\in{{\mathbb{R}}^{d}}$

$\displaystyle\|V_{s,t,x}-V_{s^{\prime},t^{\prime},x^{\prime}}\|_{L^{m}(\Omega)}\lesssim|s-s^{\prime}|^{1/2}+|t-t^{\prime}|^{\alpha}+|x-x^{\prime}|,$
and $\displaystyle\|V_{s,t,x}\|_{L^{m}(\Omega)}\lesssim\|b^{n}-b\|.$

Hence,

$\displaystyle\|V_{s,t,x}-V_{s^{\prime},t^{\prime},x^{\prime}}\|_{L^{m}(\Omega)}\lesssim(|s-s^{\prime}|^{1/2}+|t-t^{\prime}|^{\alpha}+|x-x^{\prime}|)^{1/2}\|b^{n}-b\|^{1/2}.$

Applying Garsia–Rodemich–Rumsey inequality, for every compact set
$K\subset{{\mathbb{R}}^{d}}$, every $m\geqslant 1$, we have

$\displaystyle\|\sup_{(s,t,x)\in[0,T]^{2}_{\leqslant}\times
K}|V_{s,t,x}|\|_{L^{m}(\Omega)}\leqslant C_{m,T,K}\|b^{n}-b\|^{1/2}$ (5.12)

for some finite constant $C_{m,T,K}$.

Let $(K^{i})$ be an increasing sequence of compacts such that
$\bigcup_{i\in\mathbb{N}}K^{i}=\mathbb{R}^{d}$. By definition of summable
convergence, we can choose $m\geqslant 2$ so that

$\displaystyle\sum_{n}\|b^{n}-b\|^{\frac{m}{4}}<\infty.$

Using (5.12) and Markov’s inequality, we have for every $i$

$\displaystyle\mathbb{P}\left(\sup_{(s,t,y)\in[0,T]^{2}_{\leqslant}\times
K^{i}}|X^{s,y,n}_{t}-X^{s,y}_{t}|>\|b^{n}-b\|^{\frac{1}{4}}\right)\leqslant
C_{m,T,K^{i}}^{m}\|b^{n}-b\|^{\frac{m}{4}}.$

Applying Borel–Cantelli lemma, we get an event ${\Omega}^{i}_{(b^{n}),b,T}$ of
full measure on which $X^{n}(\omega)$ converges to $X(\omega)$ uniformly on
$[0,T]^{2}_{\leqslant}\times K^{i}$. Putting
${\Omega^{\prime}}_{(b^{n}),b,T}=\cap_{i}{\Omega}^{i}_{(b^{n}),b,T}$, it
follows that (5.11) holds for every
$\omega\in{\Omega^{\prime}}_{(b^{n}),b,T}$. ∎

######  Remark 5.3.

The fact that for almost every $\omega$, for every $s,x$,
$(X^{s,x}_{t}(\omega))_{t}$ is a solution to (1.5) is crucial for the proof of
path-by-path uniqueness. When $b$ is not continuous, this property becomes
highly non-trivial. It was taken for granted without any justifications in
[Sha16, Sha17]. We achieved this property in Proposition 2.1 by utilizing the
regularizing estimates from Lemma 4.3. This issue is irrelevant to [Dav07]
because of its different arguments. We take this chance to note that the
exceptional null events in [Dav07] depend on the initial condition. This
dependence can be removed following our arguments herein. In fact, one just
replaces Proposition 3.1 by Davie’s basic estimate, then the rest of the
arguments follows with minimal adjustments. In particular, Propositions 2.1,
2.2 and 2.4 hold with $p=q=\infty$.

##  Appendix A ODE uniqueness

Most of the arguments in Section 5 are independent from the probability space,
and hence, independent from the probability law of the driving noise. This
forms a uniqueness criterion for ODE (1.1) which does not require any
regularity on the vector field, but instead relies on the regularizing effect
of the driving signal $\gamma$ with Lemma 4.3 being a prototype. This
regularizing effect is formalized by the following definition.

