# Geodesic Lévy Flights and Expected Stopping Time for Random Searches

Yann Chaubet  Cambridge University, Faculty of Mathematics, Wilberforce Road,
Cambridge CB3 0WB, United Kingdom<EMAIL_ADDRESS>,  Yannick
Guedes Bonthonneau  Institut Galilée, Université Paris 13, avenue Jean-
Baptiste Clément 93430 - Villetaneuse<EMAIL_ADDRESS>,
Thibault Lefeuvre  Université de Paris and Sorbonne Université, CNRS, IMJ-PRG,
F-75006 Paris, France<EMAIL_ADDRESS>and  Leo Tzou  University of
Amsterdam, Korteweg-de Vries Institute, Science Park, 1098XH Amsterdam,
Netherlands<EMAIL_ADDRESS>
###### Abstract.

We give an analytic description for the infinitesimal generator constructed in
[AE00] for Lévy flights on a broad class of closed Riemannian manifolds
including all negatively-curved manifolds, the flat torus and the sphere.
Various properties of the associated semigroup and the asymptotics of the
expected stopping time for Lévy flight based random searches for small
targets, also known as the “narrow capture problem”, are then obtained using
our newfound understanding of the infinitesimal generator. Our study also
relates to the _Lévy flight foraging hypothesis_ in the field of biology as we
compute the expected time for finding a small target by using the Lévy flight
random search. A similar calculation for Brownian motion on surfaces was done
in [NTTT22].

##  1\. Introduction

The _Lévy flight foraging hypothesis_ is a well-known hypothesis in the field
of biology asserting that animals foraging behaviours should be modelled by
Lévy flights insofar as they may optimize search efficiencies. While this
hypothesis has been around for more than twenty years, it is still
controversial and subject to many research articles investigating whether
Brownian motion or Lévy flights are optimal search strategies [PCM14, SK86,
VDLRS11, BN13, BLMV11, DGV22]. The purpose of this article is to shed a new
theoretical light on this question by means of a precise mathematical study.

More precisely, we will investigate the _narrow capture problem_ which
consists in finding a small target in space for a motion whose law is that of
a Lévy flight. The interesting quantity to understand then is the _expected
capture time_ , namely, the expected time that a process starting at a given
point $p$ will eventually find the target. This small target typically models
a prey hunted by a predator whose foraging behaviour is modelled by the Lévy
process. The Lévy flight foraging hypothesis can then be phrased as follows:
_is the expected capture time significantly lower if one uses a search based
on Lévy flights rather than on Brownian motion?_

For bounded domains in the Euclidian space, there are various search
strategies based on Brownian motion and in this case an important set of
literatures already investigated the expected time of finding small targets
[SSH08, SSH06, GC15, CWS10, CF11, AKKL12]. However, while the background
geometry of many animal foraging behaviours and constraint optimization
searches are naturally curved, we have only recently started addressing the
question of expected stopping time for Brownian motions on Riemannian
manifolds [NTTT22, NTT21b, NTT21a]. Thus far, nothing has been done for
stopping time for Lévy flight based searches even in flat geometry. We address
this question here for a class of isotropic pure jump Lévy processes
introduced by Applebaum–Estrade [AE00]. In particular, we investigate the
asymptotics of the expected stopping time for a Lévy flight based random
search to find a target the size of a small geodesic ball whose radius
converges to zero.

###  1.1. Main result

We assume throughout that $(M,g)$ is a smooth closed (that is, compact without
boundary) connected $n$-dimensional Riemannian manifold with $n\geqslant 2$.
We let $(X_{t})_{t\geqslant 0}$ be a cadlag semi-martingale on $M$ which is an
isotropic Lévy process in the sense of [AE00], induced by the isostropic Lévy
measure

(1.1)
$\nu_{p}(A)=C(n,\alpha)\int_{A}\frac{\mathrm{d}T_{p}(v)}{|v|_{g}^{n+2\alpha}},\quad
A\subset T_{p}M,\quad\alpha\in(0,1)$

on each tanget space. Here $T_{p}$ is the volume form on $T_{p}M$ induced by
the metric $g|_{T_{p}M}$ and $C(n,\alpha)$ is the constant111This constant is
chosen to be consistent with the definition of the fractional Laplacian on
$\mathbb{R}^{n}$, which is the infinitesmial generator of $2\alpha$-stable
isotropic Lévy processes in Euclidean space.

(1.2) $\displaystyle
C(n,\alpha):=\frac{4^{\alpha}\Gamma(n/2+\alpha)}{\pi^{n/2}|\Gamma(-\alpha)|}.$

Fix $p_{0}\in M$ and let $B_{\varepsilon}(p_{0})$ be the open geodesic ball of
radius $\varepsilon>0$ centered at $p_{0}$. We define the _expected stopping
time_ as:

(1.3) $\tau_{\varepsilon}=\inf\left\\{t\geqslant 0\mid
X_{t}\in\overline{B_{\varepsilon}(p_{0})}\right\\}\quad\text{and}\quad
u_{\varepsilon}(q)={\mathbb{E}}\left(\tau_{\varepsilon}\mid X_{0}=q\right),$

for each $q\in M$.

Throughout the paper, we will denote by $\mathbb{S}^{n}$ the Riemannian
$n$-dimensional sphere equipped with the round metric and by
$\mathbb{T}^{n}:=\mathbb{R}^{n}/\mathbb{Z}^{n}$ the $n$-dimensional torus with
the flat metric. We say that a manifold is _Anosov_ if its geodesic flow is
Anosov on its unit tangent bundle, see §1.2 and §2.3 for further details. In
particular, this includes all negatively-curved manifolds.

We will prove the following result.

######  Theorem 1.1.

Assume that $M=\mathbb{S}^{n},{\mathbb{T}^{n}}$ or is Anosov. Then the
following holds.

  1. _(i)_

There is $c(n,\alpha)>0$ such that the average of $u_{\varepsilon}$ has the
expansion

$\frac{1}{|M|}\int_{M}u_{\varepsilon}{\rm
dvol}_{g}\sim\frac{|M|c(n,\alpha)}{\varepsilon^{n-2\alpha}},\quad\varepsilon\to
0.$

  2. _(ii)_

For each $p\neq p_{0}\in M$ (and $p\neq-p_{0}$ if $M=\mathbb{S}^{n}$),

$u_{\varepsilon}(p)-\frac{1}{|M|}\int_{M}u_{\varepsilon}{\rm
dvol}_{g}\to|M|\mathrm{G}_{\mathscr{A}}(p,p_{0}),\quad\varepsilon\to 0,$

where $\mathrm{G}_{\mathscr{A}}$ is the Green’s function of the generator of
$(X_{t})_{t\geqslant 0}$ (see Theorem 1.6 and Corollary 1.2.5).

  3. _(iii)_

If $M=\mathbb{S}^{n}$, $n>1+4\alpha$ and $1>(n-4)\alpha$ then for some
$\tilde{c}(n,\alpha)\neq 0$,

$\left|u_{\varepsilon}(-p_{0})-\frac{1}{M}\int_{M}u_{\varepsilon}{\rm
dvol}_{g}\right|\sim\frac{|M|\tilde{c}(n,\alpha)}{\varepsilon^{n-1-4\alpha}},\quad\varepsilon\to
0.$

We will state below more precise results (Theorems 1.8 and 1.10) giving an
explicit expression of the constants and the size of the remainders. While
such results exist for Brownian motions in Euclidean domains [SSH08, SSH06,
GC15, CWS10, CF11, AKKL12] and on general manifolds [NTTT22, NTT21b, NTT21a],
this is the first such detailed analytical calculation for Lévy flights for
such a broad class of geometries.

We emphasize that Theorem 1.1 shows that the asymptotics of the deviation of
the expected stopping time from its average heavily depends on the underlying
geometry. In particular on the sphere, antipodal points are conjugate222On the
sphere, conjugate points correspond to pair of points that may be connected by
a non-trivial continuous one-parameter family of geodesic paths., and this
leads to a singular behavior of the expected stopping time at those points.
Nevertheless, we expect that point __(i)__ of Theorem 1.1 should remain valid
for general Riemannian manifolds, regardless of the geometry.

The work of [Get61] showed that on $\mathbb{R}^{n}$ the expected time for a
Lévy process to exit the unit ball satisfies an integral equation involving
the fractional Laplacian and we will derive an analogous statement in our
setting, see Proposition 4.1 below. Theorem 1.1 follows from a detailed study
of the analytic properties of the generator $\mathscr{A}$ of the Lévy process,
see Theorems 1.6 and 1.5 below.

We finally observe that in the physical dimensions $n=2$, the expected
stopping time for the Brownian motion was shown to be of size
$\mathcal{O}(|\log\varepsilon|)$ in [NTTT22] whereas it is here of size
$\mathcal{O}(\varepsilon^{-(2-2\alpha)})$ by Theorem 1.1.

###  1.2. Results on the generator

While it is well understood that the infinitesimal generator for
$2\alpha$-stable jump processes on Euclidean spaces are precisely the
fractional powers of the Laplacian, the same may not hold for Lévy processes
on closed compact Riemannian manifolds. In fact it was shown in [AE00] that if
$(X_{t})_{t\geqslant 0}$ is a cadlag semi-martingale valued in a Riemannian
manifold $(M,g)$, then it is an isotropic Lévy process iff it is a Feller
process with infinitesimal generator $a\Delta_{g}+\mathscr{A}$ for some
constant $a\geqslant 0$ and for $u\in C^{\infty}(M)$,

(1.4) ${\mathscr{A}}u(p):=\mathrm{p.v.}\int_{v\in T_{p}M\setminus
0}\left(u(\exp_{p}(v))-u(p)\right)\nu_{p}(\mathrm{d}v).$

Here $\mathrm{p.v.}$ means that we take the principal value of the integral,
$\\{\nu_{p}\\}_{p\in M}$ is a field of measures on $T_{p}M$ induced from an
isotropic Lévy measure $\nu$ on $\mathbb{R}^{n}$ by
$\nu_{p}(A)=\nu(r^{-1}(A))$ whenever $\pi(r)=p$ and $r\in{\mathcal{O}}(M)$ is
an element of the orthonormal frame bundle over $M$. Alternatively, one can
re-write the principal value of the integral (1.4) as

$\frac{1}{2}\int_{v\in T_{p}M\setminus
0}\left(u(\exp_{p}(v))+u(\exp_{p}(-v))-2u(p)\right)\nu_{p}(\mathrm{d}v).$

Note that thanks to the isotropic assumption on $\nu$, this definition is
independent of the choice of $r\in{\mathcal{O}}(M)$.

When the leading term in the generator is $a\Delta_{g}$ (i.e. $a>0$), some
mapping properties were analyzed in [AB21]. However, not much is known about
the case when $a=0$ (i.e. the process is ”pure jump”). This is due to the fact
that (1.4) is now the dominant driver of the process and integrating the
exponential map is difficult to control beyond the injectivity radius on a
general Riemannian manifold. We address this challenge for a broad class of
Riemannian manifolds.

Throughout the article, we make the choice

(1.5)
$\displaystyle\nu(A)=C(n,\alpha)\int_{A}\frac{1}{|v|^{n+2\alpha}}\mathrm{d}v,\quad\alpha\in(0,1),$

for the Lévy measure, which is motivated by the fact that such processes on
$\mathbb{R}^{n}$ are generated by the fractional Laplacian on the Euclidean
space. Note that after pulling back by an element of the fiber of the
orthonormal frame bundle $\mathcal{O}(M)$ over $p\in M$, this measure becomes
the Lévy measure on $T_{p}M$ described earlier in (1.1).

We will prove:

######  Theorem 1.2.

Let $(X_{t})_{t\geqslant 0}$ be a cadlag semi-martingale valued on a
Riemannian manifold $(M,g)$ which is either $\mathbb{S}^{n},\mathbb{T}^{n}$ or
Anosov. If it is an isotropic Lévy process with pure jump induced by the Lévy
measure (1.1), then its infinitesimal generator $\mathscr{A}$ is a non-
positive Fredholm operator

$\mathscr{A}:W^{s,m}(M)\to W^{s-2\alpha,m}(M),$

for all $s\in\mathbb{R},m\in(1,\infty)$, that has discrete spectrum with one
dimensional null-space and co-kernel.

We now give more details on our results on the generator on this Lévy process.

####  1.2.1. Dirichlet form of the generator

The explicit presence of the exponential map in (1.4) suggests that the
behaviour of $\mathscr{A}$ depends more on the geometry and the dynamics of
geodesics than the fractional Laplacian. Therefore, we will take a dynamical
systems approach. As an example of the advantages of taking this point of
view, we can quickly see that $\mathscr{A}$ is always formally given by a
Dirichlet form, as follows.

######  Proposition 1.3.

Let $(M,g)$ be a closed connected Riemannian manifold. There exists an
operator ${\mathscr{D}}:{\rm Lip}(M)\to L^{2}(\mathbb{R}\times SM)$, with
kernel given by the constant functions
$\mathrm{Ker}({\mathscr{D}})=\mathbb{C}\cdot\mathbf{1}$, such that for all
$u,v\in C^{\infty}(M)$,

$-4\int_{M}u\,\mathscr{A}v\,{\rm
dvol}_{g}=\int_{\mathbb{R}}\int_{SM}{\mathscr{D}}u\,{\mathscr{D}}v\,\mathrm{d}\mathrm{L}\mathrm{d}t.$

Here $SM$ denotes the unit sphere bundle and $\mathrm{L}$ is the Liouville
measure on $SM$ invariant under the geodesic flow. Consequently, $\mathscr{A}$
is a non-positive operator that admits an extension as an operator ${\rm
Lip}(M)\to{\mathcal{D}}^{\prime}(M)$ whose kernel consists of constant
functions.

We make the following remarks regarding the previous result:

  1. (i)

When the transition probability can be obtained by solving the heat equation
with infinitesimal generator $\mathscr{A}$, Proposition 1.3 implies that there
is only one differentiable equilibrium state;

  2. (ii)

When $(M,g)$ is the round sphere, the flat torus, or Anosov, we can drop the
differentiability a-priori assumption in Proposition 1.3.

It is natural to ask how similar/different is $\mathscr{A}$ to the fractional
Laplacian (defined spectrally) when $(M,g)$ is not Euclidean. We address this
question for various manifolds below.

####  1.2.2. Generator on the torus

If $M=\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$ is the flat torus, the
operator $\mathscr{A}$ happens (not surprisingly) to be the fractional
Laplacian:

######  Theorem 1.4.

If $(M,g)$ is the torus $\mathbb{T}^{n}$, the infinitesimal generator given by
(1.4) is

$-\mathscr{A}=(-\Delta)^{\alpha},$

where $\Delta$ is the non-positive Laplace operator on $\mathbb{T}^{n}$. In
particular, $\mathscr{A}$ is an elliptic, classical, pseudo-differential
operator of order $2\alpha$.

This result is byproduct of [AH14, Example 1] but can also be obtained by a
simple explicit computation, which we provide in §3.2. Obviously, for general
Riemannian manifolds, such an explicit computation will not be available.

####  1.2.3. Generator on the sphere

It turns out that in the case of the round unit sphere, $\mathscr{A}$ does not
in fact resemble the fractional powers of the Laplacian. It is rather an
object belonging to a more general class of operators called Fourier Integral
Operators introduced by Hörmander [DH72]. For a background on microlocal
analysis and pseudodifferential operators, we refer the reader to §2.1. The
consequences for the mapping properties of $\mathscr{A}$ on functional spaces
are described in §1.2.5 below. On the unit sphere with round metric, we denote
by $\mathscr{J}:{\mathcal{D}}^{\prime}(M)\to{\mathcal{D}}^{\prime}(M)$ the
pullback by the antipodal map. We will prove that the following result holds.

######  Theorem 1.5.

If $(M,g)$ is the sphere $\mathbb{S}^{n}$, the infinitesimal generator given
by (1.4) can be written

$\displaystyle\mathscr{A}=\mathscr{A}_{2\alpha}+\mathscr{A}_{0}+\mathscr{A}_{-1}\mathscr{J}$

where for each $\ell=2\alpha,0,-1$,
$\mathscr{A}_{\ell}\in\Psi^{\ell}_{\mathrm{cl}}(M)$ is a classical formally
selfadjoint pseudodifferential operator of order $\ell$. The operators
$\mathscr{A}_{2\alpha}$ and $\mathscr{A}_{-1}$ have principal symbols
$\sigma_{\mathscr{A}_{2\alpha}}(x,\eta)=-|\eta|_{g}^{2\alpha}$ and
$\sigma_{\mathscr{A}_{-1}}(x,\eta)=c(n)|\eta|_{g}^{-1}$, for some constant
$c(n)>0$. All operators commute with the operator $\mathscr{J}$.

We shall see that since the integral kernel of $\mathscr{A}$ has singularities
at both $p=q$ and $p=-q$ (antipodal point), it cannot be the fractional
Laplacian.

####  1.2.4. Generator on Anosov manifolds

It is natural to deduce that the complications arising on the sphere are due
to geodesics focusing at a single point (i.e. conjugate points). If we make
assumptions about the manifold $(M,g)$ as to rule out such behaviour, we
should expect $\mathscr{A}$ to have a simpler expression. This is indeed the
case if we assume that $(M,g)$ is _Anosov_. The class of Anosov Riemannian
manifolds is a very large class333In fact, $(M,g)$ is Anosov if and only if
$(M,g)$ lies in the $C^{2}$ interior of the set of metrics without conjugate
points [Rug91]. including in particular all negatively-curved manifolds, see
§2.3 or [Ano69, Kni02] for further details. We will prove the following
result.

######  Theorem 1.6.

If $(M,g)$ is a closed connected Anosov Riemannian manifold, the infinitesimal
generator given by (1.4) can be written

$\mathscr{A}=\mathscr{A}_{2\alpha}+\mathscr{A}_{0}$

where for each $\ell=2\alpha,0$,
$\mathscr{A}_{\ell}\in\Psi^{\ell}_{\mathrm{cl}}(M)$ is a classical formally
selfadjoint pseudodifferential operator of order $\ell$.

More precisely, for each $\chi\in C^{\infty}_{c}(\mathbb{R},[0,1])$ such that
$\chi(t)=1$ for $t$ near $0$ and
$\mathrm{supp}(\chi)\subset[0,r_{\mathrm{inj}}^{2}/2]$, where
$r_{\mathrm{inj}}$ is the injectivity radius of $(M,g)$, the operator (1.4)
writes

(1.6) $\displaystyle{\mathscr{A}}u(p)$
$\displaystyle=C(n,\alpha)~{}\mathrm{p.v.}\int_{M}\chi(\mathrm{dist}_{g}(p,q)^{2})\frac{u(q)-u(p)}{\mathrm{dist}_{g}(p,q)^{n+2\alpha}}J(p,q){\rm
dvol}_{g}(q)$ $\displaystyle+w(p)u(p)+\int_{M}K(p,q)u(q){\rm dvol}_{g}(q)$

for some smooth functions $w\in C^{\infty}(M)$ and $K\in C^{\infty}(M\times
M)$. Here we set $J(p,q)=\det\mathrm{d}_{q}\exp_{p}^{-1}$.

An immediate observation is that when $(M,g)$ is Anosov, the result of Theorem
1.6 implies that the operator $\mathscr{A}$ is an elliptic pseudodifferential
operator with principal symbol
$\sigma_{\mathscr{A}}(x,\xi)=-|\eta|_{g}^{2\alpha}$ if $\alpha\geqslant 1/2$
and
$\sigma_{\mathscr{A}}(x,\xi)=-|\eta|_{g}^{2\alpha}+\sigma_{\mathscr{A}_{0}}(x,\xi)$
if $\alpha<1/2$. Also remark that, when $(M,g)$ is Anosov, the trace formula
of Duistermaat-Guillemin [DG75] implies that the spectrum of $\mathscr{A}$
determines uniquely the lengths of periodic geodesics.

####  1.2.5. Mapping properties of $\mathscr{A}$

We will see that Theorems 1.4, 1.5 and 1.6 imply the following

######  Corollary 1.6.

If $(M,g)$ is $\mathbb{S}^{n},\mathbb{T}^{n}$ or Anosov, the following holds.

  1. _(i)_

$-\mathscr{A}$ extends to a formally selfadjoint Fredholm operator

$-\mathscr{A}:W^{s,m}(M)\to W^{s-2\alpha,m}(M),$

for all $s\in\mathbb{R},m\in(1,\infty)$, with non-negative discrete spectrum
and smooth eigenfunctions for all $s\in\mathbb{R}$. The null-space consists of
only constant functions.

  2. _(ii)_

There exists $\mathscr{A}^{+}:W^{s,m}(M)\to W^{s+2\alpha,m}(M)$ with
$\mathrm{Ker}(\mathscr{A}^{+})=\mathbb{C}\cdot\mathbf{1}$ and
$\mathrm{Ran}(\mathscr{A}^{+})\perp\mathbb{C}\cdot\mathbf{1}$ such that

$\mathscr{A}^{+}\mathscr{A}=\mathscr{A}\mathscr{A}^{+}=I-P$

where $P$ is the $L^{2}$ orthonormal projection to the space of constant
functions. The Schwartz kernel of $\mathscr{A}^{+}$,
$\mathrm{G}_{\mathscr{A}}(p,q)$, which we will call the Green’s function,
satisfies, for each $p\in M$ and $u\in C^{\infty}(M)$,

$\int_{M}\mathrm{G}_{\mathscr{A}}(p,\cdot)\mathscr{A}u\,{\rm
dvol}_{g}=u(p)-|M|^{-1}\int_{M}u{\rm dvol}_{g}$

####  1.2.6. Domain of the generator

We will see later that in the sphere, torus and Anosov case, the ”heat kernel”
$e^{t\mathscr{A}}$ has bounded integral kernel (Lemma 5.3.1) for all $t>0$ and
is therefore trace class. Since the spectrum is discrete and the operator is
semidefinite, the solutions of the heat equation converge exponentially in
$L^{2}(M)$ to the constant function because
$\mathrm{Ker}(\mathscr{A})=\mathbb{C}\cdot\mathbf{1}$. At last, we have the
Poincaré inequality on on the sphere, torus, and Anosov case:

(1.7) $\displaystyle-\int_{M}u\,\mathscr{A}u\,{\rm dvol}_{g}\geqslant
c\|u\|^{2}_{L^{2}}.$

for all $u\perp\mathbb{C}\cdot\mathbf{1}$.