######  Definition A.1.

Let $f:[0,T]\times{{\mathbb{R}}^{d}}\to{{\mathbb{R}}^{d}}$ be a measurable
function. We say that $\gamma$ is $(1-\varepsilon,\alpha)$-regularizing for
$f$ if there exist a control $\eta$ and constants $\Xi_{T,K}$ for each compact
set $K\subset{{\mathbb{R}}^{d}}$ such that

$\displaystyle\Big{|}\int_{s}^{t}[f(r,\gamma_{r}+\psi_{r})-f(r,\gamma_{r}+\psi_{s})]\,\mathrm{d}r\Big{|}\leqslant\Xi_{T,K}[\psi]_{\mathrm{var};[s,t]}^{1-\varepsilon}\eta(s,t)^{\alpha}$

for every $(s,t)\in[0,T]^{2}_{\leqslant}$ and every $\psi:[0,T]\to K$ of
finite variation.

######  Theorem A.2.

Let $K\subset{{\mathbb{R}}^{d}}$ be a compact set which contains $0$. Suppose
that $\gamma$ is $(1-\varepsilon,\alpha)$-regularizing for $b,|b|$. Let $y$ be
a solution to (1.1). Suppose that $\phi^{s,y_{s}}_{t}-\gamma_{t}$ belongs to
$K$ for every $(s,t)\in[0,T]^{2}_{\leqslant}$. Then, there exists a control
$w$ which depends only on $y,\Xi_{T,K},\eta$ such that

$\displaystyle|y_{t}-\phi^{s,y_{s}}_{t}|\leqslant w(s,t)^{{\mathord{{\rm
min}}}(1-\varepsilon+\alpha,\frac{\alpha}{\varepsilon})}\quad\forall(s,t)\in[0,T]^{2}_{\leqslant}.$
(A.1)

Suppose furthermore that $\phi$ is locally $\kappa$-Hölder continuous for some
$\kappa\in(0,1]$ such that

$\displaystyle\kappa\cdot{\mathord{{\rm
min}}}(1-\varepsilon+\alpha,\frac{\alpha}{\varepsilon})>1.$

Then $(y_{t})_{t\in[0,T]}$ is identical to $(\phi^{0,y_{0}}_{t})_{t\in[0,T]}$.

The proof of this result follows analogous arguments used in proving
Propositions 5.2 and 2.2 and hence is omitted.

##  Appendix B Adapted solutions and path-by-path solutions

This section is devoted to summarize different concepts of solutions and
uniqueness to (1.3) where
$b:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is measurable. Even
though the concept of adapted solutions is by now well covered in textbooks,
we also recall these notions in order to make a clear comparison to path-by-
path solutions and path-by-path uniqueness. In particular, we state and prove
Theorem B.3 to point out the connection between path-by-path uniqueness and
adaptedness of solutions.

######  Definition B.1 (Existence).

newline

  1. (i)

If there exists a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$ equipped with
a Brownian motion $(W_{t})_{t\in[0,T]}$ and an $(\mathcal{F}_{t})$-adapted
process $(X_{t})_{t\in[0,T]}$ such that
$\int_{0}^{T}|b(s,X_{s})|\,\mathrm{d}s<\infty$ and $(X,W)$ fulfills (1.3)
almost surely, we say that $(X,W)$ is a _weak solution_ to (1.3). If the
choice of $W$ is clear from the context, we write that $X$ is a weak solution.

  2. (ii)

We call $X$ a _strong solution_ if $X$ is a weak solution and
$(X_{t})_{t\in[0,T]}$ is adapted to the filtration
$(\mathcal{F}_{t}^{W})_{t\in[0,T]}$.

  3. (iii)

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space on which a
Brownian motion $W$ is defined. We call a mapping
$X\colon\Omega\rightarrow\mathcal{C}([0,T])$ a _path-by-path solution_ to
(1.3) if there exists a set $\tilde{\Omega}\subset\Omega$ of full measure such
that, for every $\omega\in\tilde{\Omega}$, $X(\omega)$ is a solution to (1.5).

######  Definition B.2 (Uniqueness).

newline

  1. (i)

We say that _pathwise uniqueness_ for (1.3) holds if for any two weak
solutions $(X,W)$, $(\tilde{X},W)$ defined on the same filtered probability
space with the same Brownian motion $W$ and the same initial condition, $X$
and $\tilde{X}$ are indistinguishable.