Our detailed knowledge of $\mathscr{A}$ will yield some insight into the
probabilitic aspects of the process $(X_{t})_{t\geqslant 0}$. For each
$m\in(1,\infty)$, we define the domain $D_{L^{m}}(\mathscr{A})\subset
L^{m}(M)$ to be the set

$\left\\{u\in L^{m}(M)~{}\Big{|}~{}\lim\limits_{t\to
0^{+}}\left(\mathbb{E}(u(X_{t})\mid X_{0}=\cdot)-u\right)/t\ {\text{exists in
}}L^{m}(M)\right\\}.$

We have a precise description of $D_{L^{m}}(\mathscr{A})$:

######  Proposition 1.7.

When $(M,g)$ is $\mathbb{S}^{n},\mathbb{T}^{n}$ or Anosov, then
$D_{L^{m}}(\mathscr{A})=W^{2\alpha,m}(M)$ when $m\in(1,\infty)$.

This proposition actually comes as an intermediate step (Lemma 5.3.2) in the
calculation of the expected time it takes to find a small target using a
random search which we now describe in detail.

###  1.3. Applications to random searches

As in §1.1, let $(X_{t})_{t\geq 0}$ be a cadlag semi-martingale that is an
isotropic Lévy process with infinitesimal generator $\mathscr{A}$ defined by
(1.4). Let $B_{\varepsilon}(p_{0})$ be the open geodesic ball of radius
$\varepsilon>0$ centred at $p_{0}$. We define $\tau_{\varepsilon}$ and
$u_{\varepsilon}$ by (1.3). Let $c(n,\alpha)$ be the constant given by

(1.8)
$c(n,\alpha):=\begin{cases}\dfrac{2^{-2\alpha}(1-\alpha)\Gamma(1-\alpha)^{2}}{\pi^{2}},&\text{if
$n=2$,}\vspace{0.3cm}\\\
\dfrac{2^{1-2\alpha}\Gamma(n/2-\alpha)\Gamma(n/2-\alpha+1)}{\pi^{n/2}(n-2)\Gamma(n/2-1)},&\text{if
$n\geqslant 3$.}\end{cases}$

Then we have the following result, which is a more precise version of Theorem
1.1, involving remainder terms.

######  Theorem 1.8.

If $(M,g)$ is a closed connected Anosov Riemannian manifold, then:

  1. _(i)_

As $\varepsilon\to 0$, the average of $u_{\varepsilon}$ over $M$ has expansion

$\dfrac{1}{|M|}\int_{M}u_{\varepsilon}{\rm
dvol}_{g}=\varepsilon^{2\alpha-n}|M|c(n,\alpha)(1+\mathcal{O}(E(\alpha,\varepsilon))),$

where the error term $E(\alpha,\varepsilon)$ is given by

(1.9) $\displaystyle
E(\alpha,\varepsilon)=\begin{cases}\varepsilon^{2\alpha},\quad&\text{if}\
\alpha<1/2,\\\ \varepsilon|\log\varepsilon|,\quad&\text{if}\ \alpha=1/2,\\\
\max(\varepsilon,\varepsilon^{n-2\alpha}),\quad&\text{if}\
\alpha>1/2.\end{cases}$

  2. _(ii)_

For all $\varepsilon>0$, $u_{\varepsilon}\in C^{\infty}(M\setminus\partial
B_{\varepsilon}(p_{0}))\cap L^{\infty}(M)$. Moreover, for all $p\neq p_{0}$,
we have as $\varepsilon\to 0$

(1.10) $u_{\varepsilon}(p)-\dfrac{1}{|M|}\int_{M}u_{\varepsilon}{\rm
dvol}_{g}=|M|\mathrm{G}_{\mathscr{A}}(p,p_{0})+\mathcal{O}(E(\alpha,\varepsilon))$

where $\mathrm{G}_{\mathscr{A}}(p,q)$ is the Green’s function of $\mathscr{A}$
given by (iii) of Theorem 1.6.

For the torus, the same result holds, up to changing the error term:

######  Theorem 1.9.

If $(M,g)$ is $\mathbb{T}^{n}$, then the conclusions of Theorem 1.8 hold if we
replace the error term (1.9) by

(1.11)
$E(\alpha,\varepsilon)=\begin{cases}\max(\varepsilon,\varepsilon^{n-2\alpha}),\quad&\text{if}\
\alpha\neq 1/2,\\\ \varepsilon|\log\varepsilon|,\quad&\text{if}\
\alpha=1/2.\end{cases}$

These asymptotics are similar to the ones computed in [NTT21b, NTTT22] for the
Brownian motion. When $\alpha>0$ is small the situation on the sphere is quite
different from that of Anosov manifolds. Due to the singularity structure of
$\mathscr{A}$ when $M=\mathbb{S}^{n}$, a propagation phenomena occurs from
$p_{0}$ to $-p_{0}$ to create, as $\varepsilon\to 0$, a blowup of the quantity

$\left|u_{\varepsilon}(-p_{0})-|M|^{-1}\int_{M}u_{\varepsilon}{\rm
dvol}_{g}\right|.$

We will prove that the following holds:

######  Theorem 1.10.

If $(M,g)$ is $\mathbb{S}^{n}$, then:

  1. _(i)_

The average value of $u_{\varepsilon}$ over $M$ is the same as in Theorem 1.8.

  2. _(ii)_

For all $\varepsilon>0$, we have

$u_{\varepsilon}\in C^{\infty}\left(M\setminus\left(\partial
B_{\varepsilon}(p_{0})\cup\partial B_{\varepsilon}(-p_{0})\right)\right)\cap
L^{\infty}(M)$

and (1.10) holds whenever $p\notin\\{p_{0},-p_{0}\\}$ where
$\mathrm{G}_{\mathscr{A}}$ is given by Corollary 1.2.5.

  3. _(iii)_

If $n>1+4\alpha$ and $1>(n-4)\alpha$, then at $p=-p_{0}$ we have

(1.12)
$\displaystyle\left|u_{\varepsilon}(-p_{0})-\dfrac{1}{|M|}\int_{M}u_{\varepsilon}{\rm
dvol}_{g}\right|=\frac{\tilde{c}(\alpha,n)|M|}{\varepsilon^{n-1-4\alpha}}+o(\varepsilon^{-n+1+4\alpha})$

for some $\tilde{c}(\alpha,n)>0$, which we do not make explicit.

Following Theorem 1.10, it would interesting to understand the generator
$\mathscr{A}$ and the narrow capture problem in other settings than the sphere
where conjugate points appear, like Zoll manifolds for instance. This is left
for future investigation.

###  1.4. Structure of the paper

In §2, we recall some general facts of microlocal analysis and Riemannian
geometry. In §3, we study the generator $\mathscr{A}$ and prove the results
announced in §1.2. In §4, we prove the results announced in §1.3 on the
expected stopping time, omitting technical results on the solutions of the
integral equation $\mathscr{A}u_{\varepsilon}=-1$ on $M\setminus
B_{\varepsilon}(p_{0})$, which we leave until §5.

### Acknowledgements

The authors wish to thanks David Applebaum, Sonja Cox, and Frank Redig for the
useful discussions and encouragement during the writing of this article.

##  2\. Preliminaries

In this section, we detail some tools needed throughout the paper.

###  2.1. Microlocal analysis

We refer to [GS94, Hör15] for a general treatment.

####  2.1.1. Pseudodifferential operators

Let $M$ be a closed $n$-dimensional manifold. For $k\in\mathbb{R}$, we define
$S^{k}(T^{*}M)\subset C^{\infty}(T^{*}M)$, the space of symbols of order $k$,
as the set of smooth functions $a$ satisfying the following bounds, in any
coordinate chart $U\subset\mathbb{R}^{n}$: for all
$\gamma,\beta\in\mathbb{N}^{n}$, there exists $C:=C(U,\alpha,\beta)>0$ such
that

(2.1) $\forall(x,\xi)\in
T^{*}U\simeq\mathbb{R}^{n}\times\mathbb{R}^{n},\qquad|\partial^{\gamma}_{\xi}\partial^{\beta}_{x}a(x,\xi)|\leqslant
C\langle\xi\rangle^{k-|\gamma|}.$

It can be checked that (2.1) is invariant by diffeomorphism, which implies
that $S^{k}(T^{*}M)$ is intrinsically defined on $M$.

We define $\Psi^{-\infty}(M)$, the set of _smoothing operators_ , as the space
of linear operators on $M$ with smooth Schwartz kernel. Denote by
$\mathrm{Op}$ a quantization procedure on $M$, given in a local coordinate
patch $U\subset\mathbb{R}^{n}$ by:

$\mathrm{Op}(a)f(x)=\dfrac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}_{\xi}}\int_{\mathbb{R}^{n}_{y}}e^{i\xi\cdot(x-y)}a(x,\xi)f(y)\mathrm{d}y\mathrm{d}\xi,$

where $a\in S^{k}(T^{*}U)$ and $f\in C^{\infty}_{c}(U)$. The set of
_pseudodifferential operators_ of order $k\in\mathbb{R}$ is then defined as

$\Psi^{k}(M):=\left\\{\mathrm{Op}(a)+R~{}|~{}a\in
S^{k}(T^{*}M),R\in\Psi^{-\infty}(M)\right\\}.$

It can be checked that $\Psi^{k}(M)$ is intrinsically defined and independent
on the choice of quantization $\mathrm{Op}$.

There exists a well-defined _principal symbol map_

$\sigma:\Psi^{k}(M)\to S^{k}(T^{*}M)/S^{k-1}(T^{*}M)$

such that we have the following exact sequence:

$0\longrightarrow\Psi^{k-1}(M)\longrightarrow\Psi^{k}(M)\longrightarrow
S^{k}(T^{*}M)/S^{k-1}(T^{*}M)\longrightarrow 0.$

The elliptic set $\mathrm{ell}(A)\subset T^{*}M\setminus\left\\{0\right\\}$ of
an operator $A\in\Psi^{k}(T^{*}M)$ is defined as the (open) conic set of
points $(x_{0},\xi_{0})\in T^{*}M\setminus\left\\{0\right\\}$ such that there
exists a constant $C>0$ such that the following holds:

(2.2) $\left(|\xi|\geqslant C\text{ and
}d_{S^{*}M}\left((x,\xi/|\xi|),(x_{0},\xi_{0}/|\xi_{0}|)\right)<1/C\right)\implies|\sigma_{A}(x,\xi)|\geqslant\langle\xi\rangle^{k}/C.$

Here $d_{S^{*}M}$ is any metric on the cosphere bundle
$S^{*}M:=T^{*}M/\mathbb{R}_{+}$, where the $\mathbb{R}_{+}$-action is given by
radial dilation in the fibers of $T^{*}M$. An operator is said to be
_elliptic_ if $\mathrm{ell}(A)=T^{*}M\setminus\left\\{0\right\\}$. The
_characteristic set_ $\Sigma(A)$ of an operator is the closed conic subset
defined as the complement of the elliptic set in $T^{*}M$. The important
property of elliptic operators (on $T^{*}M$) is that they are invertible
modulo smoothing remainders, that is, one can find $B\in\Psi^{-k}(T^{*}M)$ and
$R\in\Psi^{-\infty}(M)$ such that

$BA=\mathbf{1}+R.$

Such an operator $B$ is called a _parametrix_ for $A$.

####  2.1.2. Wavefront set of distributions

The _wavefront set_ ${\rm WF}(A)$ (or the _microsupport_) of an operator
$A\in\Psi^{k}(M)$ is the (closed) conic subset of
$T^{*}M\setminus\left\\{0\right\\}$ satisfying the following property:
$(x_{0},\xi_{0})\notin{\rm WF}(A)$ if and only if for all $m\in\mathbb{R}$,
for all $b\in S^{m}(T^{*}M)$ supported in a small conic neighborhood of
$(x_{0},\xi_{0})$, one has $A\mathrm{Op}(b)\in\Psi^{-\infty}(M)$. In other
words, the complement of the wavefront set of $A$ is the set of codirections
where $A$ behaves as a smoothing operator.

The wavefront set ${\rm WF}(u)$ of a distribution
$u\in\mathcal{D}^{\prime}(M)$ is the (closed) conic subset of
$T^{*}M\setminus\left\\{0\right\\}$ satisfying the following property:
$(x_{0},\xi_{0})\notin{\rm WF}(u)$ if and only if there exists a small open
conic neighborhood $V$ of $(x_{0},\xi_{0})$ such that for all
$k\in\mathbb{R}$, for all $A\in\Psi^{k}(M)$ with wavefront set contained in
$V$, one has $Au\in C^{\infty}(M)$. In particular, a distribution/function $u$
is smooth if and only if ${\rm WF}(u)=\emptyset$. Equivalently, the wavefront
set of a distribution can be characterized as follows: taking
$(x_{0},\xi_{0})\in T^{*}M\setminus\left\\{0\right\\}$, the point
$(x_{0},\xi_{0})$ is not in the wavefront set of $u$ if we can find $\chi,S\in
C^{\infty}(M)$ such that $\chi$ has support near $x_{0}$, $dS\neq 0$ on the
support of $\chi$, $dS(x_{0})=\xi_{0}$, and

(2.3) $\langle u,e^{-iS/h}\chi\rangle=\mathcal{O}(h^{\infty}).$

####  2.1.3. Functional spaces

We now introduce the functional spaces we will be working with. We denote by
$\Delta_{g}\leqslant 0$ the negative Hodge Laplacian acting on functions. For
all $s\in\mathbb{R}$, the operator $(\mathbf{1}-\Delta)^{s}$ defined using the
spectral theorem (applied to the selfadjoint operator $\Delta_{g}$ on
$L^{2}(M,\mathrm{vol}_{g})$) is an invertible pseudodifferential operator of
order $2s$.

For $s\in\mathbb{R}$, $m\in(1,\infty)$ and $u\in C^{\infty}(M)$, we set

(2.4) $\|u\|_{W^{s,m}}:=\|(\mathbf{1}-\Delta)^{s/2}u\|_{L^{m}},$

and define $W^{s,m}(M)$ to be the completion of $C^{\infty}(M)$ with respect
to the norm (2.4). Taking $m=2$, we retrieve the usual Sobolev spaces which we
will rather denote by $H^{s}(M):=W^{s,2}(M)$. Note that the spaces
$W^{s,m}(M)$ intrinsically defined, that is, they are independent of the
choice of metric $g$, and changing the metric only replaces the norm (2.4) by
an equivalent norm.

The following boundedness result for pseudodifferential operators holds: for
all $k\in\mathbb{R},A\in\Psi^{k}(M)$ and $s\in\mathbb{R}$, $m\in(1,\infty)$,

(2.5) $A:W^{s+k,m}(M)\to W^{s,m}(M)$

is bounded.

Eventually, given $\Omega\subset M$ be an open subset with non-empty smooth
boundary, we define, for $s\in\mathbb{R}$ and $m\in[1,\infty)$, the spaces

$\dot{W}^{s,m}(\overline{\Omega}):=\\{u\in
W^{s,m}(M)\mid\text{supp}(u)\subset\overline{\Omega}\\}$

and

$\overline{W}^{s,m}(\Omega):=\\{u|_{\Omega}~{}|~{}u\in W^{s,m}(M)\\}.$

###  2.2. Riemannian geometry of the unit sphere bundle

Let $(M,g)$ be a smooth closed connected $n$-dimensional Riemannian manifold.
The unit sphere bundle $SM$ over $M$ is defined by

(2.6) $\displaystyle SM:=\\{(p,v)\in TM\mid|v|_{g}=1\\}.$

Since $M$ has dimension $n$, $SM$ is a manifold of dimension $2n-1$. A generic
point in $SM$ will be denoted by $z$.

Associated to this operator is the pull-back operator
$\pi^{*}:\mathcal{D}^{\prime}(M)\to\mathcal{D}^{\prime}(SM)$, which, when
restricted to $u\in C^{\infty}(M)$ takes the form $(\pi^{*}u)(p,v):=u(p)$.
Note that $\pi^{*}:C^{\infty}(M)\to C^{\infty}(SM)$. It has an adjoint, the
push-forward $\pi_{*}:\mathcal{D}^{\prime}(SM)\to\mathcal{D}(M)$ defined via
the adjoint relation

$\langle\pi_{*}u,\varphi\rangle:=\langle u,\pi^{*}\varphi\rangle.$

When $u\in C^{\infty}(SM)$ the push-forward can be written explicitly as

$(\pi_{*}u)(p)=\int\limits_{v\in S_{p}M}u(p,v)\mathrm{d}S^{n-1}(v).$

Let $X_{g}$ be the geodesic vector field on $SM$ and $\phi_{t}(\cdot)$,
$t\in\mathbb{R}$ be the geodesic flow generated by this vector field. Using
the canonical projection $\pi:SM\to M$ which maps $\pi:(p,v)\mapsto p$, we
define the vertical bundle ${\mathbf{V}}\subset TSM$ by:

(2.7) ${\mathbf{V}}=\\{(p,v,V)\in TSM~{}|~{}\mathrm{d}\pi_{(p,v)}V=0\\}.$

The tangent space to $SM$ then splits as

(2.8) $TSM=\mathbb{R}X_{g}\oplus{\mathbf{V}}\oplus{\mathbf{H}},$

where ${\mathbf{H}}$ is the horizontal bundle, which can be defined as the
horizontal space of the Levi-Civita connection induced by $g$, see [Pat99] for
instance. The metric $g$ induced a natural metric on $SM$, called the Sasaki
metric and denoted by $G$, for which the splitting (2.8) is orthogonal.

The Riemannian measure induced by the Sasaki metric $G$ is called the
Liouville measure $\mathrm{d}\mathrm{L}$. It is invariant under the flow, that
is,

(2.9)
$\displaystyle\int_{SM}u(\phi_{t}(z))w(z)\mathrm{d}\mathrm{L}(z)=\int_{SM}u(z)w(\phi_{-t}(z))\mathrm{d}\mathrm{L}(z)$

for all $u,w\in C(SM)$ and $t\in\mathbb{R}$. There is a convenient way to
describe the measure $\mathrm{d}\mathrm{L}$ locally as the product of the
measure on the round unit sphere $S^{n-1}$ and the Riemannian volume on $M$:

(2.10) $\displaystyle\mathrm{d}\mathrm{L}(p,v)=\mathrm{d}S^{n-1}(v)\wedge{\rm
dvol}_{g}(p).$

###  2.3. Overview of Anosov manifolds

We briefly recall the definition and some basic properties of Anosov
manifolds. Let $(M,g)$ be a closed compact manifold with geodesic vector field
$X_{g}$ whose flow $\phi_{t}(\cdot)$ is complete. We say that the manifold
$(M,g)$ is Anosov if its geodesic flow $\phi_{t}(\cdot)$ is Anosov, that is,
if there is a continuous splitting of invariant bundles ${E_{u}}$ and
${E_{s}}$:

(2.11) $\displaystyle TSM={E_{s}}\oplus{E_{u}}\oplus\mathbb{R}X_{g}$

and there exists $C>0$ and $0<\rho<1<\eta$ such that for all $t>0$

(2.12) $\|\mathrm{d}\phi_{-t}|_{E_{u}}\|<C\eta^{-t},\
\|\mathrm{d}\phi_{t}|_{{E_{s}}}\|<C\rho^{t}.$

The norms $\|\bullet\|$ in (2.12) are computed with respect to an arbitrary
auxiliary smooth metric on $SM$ but both properties (2.11) and (2.12) are
independent of this choice of auxiliary metric. Throughout we will use $p$ to
denote points on $M$ and $v\in S_{p}M$. Sometimes it is convenient to denote
$(p,v)$ as a single point in $SM$ in which case we will write $z=(p,v)$.

Associated with the flow $\phi_{t}$ is the symplectic lift
$\Phi_{t}:T^{*}SM\to T^{*}SM$ defined by

$\Phi_{t}(z,\xi)=(\phi_{t}(z),\mathrm{d}_{z}\phi_{t}^{-\top}\xi).$

The dual splitting of (2.11) is given by invariant vector bundles $E_{s}^{*}$
and $E_{u}^{*}$

(2.13) $\displaystyle T^{*}SM=E_{s}^{*}\oplus
E_{u}^{*}\oplus\mathbb{R}\vartheta,$

where $E_{s}^{*}=({E_{s}}\oplus\mathbb{R}X_{g})^{\perp}$,
$E_{u}^{*}=({E_{u}}\oplus\mathbb{R}X_{g})^{\perp}$, while $\vartheta$ is the
Liouville 1-form on $SM$.

The covertical bundle is defined by:

(2.14) ${\mathbf{V}}^{\perp}=\\{(p,v,\xi)\in T^{*}SM~{}|~{}\xi(V)=0,\ \forall
V\in\mathbf{V}\\}.$

Assuming that $X_{g}$ generates an Anosov flow, we have by [Mañ87, Proposition
II.2]:

(2.15) ${\mathbf{V}}^{\perp}\cap E_{s}^{*}={\mathbf{V}}^{\perp}\cap
E^{*}_{u}=\\{0\\}.$

Moreover, Anosov manifolds are free of conjugate points [Kli74] which implies
that

(2.16) $d\phi_{t}({\mathbf{V}}^{\perp})\cap{\mathbf{V}}^{\perp}=E_{0}^{*},$

for all $t\neq 0$.

The bundles $E_{s}\oplus E_{0}$ and $E_{u}$ are integrable and tangent to a
foliation which consists of _central stable_ and _unstable_ leaves. We denote
the leaves by $W^{s0}(z)$ and $W^{u}(z)$. They are smooth immersed submanifold
in $SM$. They can be defined alternatively by:

(2.17) $W^{u}(z)=\left\\{z^{\prime}\in
SM~{}|~{}\mathrm{d}_{SM}(\phi_{-t}(z),\phi_{-t}(z^{\prime}))\to_{t\to+\infty}0\right\\},$

(and similarly for $W^{s}(z)$ by changing $-t$ to $t$) and

$W^{s0}(z)=\bigcup_{t\in\mathbb{R}}\phi_{t}(W^{s}(z)).$

Note that the convergence in (2.17) is actually exponentially fast. We will
use the notation $W^{s0}_{\mathrm{loc}}(z)$ (resp. $W^{u}_{\mathrm{loc}}(z)$)
to denote the intersection of $W^{s0}(z)$ (resp. $W^{u}_{\mathrm{loc}}(z)$)
with a small ball $B_{\varepsilon_{0}}(z)$, where $\varepsilon_{0}>0$ is some
small fixed constant.