  2. (ii)

We say that _path-by-path uniqueness_ for (1.3) holds if for any probability
space on which a Brownian motion $W$ is defined, there exists a set of full
measure $\tilde{\Omega}$, such that for any $x_{0}\in\mathbb{R}^{d}$ and any
$\omega\in\tilde{\Omega}$, there exists a unique solution on $[0,T]$ to

$X_{t}(\omega)=x_{0}+\int_{0}^{t}b(s,X_{s}(\omega))\,\mathrm{d}s+W_{t}(\omega).$

Note that in the literature the definition of path-by-path uniqueness
sometimes allows for the set of full measure to depend on the initial
condition $x_{0}$, see for example [Fla11].

The following implications follow directly from the definitions:

strong existenceweak existencepath-by-path existencepath-by-path
uniquenesspathwise uniqueness

The following theorem can be summarized in the following way: For drifts for
which pathwise uniqueness is known to hold, losing path-by-path uniqueness
implies existence of a non-adapted solution.

######  Theorem B.3.

Let $b:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ be measurable.
Assume that

  * (i)

pathwise uniqueness to (1.3) holds,

  * (ii)

all path-by-path solutions to (1.3) are weak solutions.

Then path-by-path uniqueness holds.

###### Proof.

Assume that path-by-path uniqueness does not hold. Let
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space on which a Brownian
motion $W$ is defined. Then there exists a set $A\subset\Omega$ of positive
measure such that for $\omega\in A$, there exist multiple solutions to

$\displaystyle
X_{t}(\omega)=X_{0}+\int_{0}^{t}b(s,X_{s}(\omega))\,\mathrm{d}s+W_{t}(\omega),$

with the same initial condition. Then we can define two path-by-path solutions
$X^{1}$ and $X^{2}$ to (1.3) by letting them agree with the unique strong
solution on $A^{\mathsf{c}}$ and letting $X^{1}(\omega)\neq X^{2}(\omega)$ on
$A$, which is possible by the above and the axiom of choice. By assumption,
both $X^{1}$ and $X^{2}$ are adapted w.r.t. filtrations
$(\mathcal{F}^{1}_{t})$ and $(\mathcal{F}^{2}_{t})$ such that $W$ is a
Brownian motion w.r.t. these filtrations. Hence, $W$ is also a Brownian motion
with respect to $(\mathcal{F}^{1}_{t}\cup\mathcal{F}^{2}_{t})$. Therefore we
constructed two solutions on the same filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}^{1}_{t}\cup\mathcal{F}^{2}_{t})_{t\in[0,T]},\mathbb{P})$.
By construction $X^{1}\neq X^{2}$ on a set of positive measure. This
contradicts pathwise uniqueness. ∎

######  Remark B.4.

There are examples of drifts $b$ not fulfilling (1.4) such that there exist
multiple (non-adapted) path-by-path solutions to (1.3), even though pathwise
uniqueness holds (see [SW22] for $d>1$ and [Anz22] for $d=1$). These
counterexamples heavily rely on the time-dependence of the drift. This is
leading to non-uniqueness globally (i.e. considering the equation on the whole
interval $[0,T]$. We are not aware of examples of drifts for which such a
phenomenon occurs locally. Finally, note that Theorem 2.2 prevents such
counterexamples for drifts fulfilling (1.4) since path-by-path solutions are
identified on a set of full measure.

## Acknowledgments

The authors thank Toyomu Matsuda for numerous suggestions which improve the
quality of the paper and Oleg Butkovsky for suggesting Appendix A. Discussion
on strong completeness with Prof. Scheutzow is acknowledged.

## Funding

LA is supported by the Labex de Mathématique Hadamard, CL is supported by the
Austrian Science Fund (FWF) Stand-Alone programme P 34992. CL also
acknowledges the financial support from DFG via Research Unit FOR 2402 when
this project started.

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implicit-parser-supp

# Meaning Representation of Numeric Fused-Heads in UCCA

Ruixiang Cui  Daniel Hershcovich
Department of Computer Science
University of Copenhagen
{rc<EMAIL_ADDRESS>