The Anosov foliation in central stable/unstable leaves is not smooth (it is
only transversally continuous to the leaves). Nevertheless, we can introduce
the algebra of functions $C^{\infty}_{\mathrm{cs}}(SM)$ of functions such that
their restriction to all central stable leaves are smooth and vary
continuously in the unstable direction, see [dlLMM86, Section 2]. Similarly,
we can construct vector fields which are tangent to every central stable
leaves, smooth in restriction to each leaf and vary continuously transversally
in the unstable direction. The following lemma connects the wavefront set
property of a function/distribution to the smoothness of its restriction to
every leaf of this foliation. Its proof is a standard calculation, see [BGW21,
Lemma 1.9] for instance.

######  Lemma 2.0.

Let $f\in C^{0}(SM)$ be a continuous function. Fix $z_{0}\in SM$ and consider
a family of vector fields $S_{0}=X_{g},S_{1},...,S_{n-1}\in
C^{\infty}_{\mathrm{cs}}(SM,E_{s}\oplus E_{0})$ spanning locally $E_{s}\oplus
E_{0}$ near $z_{0}$. If $S^{\beta}f\in C^{0}(SM)$ for all multi-indices
$\beta\in\mathbb{N}^{n}$, where
$S^{\beta}:=S_{0}^{\beta_{0}}S_{1}^{\beta_{1}}...S_{n-1}^{\beta_{n-1}}$, then
${\rm WF}(f)\subset E_{s}^{*}$.

Another crucial property of the Anosov foliation is that it is _absolutely
continuous_ , that is, we can disintegrate smooth measures along
stable/unstable leaves and the disintegrated measures are themselves smooth.
In other words, if $\mathrm{d}\mu$ is a smooth measure on $SM$, given $f\in
C^{\infty}(U)$ where $U\subset SM$ is an open subset, we have:

(2.18)
$\int_{U}f(z)\mathrm{d}\mu(z)=\int_{W^{u}_{\mathrm{loc}}(p)}\left(\int_{W^{cs}_{\mathrm{loc}}(x)}f(y)\delta_{x}(y)\mathrm{d}m^{cs}_{x}(y)\right)\mathrm{d}m^{u}_{p}(x),$

where $p\in U$, $m^{cs}_{x}$ is the smooth Riemannian measure induced by the
restriction of the Sasaki metric $G$ to the leaf $W^{cs}_{\mathrm{loc}}(x)$,
$m^{u}_{p}$ is the smooth Riemannian measure induced by the Sasaki metric
restricted to $W^{u}_{\mathrm{loc}}(p)$, and $y\mapsto\delta_{x}(y)$ are
smooth in restriction to every leaf, and continuous transversally in the
unstable direction, that is, $\delta\in C^{\infty}_{\mathrm{cs}}(U)$. We refer
to [BGW21, Proposition 1.6] for a proof.

###  2.4. Markov and Feller processes. Capture problem

In order relate the functional analytic properties of the generator
$\mathscr{A}$ to the study of the expected capture time $u_{\varepsilon}$, we
will rely on the results of Geetor [Get57, Get59, Get61] showing that
$u_{\varepsilon}$ has a convenient integral representation. The aim of this
paragraph is to recall the main ingredients of this construction.

For each $t>0$, let
${\mathbf{p}}(t,\cdot,\cdot)\in\mathcal{D}^{\prime}(M\times M)$ be the
fundamental solution of the heat equation with generator $\mathscr{A}$ in the
sense that

(2.19)
$\partial_{t}\mathbf{\mathbf{p}}(t,p,\cdot)=\mathscr{A}{\mathbf{p}}(t,p,\cdot),\
\ {\mathbf{p}}(0,p,\cdot)=\delta_{p}(\cdot).$

We let $X:=(X_{t})_{t\geqslant 0}$ be a cadlag semi-martingale on $M$ which is
an isotropic Lévy process in the sense of [AE00], induced by the isostropic
Lévy measure (1.1). By construction, $X$ is a Feller process with generator
$\mathscr{A}$, see [AB21]. This means that

$U_{t}\phi(p)=\mathbb{E}\left[\phi(X_{t})\ \middle|~{}X_{0}=p\right],$

defines a semi-group satisfying the following properties:

  * •

If $f\in C^{0}(M)$, $\|U_{t}f\|_{C^{0}}\leqslant\|f\|_{C^{0}}$, and $U_{t}f\to
f$ a.e as $t\to 0$;

  * •

If $f\in C^{0}(M)$ satisfies $f\geqslant 0$, then $U_{t}f\geqslant 0$;

  * •

$U_{t}\mathbf{1}=\mathbf{1}$;

  * •

The generator has domain containing $C^{\infty}(M)$, and coincides with
$\mathscr{A}$ on $C^{\infty}(M)$444That $C^{\infty}(M)$ is contained in the
domain is not explicitly stated in [AE00]. For this, [AE00] relies on [App95,
Page 177].;

  * •

For any Borel sets $\Omega_{1},\Omega_{2}\subset M$, the transition
probability for the process $(X_{t})_{t\geqslant 0}$ is given by

${\mathbb{P}}(X_{t}\in\Omega_{1}\mid
X_{0}\in\Omega_{2})=|\Omega_{2}|^{-1}\int_{q\in\Omega_{1}}\int_{p\in\Omega_{2}}\mathbf{p}(t,p,q){\rm
dvol}_{g}(q){\rm dvol}_{g}(p).$

In our case, we will see in Lemma 5.3.1 that since $\mathscr{A}$ is elliptic
pseudo-differential (or almost so in the case of the sphere) the kernel
$\mathbf{p}$ is smooth for $t>0$. We will also see that $\mathscr{A}$ is
essentially self-adjoint. This implies in particular that
$\mathbf{p}(t,p,q)=\mathbf{p}(t,q,p)$, and that $e^{t\mathscr{A}}$ is Hilbert-
Schmidt for $t>0$.

We now review the results from [Get59]. From the aforementioned properties of
the kernel $\mathbf{p}$, the assumptions (P) and (K) from [Get59] are
satisfied and the results of [Get59, §2, 5 and 6] and [Get59, Theorem 4.1]
apply. For an open set $\Omega\subset M$ and $V:\Omega\to\mathbb{R}^{+}$ a
measurable function, Geetor introduces the operators
$T_{t}=T_{t}[V,\Omega]$555The open set $\Omega$ is called $G$ in his notation.
defined by

(2.20)
$T_{t}\phi(p)=\mathbb{E}\left[\phi(X(t))e^{-\int_{0}^{t}V(X(\tau))d\tau}\mathbf{1}_{\\{X(\tau)\in\overline{\Omega},\
0\leq\tau\leqslant t\\}}\middle|X(0)=p\right].$

We have $\|T_{t}\phi\|_{L^{\infty}(M)}\leqslant\|\phi\|_{L^{\infty}(M)}$ for
every $\phi\in C^{0}(\overline{\Omega})$. If $|\partial\Omega|=0$ and for a.e
$p\in\Omega$,

$\int_{0}^{t}\int_{\Omega}p(\tau,p,q)V(y){\rm dvol}_{g}(q)d\tau\to_{t\to 0}0,$

then [Get59, Theorem 2.1] asserts that $(T_{t})_{t\geqslant 0}$ is a strongly
continuous semi-group on $L^{2}(\Omega)$ (in particular, this is certainly
true if $V$ is $L^{\infty}(M)$). Next, according to [Get59, Theorem 5.1],
$T_{t}$ has a measurable kernel $\mathbf{k}(t,p,q)$ satisfying for every fixed
$t\geqslant 0,p\in M$

(2.21) $0\leqslant\mathbf{k}(t,p,q)\leqslant\mathbf{p}(t,p,q),\ \text{ for a.e
}\ q\in M.$

An important feature is the following approximation result [Get59, Theorem
4.1]: given an open set $\Omega$, setting
$V_{\ell}:=\ell\mathbf{1}_{M\setminus\Omega}$ for $\ell\geqslant 0$, we have
that for every $t>0$ and $\phi\in L^{2}(\Omega)$,

(2.22) $T_{t}[V_{\ell},M]\phi\to_{\ell\to+\infty}T_{t}[0,\Omega]\phi,$

where the convergence holds $L^{2}(M)$. Furthermore, in [Get59, Theorems 6.1,
6.2, 6.3 and 6.4], it is proved that for $t\geqslant 0$, $T_{t}$ is self-
adjoint, a $L^{2}(\Omega)$-contraction, Hilbert-Schmidt, positive definite,
and if $\mathscr{A}_{V,G}$ is its generator with eigenvalues
$0\geqslant\lambda_{0}\geqslant\dots\geqslant\lambda_{m}\geqslant...$, and
eigenfunctions $\\{\phi_{j}\\}_{j\geqslant 0}$, then:

(2.23) $\mathbf{k}(t,p,q)=\sum_{j\geqslant
0}e^{t\lambda_{j}}\phi_{j}(p)\phi_{j}(q).$

Finally, let us give a word on the action on $L^{m}(\Omega)$-spaces for
$m\in[1,\infty]$. From the symmetry of $T_{t}$, that is
$\mathbf{k}(t,p,q)=\mathbf{k}(t,q,p)$ for all $p,q\in M$, we deduce that for
every $t\geqslant 0,q\in M$,

(2.24) $\int_{\Omega}\mathbf{k}(t,p,q){\rm dvol}_{g}(p)=1.$

This implies that $T_{t}:L^{1}(\Omega)\to L^{1}(\Omega)$ has norm at most $1$.
By interpolation, $T_{t}$ extends as a contraction on every $L^{m}(\Omega)$,
$m\in[1,\infty]$.

We will use the previous constructions in our arguments in §5.3. More
specifically, denoting $\Omega_{\varepsilon}=M\setminus
B_{\varepsilon}(p_{0})$, we set

(2.25) $T_{t}:=T_{t}[0,\Omega_{\varepsilon}],\qquad
T_{t}^{\ell}:=T_{t}[V_{\ell},M],$

where $T_{t}[V_{\ell},M]$ are the approximating semi-groups as in (2.22). It
should also be observed that in the proof of (2.22), one can replace $L^{2}$
by $L^{m}$ without any problem.

For all $p\in\Omega_{\varepsilon}$, let

$\mathcal{T}_{\varepsilon}(t,p):={\mathbb{P}}(\tau_{\varepsilon}>t\mid
X_{0}=p)$

be the probability that a process starting at $p$ does not exit
$\Omega_{\varepsilon}$ before time $t>0$. Taking $\Omega=\Omega_{\varepsilon}$
and $V=0$ in the above definitions, we get

(2.26)
$\mathcal{T}_{\varepsilon}(t,p)={\mathbb{P}}(X_{\tau}\in\Omega_{\varepsilon},\forall
0\leqslant t\leqslant\tau\mid
X_{0}=p)=\int_{\Omega_{\varepsilon}}{\mathbf{k}}(t,p,q){\rm
dvol}_{g}(q)=T_{t}\mathbf{1}_{\Omega_{\varepsilon}},$

and thus

(2.27)
$u_{\varepsilon}(p)=\int_{0}^{+\infty}\int_{\Omega_{\varepsilon}}\mathbf{k}(t,p,q){\rm
dvol}_{g}(q)\mathrm{d}t.$

To obtain the desired results on $u_{\varepsilon}$, we will study the finer
properties of $\mathbf{k}$ using arguments very similar to [Get61] in §5.3.

##  3\. Properties of the generator

###  3.1. Dirichlet form of the generator

As a quick demonstration of the advantages of working on the level of
dynamical systems, we prove the existence of a Dirichlet form for
$\mathscr{A}$ as stated in Proposition 1.3. We will in fact prove the
existence of Dirichlet forms for operators which are slightly more general:

######  Proposition 3.1.

Let $b\in C^{\infty}(\mathbb{R})$ satisfy $0\leqslant b(t)\leqslant 1$. Then
the operator given by

${\mathscr{A}}_{b}u(p):=\int_{v\in
T_{p}M}\left(u(\exp_{p}(v))-u(p)\right)b(|v|_{g}^{2})\nu_{p}(dv)$

where $\nu_{p}$ is as in (1.1) satisfies

$-4\langle
u,\mathscr{A}_{b}v\rangle=\int_{0}^{\infty}\int_{SM}{\mathscr{D}}_{b}u\,\overline{{\mathscr{D}}_{b}v}\,\mathrm{d}\mathrm{L}\mathrm{d}t$

for some ${\mathscr{D}}_{b}:{\rm Lip}(M)\to L^{2}(\mathbb{R}\times SM)$. When
$b(t)$ is strictly positive, ${\mathscr{D}}_{b}$ is has null-space consisting
of only constant functions.

Set $I_{\varepsilon}$ to be the indicator function of
$\mathbb{R}\setminus(-\varepsilon,\varepsilon)$ and define

${\mathscr{A}}_{b,\varepsilon}u(p):=\int_{v\in
T_{p}M}I_{\varepsilon}(|v|_{g})\left(u(\exp_{p}(v))-u(p)\right)b(|v|_{g}^{2})\nu_{p}(\mathrm{d}v).$

Using the structures on $SM$ introduced above, we have the following
representation for ${\mathscr{A}}_{\varepsilon}$ which is essentially polar
coordinates (see [PSU22]). We will derive it for the convenience of the
reader:

######  Lemma 3.1.

For $u\in C^{1}(M)$ we have

(3.1)
$\displaystyle{\mathscr{A}}_{b,\varepsilon}u(p)=\frac{1}{2}\int_{-\infty}^{\infty}\int_{v\in
S_{p}M}I_{\varepsilon}(t)\frac{u(\exp_{p}(tv))-u(p)}{|t|^{1+2\alpha}}b(t^{2})\mathrm{d}S^{n-1}(v)\mathrm{d}t.$

###### Proof.

Fix $p\in M$ and denote by $\tilde{\mathscr{A}}_{b,\varepsilon}u(p)$ the
right-hand side of (3.1). First observe that we can split the time integration
into the sum of integrals on $(-\infty,0)$ and $(0,\infty)$. Applying a change
of variable $(t,y)\mapsto(-t,-y)$ to the $(-\infty,0)$ integral yields

$\tilde{\mathscr{A}}_{b,\varepsilon}u(p)=\int_{0}^{\infty}\int_{v\in
S_{p}M}I_{\varepsilon}(t)\frac{u(\exp_{p}(tv))-u(p)}{|t|^{1+2\alpha}}b(t^{2})\mathrm{d}S^{n-1}(v)\mathrm{d}t.$

The change of variable $\tilde{v}=tv$ yields (see [PSU22, Lemma 8.1.8])

$\tilde{\mathscr{A}}_{\varepsilon}u(p)=\int_{v\in T_{p}M\setminus
0}I_{\varepsilon}(|v|_{g})\frac{u(\exp_{p}(v))-u(p)}{|v|_{g}^{n+2\alpha}}b(|v|_{g}^{2})\mathrm{d}T_{p}(\tilde{v})$

where $\mathrm{d}T_{p}(v)$ is the volume form on $T_{p}M$ for the metric
$g_{p}$. So we have that
$\tilde{\mathscr{A}}_{\varepsilon}={\mathscr{A}}_{\varepsilon}$. ∎

We now define the operator ${\mathscr{D}}_{b}:{\rm Lip}(M)\to
L^{2}(\mathbb{R}\times SM)$ by

(3.2)
${\mathscr{D}}_{b}u(t,p,v):=\sqrt{b(t^{2})}|t|^{-\frac{1+2\alpha}{2}}\left(u(\exp_{p}(tv)-u(p)\right),\quad
t\neq 0,\quad(p,v)\in SM.$

Observe that if $b(t)$ is strictly positive,

(3.3) $\mathrm{Ker}({\mathscr{D}}_{b})=\mathbb{C}\cdot\mathbf{1}$

since any two points on $M$ can be joined by a geodesic. We are now in a
position to prove Proposition 1.3:

###### Proof of Proposition 3.1.

Without loss of generality we may assume that $u,w\in C^{\infty}(M)$ re both
supported in a single coordinate patch. We first define

$N_{\varepsilon}:=\int_{-\infty}^{\infty}\int_{SM}I_{\varepsilon}(t)\,{\mathscr{D}}_{b}u\,\overline{{\mathscr{D}}_{b}w}\,\mathrm{d}\mathrm{L}\mathrm{d}t$

and write

(3.4)
$\displaystyle\int_{-\infty}^{\infty}\int_{SM}{\mathscr{D}}_{b}u\overline{{\mathscr{D}}_{b}w}\mathrm{d}\mathrm{L}\mathrm{d}t=\lim\limits_{\varepsilon\to
0}N_{\varepsilon}$

Now

$\displaystyle N_{\varepsilon}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}b(t^{2})|t|^{1+2\alpha}I_{\varepsilon}(t)\int_{SM}u(\pi\circ\phi_{t}(p,v))\overline{\left(w(\pi\circ\phi_{t}(p,v))-w(p)\right)}\mathrm{d}\mathrm{L}(p,v)\mathrm{d}t$
$\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}b(t^{2})|t|^{1+2\alpha}I_{\varepsilon}(t)\int_{SM}u(p)\overline{\left(w(\pi\circ\phi_{t}(p,v))-w(p)\right)}\mathrm{d}\mathrm{L}(p,v)\mathrm{d}t.$

We now make a change of variable $(p,v)=\phi_{-t}(q,v^{\prime})$ in the first
integral and use the identity (2.9) we get

$\displaystyle N_{\varepsilon}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}b(t^{2})|t|^{1+2\alpha}I_{\varepsilon}(t)\int_{SM}u(p)\overline{\left(w(p)-w(\pi\circ\phi_{-t}(p,v))\right)}\mathrm{d}\mathrm{L}(p,y)\mathrm{d}t$
$\displaystyle-$
$\displaystyle\int_{-\infty}^{\infty}b(t^{2})|t|^{1+2\alpha}I_{\varepsilon}(t)\int_{SM}u(p)\overline{\left(w(\pi\circ\phi_{t}(p,v))-w(p)\right)}\mathrm{d}\mathrm{L}(p,v)\mathrm{d}t.$

The two integrals are essentially identical except that the time is reversed
in the first integral. This can be taken care of by a substitution
$t\mapsto-t$ to get

$N_{\varepsilon}=2\int_{-\infty}^{\infty}b(t^{2})|t|^{1+2\alpha}I_{\varepsilon}(t)\int_{SM}u(p)\overline{\left(w(p)-w(\pi\circ\phi_{t}(p,v))\right)}\mathrm{d}\mathrm{L}(p,v)\mathrm{d}t.$

Now use the splitting (2.10) we have that $N_{\varepsilon}$ writes

$2\int\limits_{p\in
M}u(p)\int_{-\infty}^{\infty}b(t^{2})|t|^{1+2\alpha}I_{\varepsilon}(t)\int\limits_{v\in
S_{p}M}\overline{\left(w(p)-w(\exp_{p}(tv))\right)}dS_{p}^{n-1}(v)\mathrm{d}t{\rm
dvol}_{g}(p).$

In particular, by Lemma 3.1, one obtains

$N_{\varepsilon}=-4\int_{M}u(p)\overline{{\mathscr{A}}_{b,\varepsilon}w(p)}{\rm
dvol}_{g}(p).$

If $w\in C^{2}$, one can show that
${\mathscr{A}}_{b,\varepsilon}w(p)\to{\mathscr{A}}_{b}w(p)$ pointwise by
Taylor expanding $w(\exp_{p}(v))$ near $v=0$ and use the fact that

$\int_{\theta\in S^{n-1}}\theta\cdot\hat{n}\,\mathrm{d}S^{n-1}(\theta)=0$

for all $\hat{n}\in S^{n-1}$ (in fact this is how the principal value integral
of (1.4) is defined). As $M$ is compact we can use the same argument to get
that for all $p\in M$ and $\varepsilon>0$, $|{\mathscr{A}}w(p)|\leqslant
C\|w\|_{C^{2}(M)}$. So dominated convergence allows us to pass the limit

$\lim_{\varepsilon\to
0}N_{\varepsilon}=-4\int_{M}u(p)\overline{{\mathscr{A}}_{b}w(p)}{\rm
dvol}_{g}(p).$

This combined with (3.4) shows that $\mathscr{A}_{b}$ is formally given by the
Dirichlet form ${\mathscr{D}}_{b}$. The fact that the nullspace of
${\mathscr{D}}_{b}$ consists of only constants comes from (3.3). ∎

###  3.2. Generator on the torus

In this section, we prove Theorem 1.4 holds, that is, $-\mathscr{A}$ coincides
with the fractional Laplacian $(-\Delta)^{\alpha}$ on
$\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$, which is defined on the
Fourier basis by

$(-\Delta)^{\alpha}e_{k}=|2\pi k|^{2\alpha}e_{k},\quad k\in\mathbb{Z}^{n},$

where $e_{k}:x\mapsto\exp 2i\pi\langle k,x\rangle.$

###### Proof of Theorem 1.4.

If $M=\mathbb{T}^{n}$, the generator of the Lévy process is given by

(3.5)
$\mathscr{A}f(x)=\dfrac{C(n,\alpha)}{2}\int_{\mathbb{R}^{n}}\left(\tilde{f}(x+v)+\tilde{f}(x-v)-2\tilde{f}(x)\right)\mathrm{d}v/|v|^{n+2\alpha}.$

Here, we set $\tilde{f}=f\circ\pi_{\mathbb{T}^{n}}$ where
$\pi_{\mathbb{T}^{n}}:\mathbb{R}^{n}\to\mathbb{R}^{n}/\mathbb{Z}^{n}$ is the
natural projection. Next, it follows from [ST10, Lemma 5.1] that for every
Schwartz function $g\in\mathscr{S}(\mathbb{R}^{n}),$

(3.6)
$(-\Delta_{\mathbb{R}^{n}})^{\alpha}\tilde{g}(x)=-\dfrac{C(n,\alpha)}{2}\int_{\mathbb{R}^{n}}\left(g(x+v)+g(x-v)-2g(x)\right)\mathrm{d}v/|v|^{n+2\alpha},$

where
$(-\Delta_{\mathbb{R}^{n}})^{\alpha}:\mathcal{S}(\mathbb{R}^{n})\to\mathcal{S}(\mathbb{R}^{n})$
is the fractional Laplacian on $\mathbb{R}^{n}$, defined by

(3.7)
$\mathcal{F}\bigl{(}{(-\Delta_{\mathbb{R}^{n}})^{\alpha}g}\bigr{)}(\xi)=|\xi|^{2\alpha}\mathcal{F}(g)(\xi),\quad
g\in\mathcal{S}(\mathbb{R}^{n}).$

Here $\mathcal{F}$ is the Fourier transform. Since
$(-\Delta_{\mathbb{R}^{n}})^{\alpha}$ is formally self-adjoint, it extends as
an operator from $\mathcal{S}^{\prime}(\mathbb{R}^{n})$ to itself. Moreover,
the right hand-side of (3.6) defines a continuous operator
$C^{\infty}_{b}(\mathbb{R}^{n})\to C(\mathbb{R}^{n})$ and by density of
$\mathcal{S}(\mathbb{R}^{n})$ in $\mathcal{S}^{\prime}(\mathbb{R}^{n})$, this
implies

(3.8) $-\mathscr{A}f=(-\Delta_{\mathbb{R}^{n}})^{\alpha}\tilde{f}$

by (3.5). Now we write $f=\sum_{k\in\mathbb{Z}^{n}}c_{k}e_{k}$ so that

(3.9) $\mathcal{F}{\tilde{f}}=\sum_{k}c_{k}\delta_{2\pi k}$

where the equality holds in $\mathscr{S}^{\prime}(\mathbb{R}^{n})$ and for
$\xi\in\mathbb{R}^{n}$, $\delta_{\xi}$ is the Dirac distribution at $\xi$.
Then

$-\mathscr{A}f=\sum_{k\in\mathbb{Z}^{n}}|2\pi k|^{2\alpha}c_{k}e_{k}$

by (3.8) and (3.9), and the right-hand side of this equality is precisely
$(-\Delta)^{\alpha}f$, which concludes the proof. ∎

###  3.3. Generator on Anosov manifolds

In this section we prove Theorem 1.6 and deduce some of their consequences.
For the remainder of this section we assume that $(M,g)$ is a Riemannian
manifold whose geodesic flow is Anosov as defined in §2.3.

####  3.3.1. Smoothing properties of averaging along the geodesic flow

Let $a\in C^{\infty}(\mathbb{R})$ be supported in $(0,+\infty)$ and assume
that all derivatives $a^{(k)}$, $k\geqslant 0$, are integrable. We define the
operator $R_{a}:C^{\infty}(SM)\to\mathcal{D}^{\prime}(SM)$ by

$(R_{a}f)(z):=\int_{0}^{\infty}f(\phi_{t}(z))a(t)\mathrm{d}t.$

We will prove

######  Theorem 3.2.

Assume that $(M,g)$ is as above an Anosov manifold. Let $a\in
C^{\infty}(\mathbb{R})$ be supported in $(0,+\infty)$. Also assume that all
derivatives $a^{(k)}$, $k\geqslant 0$, are integrable. Then
$\pi_{\ast}R_{a}\pi^{\ast}\in\Psi^{-\infty}(M)$ is a well defined smoothing
operator on $\mathcal{D}^{\prime}(M)$.

In fact, the exponential decay of correlations for Anosov manifolds [NZ15],
implies that the result is also true in the case that $a(t)=e^{\varepsilon
t}b(t)$, where $\varepsilon$ is small enough, and $b$ has all its derivatives
bounded. Before we start the proof proper, we recall the following classical
result of hyperbolic dynamics. Let $\varepsilon>0$ be fixed small enough. For
$z\in SM$, let $U_{z}$ be the ball of radius $\varepsilon$ centered at $z$ of
$W^{s}_{\mathrm{loc}}(z)\subset SM$. The following holds:

######  Lemma 3.2.

Let $f\in C^{\infty}(SM)$. Then, the map

$f_{z}:U_{z}\times(-1,+\infty)\owns(w,\tau)\mapsto f(\phi_{\tau}(w))$

is smooth, with $C^{k}$ bounds ($k\geqslant 0$), independent of $z$. More
precisely, the following holds: fix $z_{0}\in SM$ and consider a family of
vector fields $S_{1},...,S_{n-1}\in C^{\infty}_{\mathrm{cs}}(SM,E_{s})$
spanning locally $E_{s}$ near $z_{0}$. Then for all
$\beta\in\mathbb{N}^{n-1},j\geqslant 0$,

$\sup_{z\in W^{u}_{\mathrm{loc}}(z_{0})}\sup_{(w,\tau)\in
U_{z}\times(-1,+\infty)}|S^{\beta}_{w}\partial^{j}_{\tau}f_{z}(w,\tau)|\leqslant
C(\alpha,j)<\infty.$

We refer the reader to [dlLMM86, §2] for more details.

###### Proof.

We can now turn to the proof, divided into three steps.

_Step 1: the operator $\pi_{\ast}R_{a}\pi^{\ast}:C^{\infty}(M)\to
C^{\infty}(M)$ is continuous._ First, using Hörmander’s wavefront set rules
[Hör15, §8.2], one has that for $f\in\mathcal{D}^{\prime}(M)$, ${\rm
WF}(\pi^{\ast}f)\subset{\mathbf{V}}^{\perp}$ (that is, the pullback of a
function is constant in the fibers of $SM$, hence smooth in the direction of
the fibers – in other words, its singularities are conormal to it), and for
$u\in\mathcal{D}^{\prime}(SM)$,

$\displaystyle{\rm WF}(\pi_{*}u)\subset\\{(p,\xi)\in T^{*}M\setminus\\{0\\}\
|\ \exists v\in S_{p}M,\
(p,v,\underbrace{\mathrm{d}_{p}\pi^{\top}(\xi)}_{\in{\mathbf{V}}^{\perp}})\in{\rm
WF}(u)\\},$

(that is, the singularities in the direction of the fibers are killed by
integrating over it). Hence, it suffices to show that for $f\in
C^{\infty}(M)$, we have ${\rm WF}(R_{a}\pi^{*}f)\subset E_{s}^{*}$. Indeed,
this would imply that $\pi_{*}R_{a}\pi^{*}f\in C^{\infty}(M)$ according to
(2.15).

For this, we can invoke Lemma 2.3: its content is that ${\rm
WF}(R_{a}\pi^{*}f)\subset E_{s}^{*}$ provided we can prove that for each $z\in
SM$, the restriction of $R_{a}\pi^{*}f$ to a piece of local weak stable
manifold $W^{s0}_{\mathrm{loc}}(z)$ is smooth, with derivatives bounded
uniformly in $z$. Then, we observe that for any point $z$, we can rewrite

$(R_{a}\pi^{\ast}f)_{z}(w,\tau)=\int_{0}^{+\infty}(\pi^{\ast}f)_{z}(w,\tau+t)a(t)\mathrm{d}t.$

Differentiating under the integral and using Lemma 3.3.1, we deduce that
$R_{a}\pi^{\ast}f$ is indeed smooth along every weak-stable leaf, uniformly in
the leaf.

_Step 2: the operator
$\pi_{\ast}R_{a}\pi^{\ast}:\mathcal{D}^{\prime}(M)\to\mathcal{D}^{\prime}(M)$
is continuous with respect to the weak* topology on
${\mathcal{D}}^{\prime}(M)$._ Since
$\pi_{\ast}R_{a}\pi^{\ast}:C^{\infty}(M)\to C^{\infty}(M)$ is bounded, the
operator

$(\pi_{\ast}R_{a}\pi^{\ast})^{\top}:\mathcal{D}^{\prime}(M)\to\mathcal{D}^{\prime}(M)$

is also bounded. But $\pi_{\ast}R_{a}\pi^{\ast}$ is formally self-adjoint,
that is, $(\pi_{\ast}R_{a}\pi^{\ast})^{\top}=\pi_{\ast}R_{a}\pi^{\ast}$, which
proves the claim.

_Step 3: the operator $\pi_{\ast}R_{a}\pi^{\ast}:\mathcal{D}^{\prime}(M)\to
C^{\infty}(M)$ is bounded._ Equivalently, this means that the kernel of
$\pi_{\ast}R_{a}\pi^{\ast}$ is smooth, or that its wavefront set is empty. In
semi-classical terms, using the characterization (2.3), this is equivalent to
proving that for any smooth functions $S_{j}$, $\chi_{j}$, $j=1,2$, on $M$,
with

$dS_{j}\neq 0\text{ on }\mathrm{supp}(\chi_{j}),$

we have as $h\to 0$

$\langle
e^{iS_{1}/h}\chi_{1},\pi_{\ast}R_{a}\pi^{\ast}(e^{iS_{2}/h}\chi_{2})\rangle=\mathcal{O}(h^{\infty}).$

Rewriting this as

$\langle
e^{i\tilde{S}_{1}/h}\tilde{\chi}_{1},R_{a}(e^{i\tilde{S}_{2}/h}\tilde{\chi}_{2})\rangle=\mathcal{O}(h^{\infty}),$

with $\tilde{S}_{j}=\pi^{\ast}S_{j}$, and
$\tilde{\chi}_{j}=\pi^{\ast}\chi_{j}$, $j=1,2$, and using a partition of
unity, it suffices to prove the following.

######  Lemma 3.2.

For $j=1,2$, given any $z_{j}\in SM$, for any smooth $\psi_{j},\theta_{j}$
supported in a small ball near $z_{j}$, with
$d\theta_{j}\in{\mathbf{V}}^{\perp}\setminus\\{0\\}$ on the support of
$\psi_{j}$, we have the estimate

$\langle
e^{i\theta_{1}/h}\psi_{1},R_{a}(e^{i\theta_{2}/h}\psi_{2})\rangle=\mathcal{O}(h^{\infty})$

###### Proof of Lemma 3.3.1.

We start by observing that since
$d\theta_{j}\in{\mathbf{V}}^{\perp}\setminus\\{0\\}$, for some
$\varepsilon>0$,

(3.10) $\|(d\theta_{1})_{|{E_{s}}\oplus E^{0}}\|\geqslant\varepsilon.$

In particular, for some $T>0$, for all $t>T$

(3.11)
$\|d(\theta_{2}\circ\phi_{t})_{|{E_{s}}}\|\leqslant\frac{1}{2}\|(d\theta_{1})_{|{E_{s}}\oplus
E^{0}}\|.$

Let $\eta\in C^{\infty}_{c}(\mathbb{R}^{+})$ be a cutoff function, with
$\eta=1$ in $[0,T]$, and $\eta=0$ in $[T+1,+\infty)$. We can cut
$R_{a}=R_{a\eta}+R_{a(1-\eta)}$.

For the $R_{a\eta}$ term, we write the contribution to our scalar product

$\begin{split}\langle
e^{i\theta_{1}/h}\psi_{1},&R_{a\eta}(e^{i\theta_{2}/h}\psi_{2})\rangle\\\
&=\int_{0}^{T+1}a(t)\eta(t)\int_{SM}e^{(i/h)(\theta_{2}\circ\phi_{t}-\theta_{1})}\psi_{1}\psi_{2}\circ\phi_{t}\end{split}$

This is an oscillatory integral, with phase
$\theta_{2}\circ\phi_{t}-\theta_{1}$. It is stationary in the $SM$ variable if
$d\theta_{2}\circ d\phi_{t}=d\theta_{1}$. However, since
$\phi_{t}({\mathbf{V}}^{\perp})\cap{\mathbf{V}}^{\perp}\cap(E^{0})^{\perp}=\\{0\\}$
by (2.16), this is only possible at a point where both $d\theta_{1}$ and
$d\theta_{2}$ belong to $E^{0}_{\ast}$. In such a case, the phase is non
stationary in the $t$ variable. It follows from usual non-stationary phase
estimates that

$\langle
e^{i\theta_{1}/h}\psi_{1},R_{a\eta}(e^{i\theta_{2}/h}\psi_{2})\rangle=\mathcal{O}(h^{\infty}).$

Here it was important that we used all variables (stable, unstable and time).

Let us now concentrate on the $R_{a(1-\eta)}$ contribution. According to
(3.11), it is now sufficient to integrate in the weak stable direction. We can
use the absolute continuity of the weak stable foliation discussed at the end
of §2.3. Using (2.18), we can write

(3.12) $\displaystyle\langle
e^{i\theta_{1}/h}\psi_{1},R_{a(1-\eta)}(e^{i\theta_{2}/h}\psi_{2})\rangle$
$\displaystyle\hskip 56.9055pt=\int_{z\in
W^{u}_{\mathrm{loc}}(z_{1})}\Biggl{(}\underset{=I_{z}}{\underbrace{\int_{W^{s0}_{\mathrm{loc}}(z)}\overline{e^{i\theta_{1}/h}\psi_{1}}R_{a(1-\eta)}(e^{i\theta_{2}/h}\psi_{2})\mathrm{d}\mathcal{L}_{z}}}\Biggr{)}\mathrm{d}\mu(z),$

where $\mathcal{L}_{z}$ is a smooth volume measure along
$W^{s0}_{\mathrm{loc}}(z)$ and $\mu$ is a smooth measure along
$W^{u}_{\mathrm{loc}}(z_{1})$. We now work directly on
$W^{s0}_{\mathrm{loc}}(z)$ and introduce the coordinates
$\kappa_{z}(w,\tau)=\phi_{\tau}(w)\in W^{s0}_{\mathrm{loc}}(z)$ for
$(w,\tau)\in U_{z}\times(-\varepsilon,\varepsilon)$. Note that we can further
identify $U_{z}$ with an open subset $U\subset\mathbb{R}^{n-1}$. We have:

$\begin{split}I_{z}=\int\limits_{U\times(-\varepsilon,\varepsilon)\times\mathbb{R}^{+}}e^{\frac{i}{h}\left(\theta_{2}\circ\kappa_{z}(w,\tau+t)-\theta_{1}\circ\kappa_{z}(w,\tau)\right)}&a(t)(1-\eta(t))\\\
&\psi_{2}\circ\kappa_{z}(w,\tau+t)\overline{\psi_{1}\circ\kappa_{z}(w,\tau)}\rho_{z}(w,\tau)\mathrm{d}w\mathrm{d}\tau\mathrm{d}t.\end{split}$

We observe that the above expression for $I_{z}$ is of the form

$I_{z}=\int\limits_{U\times(-\varepsilon,\varepsilon)\times\mathbb{R}^{+}}e^{i\Psi(w,\tau,t)/h}b(w,\tau,t)\mathrm{d}w\mathrm{d}\tau\mathrm{d}t$

which is still an oscillatory integral, with phase

$\Psi=\left(\theta_{2}\circ\kappa_{z}(w,\tau+t)-\theta_{1}\circ\kappa_{z}(w,\tau)\right)$

and smooth amplitude $b(w,\tau,t)$ which is supported on $\\{(w,\tau,t)\mid
t\geqslant T\\}$. Note that, although not indicated in the notation, $\Psi$
and $b$ both depend on $z$ in a continuous fashion (and similarly for all
their derivatives, as follows from (2.18)). Moreover, by Lemma 3.3.1, all the
derivatives of $\Psi$ are uniformly bounded on
$U\times(-\varepsilon,\varepsilon)\times\mathbb{R}^{+}$ and all the
derivatives of $b$ are smooth and integrable on
$U\times(-\varepsilon,\varepsilon)\times\mathbb{R}^{+}$. Combining (3.10) and
(3.11), we see that for some $\delta>0$,

$\|(d\theta_{2}\circ d\phi_{t})_{|E^{0}}\|>\delta,\text{ or
}\|(d\theta_{2}\circ d\phi_{t}-d\theta_{1})_{|{E_{s}}\oplus E^{0}}\|>\delta,$

implying $|d\Psi|>\delta$. We then apply a usual non-stationary phase argument
to conclude that $I_{z}=O(h^{\infty})$ and therefore by (3.12)

$\langle
e^{i\theta_{1}/h}\psi_{1},R_{a(1-\eta)}(e^{i\theta_{2}/h}\psi_{2})\rangle=O(h^{\infty}).$

∎

This concludes the proof of Theorem 3.2. ∎

####  3.3.2. Proof of Theorem 1.6

###### Proof of Theorem 1.6.

For property i), let $r_{\mathrm{inj}}$ be the injectivity radius of $(M,g)$
and let $\chi(t)=1$ for $|t|<r_{\mathrm{inj}}^{2}/4$ and $\chi(t)=0$ for
$|t|>r_{\mathrm{inj}}^{2}/2$. We then have that
$\mathscr{A}=\mathscr{A}_{1}+\mathscr{A}_{2}$ where

(3.13)
$\displaystyle\mathscr{A}_{1}u(p):=C(n,\alpha)\mathrm{p.v.}\int\limits_{v\in
T_{p}M\setminus
0}\chi(|v|_{g}^{2})\frac{\left(u(\exp_{p}(v))-u(p)\right)}{|v|_{g}^{n+2\alpha}}\mathrm{d}T_{p}(v)$
(3.14) $\displaystyle\ \mathscr{A}_{2}u(p):=C(n,\alpha)\int\limits_{v\in
T_{p}M\setminus
0}\left(1-\chi(|v|_{g}^{2})\right)\frac{\left(u(\exp_{p}(v))-u(p)\right)}{|v|_{g}^{n+2\alpha}}\mathrm{d}T_{p}(v)$

where $\mathrm{d}T_{p}(v)$ is the volume form on $T_{p}M$ for the metric
$g_{p}$. Note that both $\mathscr{A}_{1}$ and $\mathscr{A}_{2}$ are formally
self-adjoint due to Proposition 3.1.

For $\mathscr{A}_{1}$, since $|v|^{2}_{g}<r^{2}_{inj}/2$ we can perform a
change of coordinate $q=\exp_{p}(v)$ so that

(3.15)
$\displaystyle\mathscr{A}_{1}u(p)=\mathrm{p.v.}\int_{M}\chi(\mathrm{dist}_{g}(p,q)^{2})\frac{\left(u(q)-u(p)\right)}{\mathrm{dist}_{g}(p,q)^{n+2\alpha}}J(p,q){\rm
dvol}_{g}(q)$

where $J(p,q)$ is the Jacobian determinant of the map
$q\mapsto\exp_{p}^{-1}(q)$. Note that $J(p,q)$ is jointly smooth for
$\mathrm{dist}_{g}(p,q)^{2}<r^{2}_{inj}/2$ and $J(p,p)=1$.

For $\mathscr{A}_{2}$, use polar coordinates to write

(3.16) $\displaystyle\mathscr{A}_{2}u(p)$ $\displaystyle=$
$\displaystyle\int_{v\in
S_{p}M}\int_{0}^{\infty}\left(1-\chi(t^{2})\right)\frac{\left(u(\exp_{p}(tv))-u(p)\right)}{t^{1+2\alpha}}\mathrm{d}t\mathrm{d}S^{n-1}(v)$
$\displaystyle=$
$\displaystyle\left(\pi_{*}R_{a}\pi^{*}u\right)(p)-u(p)\left(\pi_{*}R_{a}\pi^{*}1\right)(p)$

for $a(t)=(1-\chi(t^{2}))t^{-1-2\alpha}$. By Theorem 3.2,
$\pi_{*}R_{a}\pi^{*}:{\mathcal{D}}^{\prime}(M)\to C^{\infty}(M)$ so the proof
of Theorem 1.6 is complete. ∎

We conclude this subsection with a statement about the microlocal properties
of $\mathscr{A}^{+}$.

######  Lemma 3.2.

Let $\mathscr{L}\in\Psi^{-2\alpha}_{\mathrm{cl}}(M)$ be the operator given by
the integral kernel

(3.17)
$-C(n,-\alpha)\frac{\chi(\mathrm{dist}_{g}(p,q)^{2})}{\mathrm{dist}_{g}(p,q)^{n-2\alpha}}$

where $\chi\in C^{\infty}_{c}(\mathbb{R})$ is $1$ near $0$ and $0$ outside of
$(-r_{\mathrm{inj}}^{2}/2,r_{\mathrm{inj}}^{2}/2)$. Then

$\mathscr{A}^{+}=\mathscr{L}+\mathscr{R}$

with
$\mathscr{R}\in\Psi_{\mathrm{cl}}^{-2\alpha-1}(M)+\Psi^{-4\alpha}_{\mathrm{cl}}(M)$
.

###### Proof.

The operator $\mathscr{A}$ has Schwartz kernel given by (1.6) so is therefore
in $\Psi^{2\alpha}_{\mathrm{cl}}(M)+\Psi^{0}_{\mathrm{cl}}(M)$ with principal
symbol $\sigma_{2\alpha}(\mathscr{A})=-c(n,\alpha)|\eta|_{g}^{2\alpha}$. So by
standard parametrix construction [Tay13] one has that
$\mathscr{A}^{+}\in\Psi^{-2\alpha}_{\mathrm{cl}}(M)+\Psi^{-4\alpha}_{\mathrm{cl}}(M)$
with principal symbol $-|\eta|_{g}^{-2\alpha}$. The principal symbol of
$\mathscr{L}\in\Psi_{\mathrm{cl}}^{-2\alpha}(M)$ is also
$-|\eta|_{g}^{-2\alpha}$ as it can be verified by a direct computation. So we
have that
$\mathscr{R}:=\mathscr{A}^{+}-\mathscr{L}\in\Psi_{\mathrm{cl}}^{-2\alpha-1}(M)+\Psi^{-4\alpha}_{\mathrm{cl}}(M)$.
∎

###  3.4. Generator on the sphere

Compared to that of Anosov manifolds, the case of the sphere is easier to
compute but yields an operator which is more complicated.