##  1 Meaning Representations

Despite the usefulness and popularity of syntactic representation frameworks
such as Universal Dependencies (Nivre et al., 2020), researchers have pointed
out their sole focus on surface-syntactic information (Hershcovich et al.,
2019; Žabokrtský et al., 2020) and proposed to design linguistic schemes on a
deep-syntactic or semantic level. Meaning representation (MR) frameworks aim
at capturing shared semantic principles in one or more languages and
representing sentences in a structured way (Žabokrtský et al., 2020).
Frameworks including as Elementary Dependency Structures (EDS; Oepen et al.,
2016), Prague Tectogrammatical Graphs (PTG; Hajič et al., 2020), Universal
Conceptual Cognitive Annotation (UCCA; Abend and Rappoport, 2013) and Abstract
Meaning Representation (AMR; Banarescu et al., 2013) are designed to represent
semantics directly and (to varying degrees) comprehensively. Whereas these
frameworks model explicit linguistic constructions, only few address implicit
phenomena, where predicates, arguments and modifiers are omitted and need to
be inferred from the context. One of them is UCCA, which annotates implicit
units as part of the meaning representation graph. Furthermore, Cui and
Hershcovich (2020) proposed refining this annotation with a layer representing
fine-grained distinctions between implicit arguments that correspond to
participants.

IA’mF42QIMPLICITCSH

Figure 1: UCCA example of a sentence with NFH.

A C D E H L P Q R S T S/A $\emptyset$ Total w/ NFH 24 80 2 6 1 0 4 68 5 8 3 1
2 206 w/o NFH 5 22 6 6 0 1 4 240 3 3 0 0 3 294

Table 1: Parser predictions for the NFH test set. The upper row represents
instances supposed to have numeric fused-heads. The lower row are instances
without numeric fused-heads. The parser predicted Quantifier (Q) for 33% of
the NFH exmaples in the test set, and 81% of the non-NFH examples in the test
set. Empty sets are failed predictions that cannot be converted from MRP
format to XML for measurements.

##  2 Numeric Fused-Heads

The linguistic construction of fused heads (FHs; Elazar and Goldberg, 2019)
corresponds to a noun phrase where the head noun is “fused” with the dependent
modifier. Numeric fused-heads (NFHs), where the phrase is a number, are an
interesting subset. For example, in the sentence ‘I’m 42’ (Figure 1), the
phrase ‘42’ refers to the age of the person in years, though years is implicit
in this context. We investigate whether the semantics of this construction is
represented in meaning representation frameworks in English, empirically with
UCCA.

##  3 NFHs in UCCA

For UCCA, the annotation guidelines Abend et al. (2020) give only a very
partial account of NFHs, and it is not clear how implicit NFHs should be
annotated, for example. However, our interpretation of the guidelines would
assign this construction the UCCA annotation of a Quantifier (Q) with an
implicit or remote Center (C), as in Figure 1. We inspect the UCCA English Web
Treebank Corpus (UCCA
EWT)111github.com/UniversalConceptualCognitiveAnnotation/UCCA_English-EWT and
the UCCA Wikipedia Corpus (UCCA
Wiki)222github.com/UniversalConceptualCognitiveAnnotation/UCCA_English-Wiki,
which are both annotated with implicit units, and find 54 and 124 instances
according to this rule, respectively. There may be other instances of this
linguistic phenomenon in the corpora, which we are in the progress of manually
identifying.

##  4 Parsing Experiments

Subsequently, we train a recently proposed transition-based neural parser,
which is able to handle implicit arguments Cui and Hershcovich (2021), on the
UCCA EWT dataset, and use its predictions on the NFH Identification test set
introduced by Elazar and Goldberg (2019), to extract NFH instances in a zero-
shot prediction experiment, without training on the associated training set.
We find that the parser failed to predict any implicit or remote unit for any
of the 206 NFH test instances in this setting. Table 1 shows the results. The
low performance is likely to result from the small number of examples for this
phenomenon in the training set, and possibly the fact that NFHs, in fact,
appear in several distinct constructions, idiosyncratically annotated in
different ways in UCCA.

##  5 Conclusion

We conclude that the implicit UCCA parser does not address NFHs consistently,
which could result either from inconsistent annotation, insufficient training
data or a modelling limitation. We are investigating which factors are
involved. We consider this phenomenon important, as it is pervasive in text
and critical for correct inference. Careful design and fine-grained annotation
of NFHs in MRs would benefit downstream tasks such as machine translation,
natural language inference and question answering, particularly when they
require numeric reasoning (e.g., Dua et al., 2019), as recovering and
categorizing them. We are investigating the treatment of this phenomenon by
other meaning representations, such as AMR. We encourage researchers in
meaning representations, and computational linguistics in general, to address
this phenomenon in future research.

## References

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