####  3.4.1. Proof of Theorem 1.5

This is given by direct computation. Let $\chi\in C^{\infty}_{c}(\mathbb{R})$
take the value $1$ when $|t|\leqslant\left(\pi/8\right)^{2}$ and value $0$ for
$|t|\geqslant\left(\pi/4\right)^{2}$ and define $\chi_{1},\chi_{2},\chi_{3}\in
C^{\infty}(\mathbb{S}^{n}\times\mathbb{S}^{n})$ by

(3.18)
$\displaystyle\quad\chi_{1}(p,q):=\chi(\mathrm{dist}_{g}(p,q)^{2}),\quad\chi_{2}(p,q):=\chi(\mathrm{dist}_{g}(p,-q)^{2}),\quad
p,q\in\mathbb{S}^{n},$

and $\chi_{3}:=1-\chi_{1}-\chi_{2}.$ We now write

(3.19) $\displaystyle C(n,\alpha)^{-1}\mathscr{A}u(p)$
$\displaystyle=\mathrm{p.v.}\int_{v\in T_{p}M\setminus
0}\chi_{1}(p,\exp_{p}(v))\frac{\left(u(\exp_{p}(v)-u(p)\right)}{|v|_{g}^{n+2\alpha}}\mathrm{d}T_{p}(v)$
$\displaystyle+\int_{v\in T_{p}M\setminus
0}\chi_{2}(p,\exp_{p}(v))\frac{\left(u(\exp_{p}(v)-u(p)\right)}{|v|_{g}^{n+2\alpha}}\mathrm{d}T_{p}(v)$
$\displaystyle+\int_{v\in T_{p}M\setminus
0}\chi_{3}(p,\exp_{p}(v))\frac{\left(u(\exp_{p}(v)-u(p)\right)}{|v|_{g}^{n+2\alpha}}\mathrm{d}T_{p}(v).$

For $j=1,2,3$, denote by $\mathcal{I}_{j}(p)$ the integral in the right-hand
side of (3.19) involving $\chi_{j}$. Observe that $\chi_{1}(p,\exp_{p}(v))=0$
whenever

$|v|_{g}\notin[0,\pi/4]\cup\bigcup_{k=1}^{\infty}[2\pi k-\pi/4,2\pi k+\pi/4].$

So for $\mathcal{I}_{1}u(p)$, we write, using spherical coordinates on
$T_{p}S^{n}$,

$\displaystyle\mathcal{I}_{1}u(p)$ $\displaystyle=$
$\displaystyle\mathrm{p.v.}\int\limits_{v\in
S_{p}(\mathbb{S}^{n})}\int_{0}^{\infty}\chi_{1}\left(p,\exp_{p}(tv)\right)\frac{\left(u(\exp_{p}(tv))-u(p)\right)}{t^{1+2\alpha}}\mathrm{d}t\mathrm{d}{S^{n-1}}(v)$
$\displaystyle=$ $\displaystyle\mathrm{p.v.}\int\limits_{v\in
S_{p}(\mathbb{S}^{n})}\int_{0}^{\pi/4}\chi_{1}\left(p,\exp_{p}(tv)\right)\frac{\left(u(\exp_{p}(tv))-u(p)\right)}{t^{1+2\alpha}}\mathrm{d}t\mathrm{d}{S^{n-1}}(v)$
$\displaystyle+$ $\displaystyle\sum\limits_{k=0}^{\infty}\int\limits_{v\in
S_{p}(\mathbb{S}^{n})}\int\limits_{7\pi/4+2\pi k}^{9\pi/4+2\pi
k}\chi_{1}\left(p,\exp_{p}(tv)\right)\frac{\left(u(\exp_{p}(tv))-u(p)\right)}{t^{1+2\alpha}}\mathrm{d}t\mathrm{d}{S^{n-1}}(v).$

Observe that each individual term of $\mathcal{I}_{1}$ is formally self-
adjoint by Proposition 3.1. Shifting the $t$ integral via a change of variable
we get

$\displaystyle\mathcal{I}_{1}u(p)$ $\displaystyle=$
$\displaystyle\mathrm{p.v.}\int\limits_{v\in
S_{p}(\mathbb{S}^{n})}\int_{0}^{\pi/4}\chi_{1}\left(p,\exp_{p}(tv)\right)\frac{\left(u(\exp_{p}(tv))-u(p)\right)}{t^{1+2\alpha}}\mathrm{d}t\mathrm{d}{S^{n-1}}(v)$
$\displaystyle+$ $\displaystyle\sum\limits_{k=1}^{\infty}\int\limits_{v\in
S_{p}(\mathbb{S}^{n})}\int\limits_{-\pi/4}^{\pi/4}\chi_{1}\left(p,\exp_{p}(tv)\right)\frac{\left(u(\exp_{p}(tv))-u(p)\right)}{(2\pi
k+t)^{1+2\alpha}}\mathrm{d}t\mathrm{d}{S^{n-1}}(v).$

Now we make the change of variable $q=\exp_{p}(tv)$ so that
$t=\operatorname{sgn}(t)\mathrm{dist}_{g}(p,q)$ for $t\in(-1/4\pi,1/4\pi)$,
and $J(p,q)={\rm det}\left(D\exp_{p}^{-1}\mid_{q}\right)$ is the Jacobian of
the change of variable, we get

$\displaystyle\mathcal{I}_{1}u(p)$ $\displaystyle=$
$\displaystyle\mathrm{p.v.}\int_{M}\chi_{1}\left(p,q\right)\frac{\left(u(q)-u(p)\right)}{\mathrm{dist}_{g}(p,q)^{n+2\alpha}}J(p,q){\rm
dvol}_{g}(q)$ $\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{\infty}\int_{M}\chi_{1}\left(p,q\right)\frac{\left(u(q)-u(p)\right)J(p,q)}{(2\pi
k-\mathrm{dist}_{g}(p,q))^{1+2\alpha}(\mathrm{dist}_{g}(p,q))^{n-1}}{\rm
dvol}_{g}(q)$ $\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{\infty}\int_{M}\chi_{1}\left(p,q\right)\frac{\left(u(q)-u(p)\right)J(p,q)}{(2\pi
k+\mathrm{dist}_{g}(p,q))^{1+2\alpha}(\mathrm{dist}_{g}(p,q))^{n-1}}{\rm
dvol}_{g}(q).$

The sum converges absolutely. So the expression $C(n,\alpha)\mathcal{I}_{1}u$
has leading term

$u\mapsto
C(n,\alpha)\,\mathrm{p.v.}\int_{M}\chi_{1}\left(p,q\right)\frac{\left(u(q)-u(p)\right)}{\mathrm{dist}_{g}(p,q)^{n+2\alpha}}J(p,q){\rm
dvol}_{g}(q)$

which is an elliptic classical pseudodifferential operator with principal
symbol $-|\eta|^{2\alpha}$. So we see that the first term of (3.19) is

$C(n,\alpha)\mathcal{I}_{1}=\mathscr{A}_{2\alpha}+\mathscr{A}_{0}$

for some formally selfadjoint $\mathscr{A}_{2\alpha}\in\Psi^{2\alpha}_{cl}(M)$
and $\mathscr{A}_{0}\in\Psi^{0}_{cl}(M)$. With the principal symbol of
$\mathscr{A}_{2\alpha}$ being $-|\eta|_{g}^{2\alpha}$.

Denote by $-p$ the antipodal point of $p\in\mathbb{S}^{n}$. Similar
calculation as for the case of $\mathcal{I}_{1}$ using $\exp_{-p}(tv)$ for
$v\in S_{-p}(\mathbb{S}^{n})$ yields that $\mathcal{I}_{2}$ and
$\mathcal{I}_{3}$ are formally selfadjoint and given by

(3.21) $\displaystyle\mathcal{I}_{2}u(p)$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{\infty}\int_{M}\chi_{2}\left(-p,q\right)\frac{\left(u(q)-u(p)\right)J(-p,q)}{(2\pi
k+\pi-\mathrm{dist}_{g}(-p,q))^{1+2\alpha}(\mathrm{dist}_{g}(-p,q))^{n-1}}{\rm
dvol}_{g}(q)$ $\displaystyle+$
$\displaystyle\sum\limits_{k=0}^{\infty}\int_{M}\chi_{2}\left(-p,q\right)\frac{\left(u(q)-u(p)\right)J(-p,q)}{(2\pi
k+\pi+\mathrm{dist}_{g}(-p,q))^{1+2\alpha}(\mathrm{dist}_{g}(-p,q))^{n-1}}{\rm
dvol}_{g}(q)$

and

$\displaystyle\mathcal{I}_{3}u(p)$ $\displaystyle=$
$\displaystyle\int\limits_{v\in T_{p}M\setminus
0}\chi_{3}(p,\exp_{p}(v))\frac{\left(u(\exp_{p}(v)-u(p)\right)}{|v|_{g}^{n+2\alpha}}\mathrm{d}T_{p}(v)$
$\displaystyle=$ $\displaystyle\int_{M}k(p,q)(u(q)-u(p)){\rm dvol}_{g}(q)$

for some $k(\cdot,\cdot)\in C^{\infty}(M\times M)$ satisfying
$k(-p,q)=k(p,-q)$.

The infinite sum in (3.21) is absolutely convergent. Let $\mathscr{J}$ be the
operator defined by pulling back by the antipodal map. A local coordinate
calculation using normal coordinates yields that it is of the form
$\mathscr{A}_{0}u+\mathscr{J}\mathscr{A}_{-1}u$ for some
$\mathscr{A}_{-1}\in\Psi^{-1}_{\mathrm{cl}}(M)$ formally selfadjopint with
principal symbol $-c(n)|\eta|^{-1}$ and
$\mathscr{A}_{0}\in\Psi^{0}_{\mathrm{cl}}(M)$.

Finally since $k(p,q)$ is smooth, (3.4.1) is of the form
$\mathscr{A}_{-\infty}u+\mathscr{A}_{0}u$ for
$\mathscr{A}_{-\infty}\in\Psi^{-\infty}(M)$ and
$\mathscr{A}_{0}\in\Psi^{0}_{\mathrm{cl}}(M)$.

All operators commute with $\mathscr{J}$ since
$\mathrm{dist}_{g}(-p,q)=\mathrm{dist}_{g}(p,-q)$, $J(p,q)=J(q,p)$,
$J(-p,q)=J(p,-q)$ by symmetry of the sphere.

Inserting the conclusions about (3.4.1), (3.21), and (3.4.1) into (3.19) we
have the assertion of Theorem 1.5.

####  3.4.2. Microlocal Properties of $\mathscr{A}$ and its inverse on
$\mathbb{S}^{n}$

We now construct a (partial) parametrix for the operator $\mathscr{A}$:

######  Lemma 3.2.

There exists operators
$\mathscr{K}\in\Psi^{-2\alpha}_{\mathrm{cl}}(\mathbb{S}^{n})+\Psi_{\mathrm{cl}}^{-4\alpha}(\mathbb{S}^{n})$
and

$\mathscr{B}\in\Psi_{\mathrm{cl}}^{-1-4\alpha}(\mathbb{S}^{n})+\Psi_{\mathrm{cl}}^{-1-6\alpha}(\mathbb{S}^{n})+\Psi^{-1-8\alpha}_{\mathrm{cl}}(\mathbb{S}^{n})$

such that

(3.23)
$(\mathscr{K}+\mathscr{B}\mathscr{J})\mathscr{A}=I+\Psi_{\mathrm{cl}}^{-2-4\alpha}(\mathbb{S}^{n})+\Psi_{\mathrm{cl}}^{-2-6\alpha}(\mathbb{S}^{n})+\Psi^{-2-8\alpha}_{\mathrm{cl}}(\mathbb{S}^{n}).$

The operator $\mathscr{K}$ has principal symbol $-|\eta|_{g}^{-2\alpha}$ and
$\mathscr{B}=-\mathscr{K}\mathscr{A}_{-1}\mathscr{K}$ has principal symbol
$\tilde{c}(n,\alpha)|\eta|_{g}^{-1-4\alpha}$ for some nonvanishing constant
$\tilde{c}(n,\alpha)$.

###### Proof.

Let
$\mathscr{K}\in\Psi^{-2\alpha}_{\mathrm{cl}}(\mathbb{S}^{n})+\Psi_{\mathrm{cl}}^{-4\alpha}(\mathbb{S}^{n})$
be a parametrix for the elliptic pseudodifferential operator
$\mathscr{A}_{2\alpha}+\mathscr{A}_{0}$. We then have that
$\mathscr{K}\in\Psi^{-2\alpha}_{\mathrm{cl}}(\mathbb{S}^{n})+\Psi_{\mathrm{cl}}^{-4\alpha}(\mathbb{S}^{n})$
with principal symbol $-|\eta|_{g}^{-2\alpha}$. Now set

$\mathscr{B}=-\mathscr{K}\mathscr{A}_{-1}\mathscr{K}\in\Psi^{-1-4\alpha}(\mathbb{S}^{n})+\Psi_{\mathrm{cl}}^{-1-6\alpha}(\mathbb{S}^{n})+\Psi^{-1-8\alpha}_{\mathrm{cl}}(\mathbb{S}^{n}),$

with principal symbol $c(n,\alpha)|\eta|_{g}^{-1-4\alpha}$. We have, using the
commuting property of $\mathscr{J}$ with $\mathscr{A}_{2\alpha}$,
$\mathscr{A}_{0}$ and $\mathscr{A}_{-1}$,

$\displaystyle(\mathscr{K}+\mathscr{B}\mathscr{J})\mathscr{A}$
$\displaystyle=(\mathscr{K}+\mathscr{B}\mathscr{J})(\mathscr{A}_{2\alpha}+\mathscr{A}_{0}+\mathscr{J}\mathscr{A}_{-1})$
$\displaystyle=I+\mathscr{B}\mathscr{A}_{-1}+\Psi^{-\infty}(\mathbb{S}^{n}).$

Simple book keeping asserts that

$\mathscr{B}\mathscr{A}_{-1}\in\Psi^{-2-4\alpha}(\mathbb{S}^{n})+\Psi_{\mathrm{cl}}^{-2-6\alpha}(\mathbb{S}^{n})+\Psi^{-2-8\alpha}_{\mathrm{cl}}(\mathbb{S}^{n})$

and we obtain (3.23). To compute the principal symbol of $\mathscr{B}$ one
simply use standard principal symbol calculus and the fact that
$\mathscr{A}_{-1}$ has principal symbol $c(n)|\eta|_{g}^{-1}$ as given in
Theorem 1.5. ∎

We have the following microlocal structure for $\mathscr{A}^{+}$:

######  Lemma 3.2.

Let $\mathscr{K}+\mathscr{B}\mathscr{J}$ be the partial parametrix for
$\mathscr{A}$ constructed in Lemma 3.4.2. Then

$\mathscr{A}^{+}=\left(\mathscr{K}+\mathscr{B}\mathscr{J}\right)+\mathscr{S}_{1}+\mathscr{J}\mathscr{S}_{2}$

where $\mathscr{S}_{1}$ is a finite sum of classical $\Psi$DOs of order
$-2-6\alpha$ or less, $S_{2}$ is a sum of classical $\Psi$DOs of order
$-3-8\alpha$ or less.

###### Proof.

We write

(3.24)
$\displaystyle\left(\mathscr{A}^{+}-\left(\mathscr{K}+\mathscr{B}\mathscr{J}\right)\right)\mathscr{A}\in\Psi_{\mathrm{cl}}^{-2-4\alpha}(\mathbb{S}^{n})+\Psi_{\mathrm{cl}}^{-2-6\alpha}(\mathbb{S}^{n})+\Psi^{-2-8\alpha}_{\mathrm{cl}}(\mathbb{S}^{n})$

Taking the adjoint of (3.23) we have

$\mathscr{A}(\mathscr{K}^{*}+\mathscr{J}^{*}\mathscr{B}^{*})=I+\mathscr{M}$

with

$\mathscr{M}\in\Psi_{\mathrm{cl}}^{-2-4\alpha}(\mathbb{S}^{n})+\Psi_{\mathrm{cl}}^{-2-6\alpha}(\mathbb{S}^{n})+\Psi^{-2-8\alpha}_{\mathrm{cl}}(\mathbb{S}^{n}).$

So if we hit (3.24) on the left with
$(\mathscr{K}^{*}+\mathscr{J}^{*}\mathscr{B}^{*})$, we get that

$\left(\mathscr{A}^{+}-\left(\mathscr{K}+\mathscr{B}\mathscr{J}\right)\right)\left(I+\mathscr{M}\right)=\mathscr{S}_{1}+\mathscr{J}\mathscr{S}_{2}$

where $\mathscr{S}_{1}$ is a finite sum of classical $\Psi$DOs of order
$-2-6\alpha$ or less, $\mathscr{S}_{2}$ is a sum of classical $\Psi$DOs of
order $-3-8\alpha$ or less. Hit the above expresson on the right by the
parametrix of $(I+\mathscr{M})$ we have the result. ∎

######  Corollary 3.2.

Let $\mathscr{L}\in\Psi^{-2\alpha}_{\mathrm{cl}}(M)$ be the operator given by
the integral kernel (3.17) where $\chi\in C^{\infty}_{c}(\mathbb{R})$ is $1$
near $0$ and $0$ outside of
$(-r_{\mathrm{inj}}^{2}/2,r_{\mathrm{inj}}^{2}/2)$. Then

$\mathscr{A}^{+}-\mathscr{L}=\mathscr{S}_{1}+\mathscr{J}\mathscr{S}_{2}$

where $\mathscr{S}_{2}$ is some pseudodifferential operator and
$\mathscr{S}_{1}$ is an element in $\Psi^{-1-2\alpha}_{\mathrm{cl}}(M)$ plus a
finite sum of classical pseudodifferential operators each of which is of order
$-4\alpha$ or less.

###  3.5. Mapping properties of the generator

In this paragraph we prove Corollary 1.2.5 about the mapping properties of
$\mathscr{A}$.

###### Proof of Corollary 1.2.5.

For __(i)__ , we first show that the null-space of $\mathscr{A}$ in
$W^{s,m}(\mathbb{S}^{n})$ consists of only constants. Suppose $\mathscr{A}u=0$
for some $u\in W^{s,m}(\mathbb{S}^{n})$ with $s\in\mathbb{R}$. Then apply
Lemma 3.4.2 we get that $(I+K)u=0$ for some $K:W^{s,m}(\mathbb{S}^{n})\to
W^{s+2+4\alpha,m}(\mathbb{S}^{n})$ for all $s\in\mathbb{R}$. This means that
$u\in C^{\infty}(\mathbb{S}^{n})$. By Proposition 1.3, $u$ is constant.

To see that $\mathscr{A}:W^{s,m}(\mathbb{S}^{n})\to
W^{s-2\alpha,m}(\mathbb{S}^{n})$ has finite dimensional cokernel, simply
observe that $\mathscr{A}$ is formally self-adjoint and use the fact that the
null-space of $\mathscr{A}$ consists of only constants.

To prove the assertion about discreteness of spectrum, simply observe that
both $K$ and the resolvent of $\mathscr{A}$ are both compact operators from
$W^{s,m}(M)\to W^{s,m}(M)$. The fact that eigenfunctions are smooth comes from
the partial parametrix constructed in Lemma 3.23.

For __(ii)__ , the construction of the operator $\mathscr{A}^{+}$ follows from
__(i)__ and basic functional analysis. ∎

##  4\. Expected Stopping Time for Random Search

The aim of this section is to prove Theorem 1.8 and Theorem 1.10.

###  4.1. Fundamental properties of the expected stopping time

Recall that, if $\Omega\subset\mathbb{R}^{n}$ is a bounded smooth domain,
$(X_{t})_{t\geqslant 0}$ denotes the Brownian motion and $\tau$ denotes the
exit time of $\Omega$ for this process, then the expected exit time

$u(p):=\mathbb{E}(\tau~{}|~{}X_{0}=p)$

solves the equation $\Delta u=-1$ in $\Omega$ with Dirichlet boundary
condition $u|_{\partial\Omega}=0$, see [Tay13, Chapter 11] for instance. We
claim that the expected stopping time for our Lévy flight satisfies similar
properties as the ones connecting the exit time of the Brownian motion to the
Dirichlet Laplacian.

######  Proposition 4.1.

If $(M,g)$ is the round sphere, the flat torus, or Anosv, the expected
stopping time satisfies the following properties ($1<m<1/\alpha$):

  1. _(i)_

In the Anosov and flat torus case,

$u_{\varepsilon}\in L^{\infty}(M)\cap C^{\infty}(M\setminus\partial
B_{\varepsilon}(p_{0}))\cap W^{2\alpha,m}(\overline{\Omega_{\varepsilon}}),$

while in the sphere case

$u_{\varepsilon}\in L^{\infty}(M)\cap C^{\infty}(M\setminus\partial
B_{\varepsilon}(\pm p_{0}))\cap
W^{2\alpha,m}(\overline{\Omega_{\varepsilon}}),$

for all $\varepsilon>0$.

  2. _(ii)_

One has the fundamental relation:

(4.1) $\mathscr{A}u_{\varepsilon}=-1,\text{ on
}\Omega_{\varepsilon}:=M\setminus\overline{B_{\varepsilon}(p_{0})},\qquad
u_{\varepsilon}=0,\text{ on }\overline{B_{\varepsilon}(p_{0})}.$

This is very similar to the equation satisfied by the expected exit time for
the Brownian motion. Note, however, that due to the nonlocality of the
generator $\mathscr{A}$, the boundary condition $u=0$ on $\partial\Omega$ in
the Laplacian case has to be replaced here by $u_{\varepsilon}=0$ on
$\overline{B_{\varepsilon}(p_{0})}$. This will actually create a lot of
troubles in the proofs and showing (4.1) will actually require some effort.

Using (4.1), we can introduce $F_{\varepsilon}\in\mathcal{D}^{\prime}(M)$ such
that

(4.2)
$\mathscr{A}u_{\varepsilon}=F_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}}.$

By construction, $F_{\varepsilon}$ is a distribution supported in
$\overline{B_{\varepsilon}(p_{0})}$. An important idea will be to study the
properties of $F_{\varepsilon}$ (and not that of $u_{\varepsilon}$ directly),
and then to deduce properties from (4.2) properties for the expected stopping
time $u_{\varepsilon}$.

Before stating the result, we need to introduce some notation. First, we
introduce rescaled geodesic coordinates centred at $p_{0}$: let
$E_{1},\dots,E_{n}\in T_{p_{0}}M$ be a orthonormal basis and define
$\psi_{\varepsilon}:{\mathbb{B}^{n}}\to B_{\varepsilon}(p_{0})$ by

(4.3) $\displaystyle\psi_{\varepsilon}(x):=\exp_{p_{0}}(\varepsilon
x_{1}E_{1}+\dots+\varepsilon x_{n}E_{n}).$

For $m\in[1,\infty]$, recall from §2.1.3 that
$\dot{L}^{m}(\overline{B_{\varepsilon}(p_{0})})$ denotes the space of
functions $f$ such that $f\in L^{m}(M)$ and
$\mathrm{supp}(f)\subset\overline{B_{\varepsilon}(p_{0})}$.

The following holds:

######  Proposition 4.2.

The distribution $F_{\varepsilon}\in\mathcal{D}^{\prime}(M)$ satisfies:

  1. _(i)_

$\mathrm{supp}(F_{\varepsilon})\subset\overline{B_{\varepsilon}(p_{0})}$ and
$F_{\varepsilon}\in C^{\infty}(M\setminus\partial B_{\varepsilon}(p_{0}))$,

  2. _(ii)_

$u_{\varepsilon}=\mathscr{A}^{+}(F_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}})+C_{\varepsilon}$
where

(4.4) $\begin{split}C_{\varepsilon}&:=|M|^{-1}\int_{M}u_{\varepsilon}(p){\rm
dvol}_{g}(p)=\varepsilon^{2\alpha-n}|M|c(n,\alpha)(1+\mathcal{O}(E(\alpha,\varepsilon))),\end{split}$

where $c(n,\alpha)$ is given by (1.8) and the error term
$E(\alpha,\varepsilon)$ is

(4.5) $\displaystyle
E(\alpha,\varepsilon)=\begin{cases}\varepsilon^{2\alpha},\quad&\text{if}\
\alpha<1/2,\\\ \varepsilon|\log\varepsilon|,\quad&\text{if}\ \alpha=1/2,\\\
\max(\varepsilon,\varepsilon^{n-2\alpha}),\quad&\text{if}\
\alpha>1/2.\end{cases}$

  3. _(iii)_

$F_{\varepsilon}\in\dot{L}^{m}(\overline{B_{\varepsilon}(p_{0})})\cap
C^{\infty}(B_{\varepsilon}(p_{0}))$ for all $m\in(1,1/\alpha)$ and in the
coordinate system (4.3), $F_{\varepsilon}$ has expansion

(4.6)
$F_{\varepsilon}(\psi_{\varepsilon}(x))=-\frac{|M|}{\varepsilon^{n}}\left(\int_{\mathbb{B}^{n}}\dfrac{\mathrm{d}x}{(1-|x|^{2})^{\alpha}}\right)^{-1}\left(\dfrac{1}{(1-|x|^{2})^{\alpha}}+\mathcal{O}_{\dot{L}^{m}(\overline{\mathbb{B}^{n}})}(E(\alpha,\varepsilon))\right).$

######  Remark 4.2.

If $(M,g)$ is $\mathbb{T}^{n}$, then the error term (4.5) can be replaced by
(1.11).

Although the statements may sound natural, the proof of Propositions 4.1 and
4.2 is actually involved, due to the nonlocality of the generator
$\mathscr{A}$ which causes trouble understanding the analytic properties of
$u_{\varepsilon},F_{\varepsilon}$ on $\partial B_{\varepsilon}(p_{0})$. We
will mainly follow the strategy of [Get61] which deals with a similar problem
in a simpler setting where $\mathscr{A}$ is the fractional Laplacian in
$\mathbb{R}^{n}$. Since they are technical, the proofs are deferred to §5
below.

###  4.2. Proof of the stopping time Theorems

We first show how Theorems 1.8, 1.9 and 1.10 can be deduced from Propositions
4.1 and 4.2.

####  4.2.1. Proof of Theorems 1.8 and 1.9

We start with the Anosov case.

###### Proof of Theorem 1.8.

First of all, observe that, using Proposition 4.2, the following holds: let
$G\in{\mathcal{D}}^{\prime}(M)$ be smooth in a neighourhood of $p_{0}$, then
for all $\varepsilon>0$ sufficiently small,

(4.7) $\int_{B_{\varepsilon}(p_{0})}F_{\varepsilon}(q)G(q){\rm
dvol}_{g}(q)=|M|G(p_{0})+\mathcal{O}(E(\alpha,\varepsilon)),$

where $E(\alpha,\varepsilon)$ is given by (4.5).

We now fix $p\neq p_{0}$ assume that $\varepsilon>0$ is sufficiently small
such that $p\notin B_{\varepsilon}(p_{0})$. Since
$u_{\varepsilon}-C_{\varepsilon}=\mathscr{A}^{+}(F_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}})$,
we have:

$\displaystyle u_{\varepsilon}(p)-C_{\varepsilon}$ $\displaystyle=$
$\displaystyle\left(\mathscr{A}^{+}\mathbf{1}_{B_{\varepsilon}(p_{0})}\right)(p)+\int_{B_{\varepsilon}(p_{0})}\mathrm{G}_{\mathscr{A}}(p,q)F_{\varepsilon}(q){\rm
dvol}_{g}(q).$ $\displaystyle=$ $\displaystyle
O(\varepsilon^{n})+\int_{B_{\varepsilon}(p_{0})}\mathrm{G}_{\mathscr{A}}(p,q)F_{\varepsilon}(q){\rm
dvol}_{g}(q).$

Since $\mathscr{A}^{+}$ is a pseudodifferential operator,
$\mathrm{G}_{\mathscr{A}}(p,q)$ is smooth away from the the set $\\{p=q\\}$.
Since we have taken $\varepsilon>0$ so that $p\notin B_{\varepsilon}(p_{0})$,
$F_{\varepsilon}(q)$ is integrated against a smooth function of $q$. We now
use the expansion produced in (4.7) to get

$u_{\varepsilon}(p)-C_{\varepsilon}=|M|\mathrm{G}_{\mathscr{A}}(p,p_{0})+\mathcal{O}(E(\alpha,\varepsilon)).$

Recalling now that $\Omega_{\varepsilon}=M\setminus B_{\varepsilon}(p_{0})$
and $C_{\varepsilon}=|M|^{-1}\int_{M}u_{\varepsilon}(q){\rm dvol}_{g}(q)$ has
expansion given in Proposition 4.2 concludes the proof of Theorem 1.8. ∎

We now pass to the torus case.

###### Proof of Theorem 1.9.

We may redo exactly the proof of Theorem 1.8, and taking into account Remark
4.1 yields the right error term. ∎

####  4.2.2. Proof of Theorem 1.10

We now deal with the case of the sphere. We will need the preliminary lemma:

######  Lemma 4.2.

We have that for any $\mu>0$,

$\frac{1}{\mathrm{dist}_{g}(\psi_{\varepsilon}(x),\psi_{\varepsilon}(y))^{n-\mu}}=\frac{1}{\varepsilon^{n-\mu}|x-y|^{n-\mu}}+\varepsilon^{2}\frac{A\left(\varepsilon,x,|x-y|,\frac{x-y}{|x-y|}\right)}{\varepsilon^{n-\mu}|x-y|^{n-\mu}}$

for some smooth function
$A:[0,\varepsilon_{0})\times\mathbb{B}\times\mathbb{R}\times
S^{n-1}\to\mathbb{R}$.

###### Proof.

This is a direct consequence of [NTT21b, Corollary 2.3]. ∎

We can now proceed with the proof of Theorem 1.10.

###### Proof of Theorem 1.10.

For $p\notin\\{p_{0},-p_{0}\\}$ computing $u_{\varepsilon}(p)-C_{\varepsilon}$
goes verbatim as in proof of Theorem 1.8 by choosing $\varepsilon>0$ small
enough so that $p\notin B_{\varepsilon}(p_{0})\cup B_{\varepsilon}(-p_{0})$.
Therefore we will not repeat it here. The interesting part is computing
$u_{\varepsilon}(-p_{0})-C_{\varepsilon}$. To do so we first recall that by
Lemma 3.4.2,

$\mathscr{A}^{+}=\mathscr{K}+\mathscr{B}\mathscr{J}+\mathscr{S}_{1}+\mathscr{S}_{2}\mathscr{J}$

where $\mathscr{K}$ and $\mathscr{B}$ are described in Lemma 3.4.2 and
$\mathscr{S}_{1},\mathscr{S}_{2}$ are pseudodifferential operators which are
more regular.

######  Lemma 4.2.

The operator $\mathscr{B}$ is of the form

$\mathscr{B}=\mathscr{L}^{\prime}+\mathscr{R}^{\prime}$

where $\mathscr{L}^{\prime}$ has Schwartz kernel
$-\chi(\mathrm{dist}_{g}(p,q)^{2})\mathrm{dist}_{g}(p,q)^{-n+1+4\alpha}$ and
$\mathscr{R}^{\prime}$ is the sum of classical pseudodifferential operators of
order at most $\max(-4\alpha-2,-6\alpha-1)$. Furthermore, as $\varepsilon\to
0$,

(4.8)
$(\mathscr{L}^{\prime}\mathscr{J}F_{\varepsilon})(-p_{0})=\frac{-|M|}{\varepsilon^{n-1-4\alpha}}\left(\int_{\mathbb{B}^{n}}\frac{\mathrm{d}x}{(1-|x|^{2})^{\alpha}}\right)^{-1}\int_{0}^{1}\frac{r^{4\alpha}}{(1-r^{2})^{\alpha}}dr+o(\varepsilon^{-n+1+4\alpha}).$

###### Proof.

The expression for $\mathscr{B}$ is a consequence of the fact that
$\mathscr{L}^{\prime}$ is a classical pseudodifferential operator with
principal symbol $-|\eta|^{-1-4\alpha}_{g}$ which is the same as the principal
symbol of $\mathscr{B}$ as given by Lemma 3.4.2.

To obtain (4.8), use a rescaled geodesic coordinate system centred at $-p_{0}$
given by

$\tilde{\psi}_{\varepsilon}(x)=\exp_{-p_{0}}(\varepsilon
x_{1}E_{1}+\dots+\varepsilon x_{n}E_{n})$

for $E_{1},\dots,E_{n}\in T_{-p_{0}}\mathbb{S}^{n}$ an orthonormal set of
vectors over $-p_{0}$. Using Lemma 4.2.2 we see that the coordinate expression
for
$\displaystyle{-\frac{\chi(\mathrm{dist}_{g}(p,q)^{2})}{\mathrm{dist}_{g}(p,q)^{n-1-4\alpha}}}$
in these coordinates is:

$\frac{1}{\mathrm{dist}_{g}(\tilde{\psi}_{\varepsilon}(x),\tilde{\psi}_{\varepsilon}(y))^{n-1-4\alpha}}=\frac{1}{\varepsilon^{n-1-4\alpha}|x-y|^{n-1-4\alpha}}+\varepsilon^{2}\frac{A\left(\varepsilon,x,|x-y|,\frac{x-y}{|x-y|}\right)}{\varepsilon^{n-1-4\alpha}|x-y|^{n-1-4\alpha}}$

for some smooth function
$A:[0,\varepsilon_{0})\times\mathbb{B}\times\mathbb{R}\times
S^{n-1}\to\mathbb{R}$. So writing out
$\left(\mathscr{L}^{\prime}\mathscr{J}F_{\varepsilon}\right)(-p_{0})$ in these
coordinates and use the expression

$\psi_{\varepsilon}^{*}{\rm
dvol}_{g}=\varepsilon^{n}(1+\varepsilon^{2}Q_{\varepsilon}(x))\mathrm{d}x$

for the volume form with $Q_{\varepsilon}(x)$ smooth and uniformly bounded in
$\varepsilon$, we get

$\displaystyle\left(\mathscr{L}^{\prime}\mathscr{J}F_{\varepsilon}\right)(-p_{0})$
$\displaystyle=$
$\displaystyle-\varepsilon^{1+4\alpha}\int_{\mathbb{B}^{n}}\frac{(\mathscr{J}F_{\varepsilon})(\tilde{\psi}_{\varepsilon}(x))}{|x|^{n-1-4\alpha}}(1+\varepsilon^{2}Q_{\varepsilon}(x))\mathrm{d}x$
$\displaystyle-$
$\displaystyle\varepsilon^{3+4\alpha}\int_{\mathbb{B}^{n}}(\mathscr{J}F_{\varepsilon})(\tilde{\psi}_{\varepsilon}(x))\frac{A\left(\varepsilon,x,|x|,\frac{x}{|x|}\right)}{|x|^{n-1-4\alpha}}(1+\varepsilon^{2}Q_{\varepsilon}(x))\mathrm{d}x.$

Clearly
$(\mathscr{J}F_{\varepsilon})(\tilde{\psi}_{\varepsilon}(x))=F_{\varepsilon}(\psi_{\varepsilon}(x))$
where $\psi_{\varepsilon}$ is the rescaled geodesic coordinate (4.3) centred
at $p_{0}$. So we have

$\displaystyle\left(\mathscr{L}^{\prime}\mathscr{J}F_{\varepsilon}\right)(-p_{0})$
$\displaystyle=$
$\displaystyle-\varepsilon^{1+4\alpha}\int_{\mathbb{B}^{n}}\frac{F_{\varepsilon}(\psi_{\varepsilon}(x))}{|x|^{n-1-4\alpha}}(1+\varepsilon^{2}Q_{\varepsilon}(x))\mathrm{d}x$
$\displaystyle-$
$\displaystyle\varepsilon^{3+4\alpha}\int_{\mathbb{B}^{n}}F_{\varepsilon}(\psi_{\varepsilon}(x))\frac{A\left(\varepsilon,x,|x|,\frac{x}{|x|}\right)}{|x|^{n-1-4\alpha}}(1+\varepsilon^{2}Q_{\varepsilon}(x))\mathrm{d}x.$

Now, plugging the expansion of $F_{\varepsilon}(\psi_{\varepsilon}(x))$ given
by __(iii)__ in Proposition 4.2 into the above integral, we get

$\displaystyle\varepsilon^{1+4\alpha}\int_{\mathbb{B}^{n}}\frac{F_{\varepsilon}(\psi_{\varepsilon}(x))}{|x|^{n-1-4\alpha}}\mathrm{d}x=-\varepsilon^{1+4\alpha-n}|M|\tilde{c}(n,\alpha)-\int_{\mathbb{B}^{n}}\frac{O_{\dot{L}^{m}(\overline{\mathbb{B}})}\left(\varepsilon^{1-n+4\alpha}E(\alpha,\varepsilon)\right)}{|x|^{n-1-4\alpha}}\mathrm{d}x,$

where

$\tilde{c}(n,\alpha)=\left(\int_{\mathbb{B}^{n}}\frac{\mathrm{d}x}{(1-|x|^{2})^{\alpha}}\right)^{-1}.$

By our assumption that $1>(n-4)\alpha$, we can choose $m\in(1,1/\alpha)$ so
that the function $x\mapsto|x|^{-n+1+4\alpha}$ lies in
$L^{m^{\prime}}(\mathbb{B})$, where $1/m+1/m^{\prime}=1$. Now by Hölder
inequality we get (4.8), which concludes the proof. ∎

Inserting the expression for $\mathscr{A}^{+}$ obtained in Lemma 4.2.2 and
Lemma 3.4.2 into the equation
$u_{\varepsilon}-C_{\varepsilon}=\mathscr{A}^{+}\left(F_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}}\right)$,
we see that

(4.9) $\displaystyle u_{\varepsilon}(-p_{0})-C_{\varepsilon}$ $\displaystyle=$
$\displaystyle\left(\mathscr{A}^{+}\mathbf{1}_{B_{\varepsilon}}\right)(-p_{0})+\left(\mathscr{L}F_{\varepsilon}\right)(-p_{0})+\left(\mathscr{R}F_{\varepsilon}\right)(-p_{0})$
$\displaystyle+$
$\displaystyle\left(\mathscr{L}^{\prime}\mathscr{J}F_{\varepsilon}\right)(-p_{0})+\left(\mathscr{R}^{\prime}\mathscr{J}F_{\varepsilon}\right)(-p_{0})$

where $\mathscr{L}$ and $\mathscr{R}$ are pseudodifferential operators and
$\mathscr{R}^{\prime}$ is the sum of classical pseudodifferential operators of
order at most $\max(-4\alpha-2,-6\alpha-1)$.

The Schwartz kernel of $\mathscr{L}$ is
$\chi(\mathrm{dist}_{g}(p,q)^{2})\mathrm{dist}_{g}(p,q)^{-n+2\alpha}$ (see
Corollary 3.4.2) which vanishes for $(p,q)$ near $(-p_{0},p_{0})$. Since
$\mathscr{R}$ is a pseudodifferential operator, its Schwartz kernel is smooth
for $(p,q)$ near $(-p_{0},p_{0})$. By Lemma 4.2.2 the operator
$\mathscr{R}^{\prime}$ has a weaker singularity than $\mathscr{L}^{\prime}$.
So (4.9) becomes

$\displaystyle u_{\varepsilon}(-p_{0})-C_{\varepsilon}$ $\displaystyle=$
$\displaystyle\left(\mathscr{L}^{\prime}\mathscr{J}F_{\varepsilon}\right)(-p_{0})+o(\varepsilon^{-n+1+4\alpha})$

Now insert the formula (4.8) for
$\left(\mathscr{L}^{\prime}\mathscr{J}F_{\varepsilon}\right)(-p_{0})$ into the
above identity we get

$\displaystyle u_{\varepsilon}(-p_{0})-|M|^{-1}\int_{M}u_{\varepsilon}{\rm
dvol}_{g}=\frac{-|M|\tilde{c}(n,\alpha)}{\varepsilon^{n-1-4\alpha}}+o(\varepsilon^{-n+1+4\alpha})$

which is the identity (1.12). This completes the proof of Theorem 1.10. ∎

##  5\. Proof of the fundamental properties of the expected stopping time

The aim of this section is to prove Propositions 4.1 and 4.2.

###  5.1. Structure of the argument

Although stated this way, Propositions 4.1 and 4.2 do not reflect the
structure of the argument which is quite involved due to the technical issues
caused by the nonlocality of the generator $\mathscr{A}$. Basically, the main
issue is that we cannot show _directly_ that $\mathscr{A}u_{\varepsilon}=-1$
on $\Omega_{\varepsilon}$. The idea, somehow, is to revert the logic of the
argument. First, let us make some quick observations. Assuming that the
fundamental equation $\mathscr{A}u_{\varepsilon}=-1$ holds on
$\Omega_{\varepsilon}=M\setminus\overline{B_{\varepsilon}(p_{0})}$ and
$u_{\varepsilon}=0$ on $\overline{B_{\varepsilon}(p_{0})}$, that is, (4.2)
holds, we get by applying $\mathscr{A}^{+}$ to both sides of (4.2) that

$\mathscr{A}^{+}\mathscr{A}u_{\varepsilon}=u_{\varepsilon}-|M|^{-1}\int_{M}u_{\varepsilon}(p){\rm
dvol}_{g}(p)=\mathscr{A}^{+}(F_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}}),$

that is
$\mathscr{A}^{+}(F_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}})=u_{\varepsilon}-C_{\varepsilon}$.
Since $u_{\varepsilon}$ vanishes on $B_{\varepsilon}(p_{0})$, we thus get:

(5.1)
$\mathscr{A}^{+}(F_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}})=-C_{\varepsilon},\qquad\text{
on }B_{\varepsilon}(p_{0}).$

Moreover, integrating (4.2) with respect to ${\rm dvol}_{g}$, we get

(5.2) $\langle F_{\varepsilon},{\rm dvol}_{g}\rangle=|\Omega_{\varepsilon}|.$

The pair of equations (5.1) - (5.2) with unknowns
$(F_{\varepsilon},C_{\varepsilon})$ will be called the _integral equation_.
The argument then goes as follows:

  1. (1)

Existence and uniqueness to the integral equation. First, we _construct_ a
pair of solution $(\widetilde{F}_{\varepsilon},\widetilde{C}_{\varepsilon})$
to the integral equation (5.1) - (5.2) such that $\widetilde{F}_{\varepsilon}$
has support in $\overline{B_{\varepsilon}(p_{0})}$ and control precisely its
analytic properties, that is, show that it satisfies the content of
Proposition 4.2. More precisely, we will show:

######  Proposition 5.1 (Existence and uniqueness of regular solutions to the
integral equation).

Let $(M,g)$ be either Anosov, the flat torus, or the round sphere. For
$\varepsilon>0$ small enough, there _exists_ a _unique_
$\widetilde{F}_{\varepsilon}\in\dot{L}^{m}(\overline{B_{\varepsilon}(p_{0})})\cap
C^{\infty}(B_{\varepsilon}(p_{0}))$ with $m\in(1,1/\alpha)$ and constant
$\widetilde{C}_{\varepsilon}$ solving (5.1) - (5.2). Moreover,
$\widetilde{C}_{\varepsilon}$ satisfies the expansion (4.4) and
$\widetilde{F}_{\varepsilon}$ satisfies the expansion (4.6) in Proposition
4.2.

Recall here that the spaces $\dot{L}^{m}$ were introduced in §2.1.3. The proof
of Proposition 5.1 is the content of §5.2. The proof of Proposition 5.1 relies
on local argument involving Fourier analysis.

  2. (2)

Uniqueness of solutions to the fundamental equation. We then set

(5.3)
$\widetilde{u}_{\varepsilon}:=\mathscr{A}^{+}(\widetilde{F}_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}})+\widetilde{C}_{\varepsilon}.$

By construction, the distribution $\widetilde{u}_{\varepsilon}$ satisfies
$\mathscr{A}\widetilde{u}_{\varepsilon}=-1$ on $\Omega_{\varepsilon}$,
$\widetilde{u}_{\varepsilon}=0$ on $B_{\varepsilon}(p_{0})$. Moreover, as a
consequence of Proposition 5.1, $\widetilde{u}_{\varepsilon}$ lies in some
Sobolev space of positive regularity, that is,
$\widetilde{u}_{\varepsilon}\in\dot{W}^{2\alpha,m}(\overline{\Omega_{\varepsilon}})$.
We will show that the following uniqueness result holds:

######  Proposition 5.2 (Uniqueness of regular solutions to the fundamental
equation).

Let $(M,g)$ be the round sphere, the torus, or Anosov. Let
$w\in\dot{W}^{2\alpha,m}(\overline{\Omega_{\varepsilon}})$, $1<m<\infty$.
Assume that $\mathscr{A}w=-1$ on $\Omega_{\varepsilon}$. Then
$w=u_{\varepsilon}$.

This uniqueness result should be compared with [Get61, Corollary 5.1]. It will
be the content of §5.3. Proposition 5.2 will therefore imply that
$u_{\varepsilon}=\widetilde{u}_{\varepsilon}\in\dot{W}^{2\alpha,m}(\overline{\Omega_{\varepsilon}})$
satisfies the fundamental equation $\mathscr{A}u_{\varepsilon}=-1$ on
$\Omega_{\varepsilon}$. The idea behind Proposition 5.2 is to use the integral
representation (2.27), relating $u_{\varepsilon}$ to the generator of a
bounded semi-group on $\Omega_{\varepsilon}$.

The proofs of Propositions 4.1 and 4.2 are then straightforward, combining
both Proposition 5.1 and 5.2.

###### Proof of Proposition 4.1.

By the previous Propositions, we have that
$u_{\varepsilon}=\widetilde{u}_{\varepsilon}\in\dot{W}^{2\alpha,m}(\overline{\Omega_{\varepsilon}})$
and $u_{\varepsilon}$ satisfies the fundamental relation (4.1). Moreover, in
the case of Anosov manifolds or the torus, $u_{\varepsilon}\in
C^{\infty}(M\setminus\partial B_{\varepsilon}(p_{0}))$ follows from standard
elliptic regularity since $u_{\varepsilon}=0$ in $B_{\varepsilon}(p_{0})$,
$\mathscr{A}u_{\varepsilon}=-1$ on $\Omega_{\varepsilon}$ and $\mathscr{A}$ is
pseudodifferential elliptic. In the sphere case, we get similarly that
$u_{\varepsilon}\in C^{\infty}(M\setminus\partial B_{\varepsilon}(\pm p_{0}))$
by elliptic regularity of $\mathscr{A}$ (up to antipodal points). Eventually,
the proof that $u_{\varepsilon}\in L^{\infty}(M)$ is deferred to Corollary
5.3.1 below and will be a consequence of the representation formula (2.27) for
$u_{\varepsilon}$. ∎

###### Proof of Proposition 4.2.

It suffices to prove that $\widetilde{F}_{\varepsilon}=F_{\varepsilon}$, where
$F_{\varepsilon}$ is defined by (4.2) and $\widetilde{F}_{\varepsilon}$ solves
the integral equation (5.1) - (5.2), and similarly that
$\widetilde{C}_{\varepsilon}=C_{\varepsilon}$. But by definition of
$\widetilde{u}_{\varepsilon}$ in (5.3), and by (4.2), one has using
Proposition 5.2:

$\widetilde{u}_{\varepsilon}=\mathscr{A}^{+}(\widetilde{F}_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}})+\widetilde{C}_{\varepsilon}=u_{\varepsilon}=\mathscr{A}^{+}(F_{\varepsilon}-\mathbf{1}_{\Omega_{\varepsilon}})+C_{\varepsilon}.$

Integrating the previous equation over $M$ yields
$C_{\varepsilon}=\widetilde{C}_{\varepsilon}$. Hitting the previous equation
with $\mathscr{A}$ then yields $F_{\varepsilon}=\widetilde{F}_{\varepsilon}$.
This concludes the proof. ∎

###  5.2. Existence of solutions to the integral equation

The aim of this paragraph is to construct a solution to (5.1) and (5.2), that
is, to prove Proposition 5.1. The first step is to rewrite a bit (5.1) more
explicitly. For that, fix $p_{0}\in M$ and let
$\varepsilon\in(0,\frac{r_{\mathrm{inj}}}{10})$ be small. When $(M,g)$ is the
torus, Anosov, or the sphere, by Theorem 1.4 (torus), Lemma 3.3.2 (Anosov),
and Corollary 3.4.2 (sphere), $\mathscr{A}^{+}$ can be written as

$\mathscr{A}^{+}=\mathscr{L}+\mathscr{R}$

where $\mathscr{R}:C^{\infty}(M)\to{\mathcal{D}}^{\prime}(M)$ is such that the
map

$u\mapsto(\mathscr{R}u)|_{B_{\varepsilon}(p_{0})}$

with domain $\dot{W}^{s,m}(\overline{B_{\varepsilon}(p_{0})})$ can be
represented by a finite sum of classical pseudodifferential operators each of
which is of order at most $\max(-4\alpha,-2\alpha-1)$. In the case of the
torus, since $\mathscr{A}$ is just the fractional Laplacian,
$\mathscr{R}\in\Psi^{-1-2\alpha}_{cl}(M)$.

As a consequence, using this decomposition and that

$\mathscr{A}^{+}\mathbf{1}=0=\mathscr{A}^{+}(\mathbf{1}_{B_{\varepsilon}(p_{0})}+\mathbf{1}_{\Omega_{\varepsilon}}),$

we get

(5.4)
$(\mathscr{L}+\mathscr{R})(F_{\varepsilon}+\mathbf{1}_{B_{\varepsilon}(p_{0})})=-C_{\varepsilon},\qquad\text{
on }B_{\varepsilon}(p_{0}).$

Observe that both functions $F_{\varepsilon}$ and
$\mathbf{1}_{B_{\varepsilon}(p_{0})}$ are supported on the small ball
$\overline{B_{\varepsilon}(p_{0})}$. The first step is to study the
boundedness properties of $\mathscr{R}$ on functions supported on
$\mathbf{1}_{B_{\varepsilon}(p_{0})}$.

####  5.2.1. Boundedness properties of $\mathscr{R}$ on small balls

We characterize the mapping property of $\mathscr{R}$ on the small ball
$B_{\varepsilon}(p_{0})$ of radius $\varepsilon>0$. We will need the following
preliminary standard result:

######  Lemma 5.2.

For $\varepsilon\geqslant 0$, let $A_{\varepsilon}\in
C^{\infty}_{c}(\mathbb{R}^{n}_{x}\times\mathbb{R}_{r}\times S^{n-1}_{\omega})$
be a family of compactly supported functions, with support uniformly bounded
in $\varepsilon\in[0,\varepsilon_{0}]$ and whose derivatives are also
uniformly bounded in $\varepsilon$. Then, the family of operators
$T_{\varepsilon}$ given by

$T_{\varepsilon}f(x):=\int\limits_{\omega\in
S^{n-1}}\int_{0}^{\infty}A_{\varepsilon}(x,r,\omega)r^{l}f(x+r\omega)drd\omega$

is a family of pseudodifferential operators of order $-1-l$, with uniform
boundedness properties in terms of $\varepsilon$. In particular,

$T_{\varepsilon}:W^{s,m}(\mathbb{R}^{n})\to W^{s+l+1,m}(\mathbb{R}^{n})$

is bounded for all $s\in\mathbb{R},m\in(1,\infty)$, with uniform bounds in
$\varepsilon\geqslant 0$.

###### Proof.

Observe that for all $(x,\xi)\in T^{*}\mathbb{R}^{n}$,

$T_{\varepsilon}(e^{ix\cdot\xi})=e^{ix\cdot\xi}\sigma_{\varepsilon}(x,\xi),$

for

$\sigma_{\varepsilon}(x,\xi):=\int_{\omega\in
S^{n-1}}\int_{0}^{+\infty}A_{\varepsilon}(x,r,\omega)r^{l}e^{ir\omega\cdot\xi}drd\omega,$

which is compactly supported in the $x$-variable. It is immediate to check
that the following estimates hold: for all
$\beta\in\mathbb{N}^{n},\gamma\in\mathbb{N}^{n}$, there exists
$C:=C(\beta,\gamma)$ such that

$|\partial^{\beta}_{x}\partial^{\gamma}_{\xi}\sigma_{\varepsilon}(x,\xi)|\leqslant
C\langle\xi\rangle^{-1-l-|\gamma|},\qquad\forall(x,\xi)\in
T^{*}\mathbb{R}^{n},\quad\forall\varepsilon\geqslant 0.$

This proves that $(T_{\varepsilon})_{\varepsilon\geqslant 0}$ is indeed a
family of pseudodifferential operators with uniform estimates, see [GS94,
Theorem 3.4]. ∎

The previous lemma has the following consequence for the boundedness of
$\mathscr{R}$ on small balls:

######  Lemma 5.2.

Let
$R_{\varepsilon}:C^{\infty}_{c}(\mathbb{B})\to{\mathcal{D}}^{\prime}(\mathbb{B})$
be the coordinate representation of $\mathscr{R}$ in the coordinate system
(4.3), that is,
$R_{\varepsilon}f:=\psi_{\varepsilon}^{*}\mathscr{R}(\psi_{\varepsilon}^{-1})^{*}$.
Then we have the following estimate

(5.5)
$\displaystyle\|R_{\varepsilon}\|_{{\dot{L}}^{m}(\overline{\mathbb{B}})\to{\overline{W}}^{\min(2\alpha+1,4\alpha),m}(\mathbb{B})}\leqslant
C\varepsilon^{2\alpha}E(\alpha,\varepsilon)$

where $E(\alpha,\varepsilon)$ is defined as (4.5), and $C>0$ is some positive
constant.

###### Proof.

By assumption the operator
$u\mapsto(\mathscr{R}u)\mid_{B_{\varepsilon}(p_{0})}$ with domain
$\dot{W}^{s,m}(\overline{B_{\varepsilon}(p_{0})})$ can be represented by an
element of
$\Psi^{-2\alpha-1}_{\mathrm{cl}}(M)+\Psi^{-4\alpha}_{\mathrm{cl}}(M)$. So, by
[Tay96, Page 285, (8.45)], in the coordinate system given by (4.3), the
Schwartz kernel of $R_{\varepsilon}$ is given, if $\alpha\neq 1/2$, by

$\varepsilon^{2\alpha+1}\frac{A_{\varepsilon}\left(x,|x-y|,\frac{x-y}{|x-y|}\right)}{|x-y|^{n-2\alpha-1}}+\varepsilon^{4\alpha}\frac{B_{\varepsilon}\left(x,|x-y|,\frac{x-y}{|x-y|}\right)}{|x-y|^{n-4\alpha}}+\varepsilon^{n}\kappa(\varepsilon
x),$

and if $\alpha=1/2$ by

$\varepsilon^{2}\frac{A_{\varepsilon}\left(x,|x-y|,\frac{x-y}{|x-y|}\right)}{|x-y|^{n-2}}+\varepsilon^{2}P(\varepsilon
x,\varepsilon(x-y)),$

where $A_{\varepsilon}(x,r,\omega)$ and $B_{\varepsilon}(x,r,\omega)$ satisfy
the hypothesis of Lemma 5.2.1 and $\kappa$ is a smooth function. The functions
$P(x,z)$ satisfy $P(x,z)\sim\sum_{j\geqslant 0}P_{j}(x,z)\log|z|$ with each
$P_{j}(x,z)\in C^{\infty}({\mathbb{B}^{n}}\times\mathbb{R}^{n})$ a homogeneous
polynomial of degree $j$ in the $z$ variable. The notation
$P(x,z)\sim\sum_{j\geqslant 0}P_{j}(x,z)\log|z|$ means that for all
$k\in\mathbb{N}$ there exists $N\in\mathbb{N}$ such that

$P(x,z)-\sum_{j=0}^{N}P_{j}(x,z)\log|z|\in
C^{k}(\mathbb{B}\times\mathbb{R}^{n}).$

We can now apply Lemma 5.2.1 to obtain the desired estimate. ∎

####  5.2.2. Solving the integral equation

Our aim is to rewrite more explicitly (5.4). For that purpose, we introduce
the operator

(5.6) $L_{\alpha}u:=-\int_{\mathbb{B}^{n}}\frac{u(y)}{|x-y|^{n-2\alpha}}dy,$

defined for $u\in C^{\infty}(\overline{\mathbb{B}^{n}})$. Using the rescaled
normal coordinates (4.3), we set

(5.7)
$\displaystyle\tilde{F}_{\varepsilon}(x):=F_{\varepsilon}(\psi_{\varepsilon}(x)).$

The following holds:

######  Lemma 5.2.

Written in the geodesic normal coordinates (4.3), the equation (5.4) is
equivalent to

(5.8)
$(L_{\alpha}+R^{\prime}_{\varepsilon})(\tilde{F}_{\varepsilon}+\mathbf{1}_{\mathbb{B}^{n}})=-\dfrac{C_{\varepsilon}\varepsilon^{-2\alpha}}{C(n,-\alpha)}\mathbf{1}_{\mathbb{B}^{n}},\quad\text{
on }\mathbb{B},$

where the operator
$R^{\prime}_{\varepsilon}:\dot{L}^{m}(\mathbb{B})\to\overline{W}^{\min(2\alpha+1,4\alpha),m}(\mathbb{B})$
satisfies the bound:

(5.9)
$\|R^{\prime}_{\varepsilon}\|_{\dot{L}^{m}\to\overline{W}^{\min(2\alpha+1,4\alpha),m}}\leqslant
CE(\alpha,\varepsilon).$

###### Proof.

Using the decomposition $\mathscr{A}^{+}=\mathscr{L}+\mathscr{R}$, we can
rewrite (5.4) as

(5.10)
$\begin{split}-C(n,-\alpha)\int_{B_{\varepsilon}}\frac{1+F_{\varepsilon}(q)}{d_{g}(p,q)^{n-2\alpha}}&{\rm
dvol}_{g}(q)+\mathscr{R}(\mathbf{1}_{B_{\varepsilon}}+F_{\varepsilon})=-C_{\varepsilon}.\end{split}$

Observe that in the geodesic normal coordinates (4.3),

(5.11) $\displaystyle\psi_{\varepsilon}^{*}{\rm
dvol}_{g}(y)=\varepsilon^{n}(1+\varepsilon^{2}Q_{\varepsilon}(y))\mathrm{d}y$

for some smooth function $Q_{\varepsilon}$ whose derivatives are also
uniformly bounded in $\varepsilon>0$. The compatibility condition (5.2)
becomes

(5.12)
$\displaystyle\varepsilon^{n}\int_{\mathbb{B}^{n}}\tilde{F}_{\varepsilon}(x)(1+\varepsilon^{2}Q_{\varepsilon}(x))\mathrm{d}x=|\Omega_{\varepsilon}|$

In the geodesic normal coordinates, using Lemma 4.2.2 and the above formulas
for the change of volume, the terms of (5.10) have the following expression:

$-\int_{B_{\varepsilon}}\frac{1}{d_{g}(p,q)^{n-2\alpha}}{\rm
dvol}_{g}(q)=\varepsilon^{2\alpha}L_{\alpha}\mathbf{1}_{\mathbb{B}^{n}}+\varepsilon^{2+2\alpha}K_{\varepsilon}\mathbf{1}_{\mathbb{B}^{n}},$

and

$-\int_{B_{\varepsilon}}\frac{F_{\varepsilon}(q)}{d_{g}(p,q)^{n-2\alpha}}{\rm
dvol}_{g}(q)=\varepsilon^{2\alpha}L_{\alpha}\tilde{F}_{\varepsilon}+\varepsilon^{2+2\alpha}K_{\varepsilon}\tilde{F}_{\varepsilon},$

where $K_{\varepsilon}$ is the operator on the unit ball $\mathbb{B}$ defined
by:

$K_{\varepsilon}u(x):=-\int_{\mathbb{B}^{n}}\dfrac{Q_{\varepsilon}(y)+A_{\varepsilon}(x,|x-y|,x-y/|x-y|)(1+\varepsilon^{2}Q_{\varepsilon}(y))}{|x-y|^{n-2\alpha}}u(y)\mathrm{d}y.$

We then set

$R^{\prime}_{\varepsilon}:=\varepsilon^{2}K_{\varepsilon}+\dfrac{\varepsilon^{-2\alpha}R_{\varepsilon}}{C(n,-\alpha)},$

so that equality (5.8) is satisfied.

It remains to show that $R^{\prime}_{\varepsilon}$ satisfies the stated bound.
The term involving $R_{\varepsilon}$ yields the bound (5.9) by Lemma 5.2.1. As
to the term involving $K_{\varepsilon}$, using Lemma 5.2.1, is is easily seen
to be of size $\mathcal{O}(\varepsilon^{2+2\alpha})$ as a bounded operator
$\dot{L}^{m}(\mathbb{B})\to\overline{W}^{\min(2\alpha+1,4\alpha),m}(\mathbb{B})$,
and this is in turn a $o(E(\alpha,\varepsilon))$. ∎

We now show the following:

######  Lemma 5.2.

The operator
$L_{\alpha}:\dot{L}^{m}(\overline{\mathbb{B}})\to{\overline{W}}^{2\alpha,m}(\mathbb{B})$
defined by (5.6) is a continuous isomorphism for $m\in(1,1/\alpha)$. Hence,
for $m\in(1,1/\alpha)$, there exists a bounded operator
$L_{\alpha}^{-1}:{\overline{W}}^{2\alpha,m}(\mathbb{B})\to\dot{L}^{m}(\overline{\mathbb{B}})$
such that $L^{-1}_{\alpha}L_{\alpha}=L_{\alpha}L_{\alpha}^{-1}=\mathbf{1}$.

###### Proof.

The operator $L_{\alpha}$ has Schwartz kernel $|x-y|^{-n+2\alpha}$ and
therefore belongs to $\Psi^{-2\alpha}_{\mathrm{cl}}(\mathbb{R}^{n})$ with full
symbol $-c|\eta|^{-2\alpha}$ for some constant $c>0$ depending on $\alpha$. It
therefore satisfies the Hörmander transmission condition on $\mathbb{B}$ with
factorization index $-\alpha$, see [Gru15, Proposition 1 and Equation (1)].
The Fredholm mapping property is then stated in [Gru15, Theorem 2].

To see injectivity, suppose $u\in\dot{L}^{m}(\overline{\mathbb{B}})$ satisfies
$L_{\alpha}u=0$. But [Kah81, Theorem 2.4] (for dimension $2$) and [Kah83,
Theorem 4.3] (for higher dimensions) then asserts that $u=0$.

To see surjectivity, recall that the dual space of
$\overline{W}^{2\alpha,m}(\mathbb{B})$ is
$\dot{W}^{-2\alpha,m^{\prime}}(\overline{\mathbb{B}})$, see discussion before
equation (1.5) and equation (1.8) of [Gru15]. So suppose
$u\in\dot{W}^{-2\alpha,m^{\prime}}(\overline{\mathbb{B}})$ is orthogonal to
the range

$L_{\alpha}\left(\dot{L}^{m}(\overline{\mathbb{B}})\right)\subset\overline{W}^{2\alpha,m}(\mathbb{B}).$

Then by the fact that $L_{\alpha}$ is formally self-adjoint, we have that
$L_{\alpha}u=0$. The operator $L_{\alpha}$ is type $-\alpha$ with respect to
$\mathbb{B}$, see definition in [Gru15, Proposition 1 and Equation (1)]. Then
[Gru15, Theorem 2] forces that $u(x)=(1-|x|^{2})^{-\alpha}v(x)$ for some $v\in
C^{\infty}(\overline{\mathbb{B}})$. Since $\alpha\in(0,1)$ we have that $u\in
L^{1}(\mathbb{B})$. Then [Kah81, Theorem 2.4] (for dimension $2$) and [Kah83,
Theorem 4.3] (for higher dimensions) forces $u=0$. ∎

We now complete the proof of Proposition 5.1.

###### Proof of Proposition 5.1.

First of all, assume that we have a solution
$(F_{\varepsilon},C_{\varepsilon})$ to (5.1) - (5.2) satisfying the properties
of Proposition 5.1. By Lemma 5.2.2, this is the same as having (5.8). Hitting
(5.8) with $L_{\alpha}^{-1}$, we then get:

$(\mathbf{1}+L_{\alpha}^{-1}R^{\prime}_{\varepsilon})(\tilde{F}_{\varepsilon}+\mathbf{1}_{\mathbb{B}^{n}})=-\dfrac{C_{\varepsilon}\varepsilon^{-2\alpha}}{C(n,-\alpha)}L_{\alpha}^{-1}\mathbf{1}_{\mathbb{B}^{n}}$

Observe that by Lemmas 5.2.2 and 5.2.2,
$L_{\alpha}^{-1}R^{\prime}_{\varepsilon}:\dot{L}^{m}(\overline{\mathbb{B}^{n}})\to\dot{L}^{m}(\overline{\mathbb{B}^{n}})$
is a bounded operator with norm $\mathcal{O}(E(\alpha,\varepsilon))=o(1)$. As
a consequence, we can invert
$\mathbf{1}+L_{\alpha}^{-1}R^{\prime}_{\varepsilon}$ by Neumann series and
this yield:

(5.13)
$\tilde{F}_{\varepsilon}+\mathbf{1}_{\mathbb{B}^{n}}=-\dfrac{C_{\varepsilon}\varepsilon^{-2\alpha}}{C(n,-\alpha)}(\mathbf{1}+L_{\alpha}^{-1}R^{\prime}_{\varepsilon})^{-1}L_{\alpha}^{-1}\mathbf{1}_{\mathbb{B}^{n}}.$

Integrating with respect to the measure
$\varepsilon^{n}(1+\varepsilon^{2}Q_{\varepsilon}(x))dx$ on $\mathbb{B}$
(rescaled Riemannian measure on the ball, see (5.11)), and using the
compatibility condition (5.2), we get

$|M|=-\dfrac{C_{\varepsilon}\varepsilon^{-2\alpha}}{C(n,-\alpha)}\int_{\mathbb{B}^{n}}\left((\mathbf{1}+L_{\alpha}^{-1}R^{\prime}_{\varepsilon})^{-1}L_{\alpha}^{-1}\mathbf{1}_{\mathbb{B}^{n}}\right)(x)\varepsilon^{n}(1+\varepsilon^{2}Q_{\varepsilon}(x))dx,$

that is

(5.14)
$C_{\varepsilon}=-\dfrac{\varepsilon^{2\alpha-n}|M|C(n,-\alpha)}{D_{\varepsilon}},$

where

$D_{\varepsilon}:=\int_{\mathbb{B}^{n}}\left((\mathbf{1}+L_{\alpha}^{-1}R^{\prime}_{\varepsilon})^{-1}L_{\alpha}^{-1}\mathbf{1}_{\mathbb{B}^{n}}\right)(x)(1+\varepsilon^{2}Q_{\varepsilon}(x))dx.$

Hence, (5.13) and (5.14) show that if such a regular solution
$(F_{\varepsilon},C_{\varepsilon})$ to (5.1) and (5.2) exists, then it must be
unique.

Now, we define $(F_{\varepsilon},C_{\varepsilon})$ by (5.13) and (5.14). By
construction, they satisfy (5.1) and (5.2) so all that remains to be shown is
that they enjoy the asymptotic expansions of Proposition 4.2. We start with
$C_{\varepsilon}$. By [Kah81, Theorem 3.1] (in dimension 2) and [Kah83,
Theorem 5.1] (in dimension $n\geqslant 3$) we have that

(5.15)
$\displaystyle(L^{-1}_{\alpha}\mathbf{1}_{\mathbb{B}^{n}})(x)=-c_{\alpha}(1-|x|^{2})^{-\alpha},$

where

$c_{\alpha}=\begin{cases}\pi^{-2}\sin((1-\alpha)\pi),\quad\text{if
}\dim(\mathbb{B})=2,\vspace{0.2cm}\\\
\displaystyle\dfrac{(n-2)\sin((1-\alpha)\pi)\Gamma(n/2-\alpha)}{\pi^{n/2+1}\Gamma(1-\alpha)}\int_{0}^{1}r^{n-3}(1-r^{2})^{-\alpha}~{}\mathrm{d}r,\quad\text{if
}\dim(\mathbb{B})\geqslant 3.\end{cases}$

Hence, inserting (5.15) in the expression of (5.14), we easily get that

(5.16)
$C_{\varepsilon}=\varepsilon^{2\alpha-n}|M|C(n,-\alpha)c_{\alpha}^{-1}\left(\int_{\mathbb{B}^{n}}\dfrac{\mathrm{d}x}{(1-|x|^{2})^{\alpha}}\right)^{-1}(1+\mathcal{O}(E(\alpha,\varepsilon))).$

We eventually claim that the constant $c(n,\alpha)$ introduced in (1.8) is
given by

$c(n,\alpha)=\frac{C(n,-\alpha)}{c_{\alpha}}\left(\int_{\mathbb{B}^{n}}\dfrac{\mathrm{d}x}{(1-|x|^{2})^{-\alpha}}\right)^{-1},$

Then, (5.16) is the content of (4.4).

Then, inserting the expansion of $C_{\varepsilon}$ into (5.13), using that

$(\mathbf{1}+L_{\alpha}^{-1}R^{\prime}_{\varepsilon})^{-1}=\mathbf{1}+\mathcal{O}_{\dot{L}^{m}\to\dot{L}^{m}}(\|L_{\alpha}^{-1}R^{\prime}_{\varepsilon}\|_{\dot{L}^{m}\to\dot{L}^{m}})=\mathbf{1}+\mathcal{O}_{\dot{L}^{m}\to\dot{L}^{m}}(E(\alpha,\varepsilon)),$

and the expression (5.15) for $L^{-1}_{\alpha}\mathbf{1}_{\mathbb{B}^{n}}$, we
get (4.6). ∎

###  5.3. Uniqueness of regular solutions to the fundamental equation

The aim of this subsection is to prove Proposition 5.2. For that, we will use
the fact that the expected stopping time
${u}_{\varepsilon}(p)=\mathbb{E}(\tau_{\varepsilon}\mid X_{0}=p)$ admits a
nice integral representation as explained in §2.4, that is, there exists a
measurable (in all variables) function $\mathbf{k}$ on $[0,+\infty)\times
M\times M$ so that

(5.17)
$u_{\varepsilon}=\int_{0}^{+\infty}\int_{\Omega_{\varepsilon}}\mathbf{k}(t,\cdot,q)\mathrm{dvol}_{g}(q).$

We claim that the following holds:

######  Proposition 5.3.

For all $t>0$, $\mathbf{k}(t,\cdot,\cdot)\in
L^{\infty}(\Omega_{\varepsilon}\times\Omega_{\varepsilon})$. Moreover, if
$\mathscr{A}^{\prime}$ is the generator of the associated semi-group
$(T_{t})_{t\geqslant 0}$, we have $D(\mathscr{A})\cap
L^{m}(\Omega_{\varepsilon})\subset D(\mathscr{A}^{\prime})\cap
L^{m}(\Omega_{\varepsilon})$ for every $1\leqslant m<\infty$. In particular,
for $1<m<\infty$ and
$u\in\dot{W}^{2\alpha,m}(\overline{\Omega}_{\varepsilon})$, one has
$\mathscr{A}^{\prime}u=\mathbf{1}_{\Omega_{\varepsilon}}\mathscr{A}u$.

The proof of Proposition 5.2 will then be a direct consequence of Proposition
5.3, see the end of §5.3.2.

####  5.3.1. Integral representation of the expected stopping time

Recall from §2.4 that $\mathbf{p}$ is the kernel of the semi-group
$(U_{t})_{t\geqslant 0}$ introduced in §2.4. We start by observing

######  Lemma 5.3.

If $(M,g)$ is Anosov, $\mathbb{T}^{n}$, or $\mathbb{S}^{n}$, then
$\mathbf{p}(t,\cdot,\cdot)\in C^{\infty}(M\times M)$ for all $t>0$.

###### Proof.

According to [Str72, Theorem 1] applied with $\nu=1$, if $P_{1}=P$ is an
elliptic self-adjoint non-negative pseudo-differential operator and
$m_{t}(x):=e^{-tx}$, we deduce that for any $a>0$ and $N>0$, $e^{-tP}$ is a
pseudodifferential operator of order $-N$, uniformly in $t\geqslant a$. This
means that the Schwartz kernel
$K_{P}(t,\cdot,\cdot)\in\mathcal{D}^{\prime}(M\times M)$ of $e^{-tP}$
satisfies (in every local coordinate patch) the usual estimates for
pseudodifferential operators of order $-N<-n$666For instance, there exists
$C>0$ such that:  $\sup_{t\geqslant a,p,q\in M}|K_{P}(t,p,q)|\leqslant
C<\infty.$ , and uniformly in time $t\geqslant a$. This implies that the
kernel $K_{P}(t,\cdot,\cdot)$ of $e^{-tP}$ is smooth, uniformly if $t\geqslant
a$.

In the case of the torus or Anosov manifold, $-\mathscr{A}$ satisfies the
assumptions above. For the sphere, we can still apply the result. Indeed,
recall from Theorem 1.5

$\mathscr{A}=\mathscr{A}_{2\alpha}+\mathscr{A}_{0}+\mathscr{J}\mathscr{A}_{-1}$

where $\mathscr{A}_{2\alpha}\in\Psi^{2\alpha}_{\mathrm{cl}}(M)$,
$\mathscr{A}_{0}\in\Psi^{0}_{\mathrm{cl}}(M)$, and
$\mathscr{A}_{-1}\in\Psi^{-1}_{\mathrm{cl}}(M)$ all commute with
$\mathcal{J}$. We use $\mathscr{J}$ to split

$L^{2}(M)=L^{2}_{\mathrm{odd}}(M)\oplus L^{2}_{\mathrm{even}}(M)$

with projections $P_{\mathrm{even}}$ and $P_{\mathrm{odd}}$. The generator
$\mathscr{A}$ must preserve this decomposition, and we get

$\mathscr{A}=P_{\mathrm{even}}[\underset{=\mathscr{A}_{\mathrm{even}}}{\underbrace{\mathscr{A}_{2\alpha}+\mathscr{A}_{0}+\mathscr{A}_{-1}}}]P_{\mathrm{even}}+P_{\mathrm{odd}}[\underset{=\mathscr{A}_{\mathrm{odd}}}{\underbrace{\mathscr{A}_{2\alpha}+\mathscr{A}_{0}-\mathscr{A}_{-1}}}]P_{\mathrm{odd}}.$

Now we can apply [Str72, Theorem 1] to $\mathscr{A}_{\mathrm{even}}$ and
$\mathscr{A}_{\mathrm{odd}}$. ∎

Since the kernel of $\mathscr{A}$ only contains the constants, we get that as
$t\to+\infty$, $\mathbf{p}(t,\cdot,\cdot)\rightharpoonup 1/|M|$ in
$\mathcal{D}^{\prime}(M)$. However, since $t\mapsto\mathbf{p}(t,\cdot,\cdot)$
also uniformly bounded as a map $\left[1,\infty\right[\to C^{\infty}(M\times
M)$, we deduce that

(5.18)
$\mathbf{p}(t,\cdot,\cdot)\underset{t\to+\infty}{\longrightarrow}1/|M|\cdot\mathbf{1}_{M\times
M},$

where the convergence holds in $C^{\infty}(M\times M)$.

Recall that $\mathbf{k}$ was introduced in §2.4 as the kernel of the semi-
group $(T_{t})_{t\geqslant 0}$ with generator $\mathscr{A}^{\prime}$ defined
by (2.20), and that according to (2.27), $u_{\varepsilon}$ can be expressed in
terms of $\mathbf{k}$.

######  Lemma 5.3.

The generator $\mathscr{A}^{\prime}$ is negative definite on
$L^{2}(\overline{\Omega_{\varepsilon}})$.

###### Proof.

We know that $\mathscr{A}^{\prime}$ has pure point spectrum accumulating at
$+\infty$, so it suffices to prove its kernel is trivial. Let thus
$\varphi\in\ker\mathscr{A}^{\prime}$. Then by (2.21), we have:

$|\varphi(p)|\leqslant\int_{\Omega_{\varepsilon}}{\mathbf{k}}(t,p,q)|\varphi(q)|{\rm
dvol}_{g}(q)\leqslant\int_{\Omega_{\varepsilon}}{\mathbf{p}}(t,p,q)|\varphi(q)|{\rm
dvol}_{g}(q).$

Take the limit $t\to\infty$ and use (5.18) to get for almost-every
$p\in\Omega_{\varepsilon}$:

$|\varphi(p)|\leqslant|M|^{-1}\int_{\Omega_{\varepsilon}}|\varphi(q)|{\rm
dvol}_{g}(q)<|\Omega_{\varepsilon}|^{-1}\int_{\Omega_{\varepsilon}}|\varphi(q)|{\rm
dvol}_{g}(q),$

which easily implies $\varphi=0$. ∎

Note that due to Lemma 5.3.1 and (2.21), we have that
$\mathbf{k}(t,\cdot,\cdot)\in L^{\infty}(M\times M)$. In fact, we have a
better estimate:

######  Lemma 5.3.

Let $\lambda_{0}>0$ be the lowest eigenvalue of $-\mathscr{A}^{\prime}$. Then
there exists a constant $C>0$ such that for all $t>1$,

$|\mathbf{k}(t,p,q)|\leqslant Ce^{-\lambda_{0}t}.$

###### Proof.

Integrating $\varphi_{j}$ agains $\mathbf{k}(t,\cdot,\cdot)$ and using (2.21)
and (2.23), we see that for $s>0$:

$e^{-s\lambda_{j}}|\varphi_{j}(p)|\leqslant\left(\int_{M}|\mathbf{p}(s,p,q)|^{2}{\rm
dvol}_{g}(q)\right)^{1/2}=C_{s}<\infty,$

due to Lemma 5.3.1. Coming back to (2.23), we observe that with $s=1$,

$\displaystyle|\mathbf{k}(t,p,q)|$
$\displaystyle\leqslant\sum_{j}e^{-\lambda_{j}t}|\varphi_{j}(p)||\varphi_{j}(q)|$
$\displaystyle\leqslant\sum_{j}e^{-\lambda_{j}(t-2s)}C_{1}^{2}=C_{1}^{2}\sum_{j}e^{-\lambda_{j}(t-2)}.$

Since the last sum converges absolutely, it is
$\mathcal{O}(e^{-\lambda_{0}t})$ as $t\to+\infty$. ∎

######  Corollary 5.3.

For all $\varepsilon>0$, $u_{\varepsilon}\in L^{\infty}(M)$.

###### Proof.

We split (2.27) into two parts

$u_{\varepsilon}(p)=\int_{0}^{1}\int_{\Omega_{\varepsilon}}{\mathbf{k}}(t,p,q){\rm
dvol}_{g}(q)\mathrm{d}t+\int_{1}^{\infty}\int_{\Omega_{\varepsilon}}{\mathbf{k}}(t,p,q){\rm
dvol}_{g}(q)\mathrm{d}t$

The second integral can be estimated using Lemma 5.3.1, while for the first
one, we may use (2.21) to get

$\left|\int_{0}^{1}\int_{\Omega_{\varepsilon}}{\mathbf{k}}(t,p,q){\rm
dvol}_{g}(q)\mathrm{d}t\right|\leqslant\left|\int_{0}^{1}\int_{M}{\mathbf{p}}(t,p,q){\rm
dvol}_{g}(q)\mathrm{d}t\right|=1,$

which concludes the proof. ∎

####  5.3.2. Study of the generator

We now show the second part of Proposition 5.3, namely, that the generator
$\mathscr{A}^{\prime}$ of $T_{t}$ can be interpreted as
$\mathbf{1}_{\Omega_{\varepsilon}}\mathscr{A}$ on sufficiently regular
functions.For this we will use the approximating semi-groups
$(T_{t}^{\ell})_{t\geqslant 0}$ introduced in (2.25). Originally in [Get59],
they were defined as semi-groups on $L^{2}(M)$, but as we mentioned, the bound
(2.24) on the kernels implies that they are actually bounded semi-groups on
each $L^{m}(M)$, $m\in[1,+\infty]$. We have the analog of [Get59, Theorem
2.1]:

######  Lemma 5.3.

For all $m\in[1,\infty)$, the semi-groups $(T_{t})_{t\geqslant 0}$,
$(U_{t})_{t\geqslant 0}$, and $(T^{\ell}_{t})_{t\geqslant 0}$ are strongly
continuous on $L^{m}(\Omega_{\varepsilon})$, $L^{m}(M)$, and $L^{m}(M)$
respectively.

###### Proof.

The proof of Geetor applies almost verbatim, replacing $L^{2}$ therein by
$L^{m}$. Its main ingredient is the fact that for $\phi\in C^{0}(\Omega)$
(with the notations of §2.4)

$T_{t}[V,\Omega]\phi(p)\to\phi(p),\ \text{ for a.e }\ p\in\overline{\Omega}.$

With boundedness on $L^{\infty}$ and dominated convergence, this implies that
$(T_{t})_{t\geqslant 0}$ is weakly continuous on all $L^{m}$ with $m<\infty$,
and hence strongly continuous. ∎

As the semigroups on $L^{m}$ are strongly continuous, they all admit
generators which are densely defined but the question then is to characterize
their domain. We first give a precise description of the domain of the
infinitesimal generator $\mathscr{A}$ of the semigroup $(U_{t})_{t\geqslant
0}$ given by (2.19) on $L^{m}(M)$. Recall that the domain of the generator of
the semigroup $U_{t}:L^{m}(M)\to L^{m}(M)$, which we denote by
$D_{L^{m}(M)}(\mathscr{A})$, is defined by

$D_{L^{m}(M)}(\mathscr{A}):=\\{u\in L^{m}(M)\mid\lim\limits_{t\to
0}\left(U_{t}u-u\right)/t\ {\text{ converges in }}L^{m}(M)\\}.$

In what follows, when the $L^{m}(M)$ space is clear within the context, we
will drop the $L^{m}(M)$ subscript and simply write $D(\mathscr{A})$.

######  Lemma 5.3.

For each $m\in(1,\infty)$, let $U_{t}:L^{m}(M)\to L^{m}(M)$ be the strongly
continuous semigroup given by Lemma 5.3.2. Then
$D(\mathscr{A})=W^{2\alpha,m}(M)$.

###### Proof.

According to [AE00] (and [App95] as mentioned in the footnote of §2.4), the
domain of the generator contains $C^{\infty}(M)$. Since the semi-group $U_{t}$
is strongly continuous and contracting, $(\mathscr{A},D(\mathscr{A}))$ must be
closed. However, since $\mathscr{A}$ is elliptic,
$(\mathscr{A},C^{\infty}(M))$ has only one closure in $L^{m}$,
$m\in(1,+\infty)$ which must be $(\mathscr{A},W^{2\alpha,m})$, see [Won91,
Theorem 12.15] for a reference. ∎

The next step will be to show that the generator $\mathscr{A}^{\prime}$ of the
semigroup $T_{t}:L^{m}(\Omega_{\varepsilon})\to L^{m}(\Omega_{\varepsilon})$
defined in (2.25) coincides with
$\mathbf{1}_{\Omega_{\varepsilon}}\mathscr{A}$:

######  Lemma 5.3.

For $m\in[1,+\infty)$, we have $L^{m}(\Omega_{\varepsilon})\cap
D(\mathscr{A})\subset D(\mathscr{A}^{\prime})$ with
$\mathscr{A}^{\prime}u=\mathbf{1}_{\Omega_{\varepsilon}}\mathscr{A}u$ for all
$u\in L^{m}(\Omega_{\varepsilon})\cap D(\mathscr{A})$.

Of course, this is only useful for $m>1$, in which case we are able to
characterize $D(\mathscr{A})$. The only ingredient in the proof is the strong
continuity of $(U_{t})_{t\geqslant 0}$ on $C^{0}(M)$.

###### Proof.

We proceed as in [Get61]. First, [Get57, Theorem 5.1] asserts that if
$\mathscr{A}_{\ell}$ is the infinitesimal generator of
$(T^{\ell}_{t})_{t\geqslant 0}$, then

(5.19)
$\displaystyle\mathscr{A}_{\ell}=\mathscr{A}-\ell\mathbf{1}_{B_{\varepsilon}(p_{0})}.$

with domain $D(\mathscr{A}_{\ell})=D(A)$. Let $(\mathscr{A}-\lambda)^{-1}$,
$(\mathscr{A}^{\prime}-\lambda)^{-1}$, and $(\mathscr{A}_{\ell}-\lambda)^{-1}$
be resolvents for the semigroups $U_{t}$, $T_{t}$, and $T_{t}^{\ell}$
respectively. Assume that $\lambda>0$, and write

$(\lambda-\mathscr{A}^{\prime})^{-1}=\int_{0}^{+\infty}e^{-t\lambda}T_{t}dt,\quad(\lambda-\mathscr{A}_{\ell})^{-1}=\int_{0}^{+\infty}e^{-t\lambda}T_{t}^{\ell}dt.$

According to (2.22), for $\phi\in L^{2}(\Omega_{\varepsilon})$,
$T_{t}^{\ell}\phi\to T_{t}\phi,\ \text{ in }L^{2}(M)$. As we have already
observed in §2.4, this result extends to $\phi\in L^{m}(\Omega_{\varepsilon})$
with convergence in $L^{m}(M)$. We will need a stronger statement, namely,
that this holds for all $\phi\in L^{m}(M)$ with convergence in $L^{m}(M)$ to
$T_{t}(\mathbf{1}_{\Omega_{\varepsilon}}\phi)$. Hence, it suffices to show
that if $\phi\in L^{m}(M\setminus\Omega_{\varepsilon})$, then
$T_{t}^{\ell}\phi\to 0$ in $L^{m}(M)$. In the proof of [Get59, Theorem 4.1],
we see that for $\phi\in C^{0}(M)$, we have the pointwise convergence

$\lim_{\ell\to\infty}T_{t}^{\ell}\phi(p)=\mathbb{E}\left[\phi(X(t))\mathbf{1}_{\\{X(\tau)\in\Omega_{\varepsilon}|0\leqslant\tau<t\\}}\middle|~{}X(0)=p\right].$

Hence, if $\phi$ is continuous and supported in
$M\setminus\Omega_{\varepsilon}$, we have the pointwise convergence
$T_{t}^{\ell}\phi(p)\to 0$ (by dominated convergence). Then, using that
$T_{t}^{\ell}$ is a contraction on $L^{m}(M)$ and approximating a function
$\phi\in L^{m}(M\setminus\Omega_{\varepsilon})$ by continuous functions
$\phi_{n}\to\phi$ in $L^{m}(M\setminus\Omega_{\varepsilon})$, we deduce that
$T_{t}^{\ell}\phi\to 0$ in $L^{m}(M)$. This shows that for all $\phi\in
L^{m}(M)$,
$T_{t}^{\ell}\phi\to_{\ell\to\infty}T_{t}(\mathbf{1}_{\Omega_{\varepsilon}}\phi)$
with convergence in $L^{m}(M)$.

Now, since $\lambda>0$, we can use dominated convergence to conclude that for
all $L^{m}(M)$ with $m<\infty$, $\phi\in L^{m}(M)$

(5.20)
$(\lambda-\mathscr{A}_{\ell})^{-1}\phi\to(\lambda-\mathscr{A}^{\prime})^{-1}\mathbf{1}_{\Omega_{\varepsilon}}\phi,\
\text{ in $L^{m}(M)$.}$

For $u\in D(\mathscr{A})\cap L^{m}(\Omega_{\varepsilon})$, we observe that

$\mathscr{A}_{\ell}u=(\mathscr{A}-\ell\mathbf{1}_{B_{\varepsilon}(p_{0})})u=\mathscr{A}u.$

In particular,

$u=(\lambda-\mathscr{A}_{\ell})^{-1}(\lambda-\mathscr{A})u\to(\lambda-\mathscr{A}^{\prime})^{-1}\mathbf{1}_{\Omega_{\varepsilon}}(\lambda-\mathscr{A})u.$

so that $u$ is in $D(\mathscr{A}^{\prime})$ and

$(\lambda-\mathscr{A}^{\prime})u=\lambda
u-\mathbf{1}_{\Omega_{\varepsilon}}\mathscr{A}u.$

∎

The previous lemma completes the proof of Proposition 5.3. In turn, we can now
conclude the proof of Proposition 5.2:

###### Proof.

Let $w\in\dot{W}^{2\alpha,m}(\overline{\Omega}_{\varepsilon})$ such that
$\mathscr{A}w=-1$ in $\Omega_{\varepsilon}$. This means that