is almost surely finite. We then use the Frostman Lemma to conclude. ∎

######  Remark 5.3.

Without involving [PW02], an alternative proof uses the bottleneck property
introduced in [DY20]. Indeed, with $r,F$ be given in [DY20, Lemma 5.4]
replacing Lemma 2.4, any two elements $g,h$ can be concatenated via some $f\in
F$ such that any path from $e$ to $gfh$ intersects $B(g,r)$. Such points are
referred to as bottleneck points on $[e,gfh]$. If a path contains a sequence
of bottleneck points with consecutive distance at most $L$, then it is said to
have the $L$-bottleneck property. A triangle with sides having the
$L$-bottleneck property satisfies the conclusion of Lemma 4.4 and Lemma 4.5.
We define similarly $S_{n,L}$ to be the set of points $x\in S_{n}$ so that
$[e,x]$ has the $L$-bottleneck property. Then Proposition 4.1 follows verbatim
the same argument with transition point replaced with bottleneck points. The
remaining modification goes as explained above.

A particular class of groups with infinitely many ends are free products of
the form $\Gamma=\Gamma_{0}*\Gamma_{1}$ where at least one the free factors
$\Gamma_{i}$ is not $\mathbb{Z}/2\mathbb{Z}$. For such groups, the authors of
[CGM12] prove that the Hausdorff dimension of the limit set in the end
boundary endowed with a visual distance is exactly
$\frac{-1}{\log\lambda}\omega_{\Gamma}(r)$. Their proof for the upper bound
applies to any group with infinitely many ends. Indeed, it consists in saying
that for any end $\xi\in\Lambda_{\mathcal{E}}(r)$, for every $n$, the
branching random walk has to visit some point $x\in\bigcup_{0\leq l\leq
l_{0}}S_{n+l}$ such that $\xi$ and $x$ are in the same infinite connected
component of the complement of $B(e,n)$. The constant $l_{0}$ only depends on
the support of the measure $\mu$. Thus, $\Lambda_{\mathcal{E}}(r)$ can be
covered with $\mathcal{P}_{n}$ sets of diameter bounded by $C\lambda^{n}$.

For the lower bound, they first show that ${\rm
Hdim}(\Lambda_{\mathcal{E}}(r),\tilde{\delta}_{e})$ is bounded from below by
some number $z^{*}$ and then prove that $z^{*}=\omega_{\Gamma}(r)$, see
[CGM12, Lemma 4.7]. However, in order to prove that
$z^{*}=\omega_{\Gamma}(r)$, they use the same invalid argument as the one
described in Section 3.5, namely that the quantity $H_{r}(n)$ is sub-
multiplicative. Proposition 5.2 fills their gap and combining it with the
proof for the upper bound described above, we get Theorem 1.4.

######  Remark 5.4.

The proof of the upper bound described above also works for the parabolic
cosets. Namely, for every $P\in\mathbb{P}$, the Hausdorff dimension of
$\Lambda_{\mathcal{E}}(r)\cap\partial_{\mathcal{E}}P$ is bounded from above by
$\omega_{P}(r)$. Thus, assuming further that the Green function has a
parabolic gap, we recover [CGM12, Corollary 3.7], i.e.

${\rm Hdim}(\Lambda_{\mathcal{E}}(r)\cap\partial_{\mathcal{E}}P)<{\rm
Hdim}(\Lambda_{\mathcal{E}}(r)).$

##  6\. An upper bound for the Hausdorff dimension of the limit set

Let $\Gamma$ be a relatively hyperbolic group and let
$\Lambda_{\mathcal{F}}(r)$ be the limit set of the branching random walk in
the Floyd boundary of $\Gamma$, endowed with the Floyd distance $\delta_{e}$.
Recall that by Theorem 2.3, there exists a map $\phi$ from the Floyd boundary
to the Bowditch boundary such that the preimage of a conical limit point is
reduced to a single point. We can thus see the set of conical limit points
$\partial_{\mathcal{B}}^{con}\Gamma$ in the Bowditch boundary as a subset of
the Floyd boundary. We set

$\Lambda^{con}_{\mathcal{F}}(r)=\Lambda_{\mathcal{F}}(r)\cap\phi^{-1}(\partial_{\mathcal{B}}^{con}\Gamma).$

We prove here the following proposition.

######  Proposition 6.1.

Let $r\leq\rho^{-1}$. Almost surely,

${\rm
Hdim}(\Lambda^{con}_{\mathcal{F}}(r),\delta_{e})\leq\frac{-1}{\log\lambda}\omega_{\Gamma}(r).$

Under additional assumption on the volume growth of parabolic subgroup, we
have the upper bound on the full limit set in Floyd boundary.

######  Corollary 6.2.

Let $r\leq\rho^{-1}$. Assume that $v_{S}(P)\leq\omega_{\Gamma}(r)$ for every
parabolic subgroup $P$. Then almost surely,

${\rm
Hdim}(\Lambda_{\mathcal{F}}(r),\delta_{e})\leq\frac{-1}{\log\lambda}\omega_{\Gamma}(r).$

###### Proof.

The Bowditch boundary consists of conical points and countably many parabolic
points and the preimage of each parabolic point is exactly the limit set of a
parabolic subgroup (see [GP13, Theorem A]). So the limit set
$\Lambda_{\mathcal{F}}(r)$ is contained in the union of
$\Lambda^{con}_{\mathcal{F}}(r)$ with countably many limit sets of parabolic
cosets $P\in\mathbb{P}$. As $P$ is quasi-convex in the Cayley graph of
$\Gamma$, any geodesic from $e$ to the limit points of $P$ is contained in a
fixed neighborhood of $P$, see for instance [DS05, Lemma 4.3]. Using a suited
covering by shadows based at $S_{n}\cap P$, we can see that ${\rm
Hdim}(\Lambda_{\mathcal{F}}P)\leq\frac{-1}{\log\lambda}v_{S}(P)$ (e.g. [PY19,
Lemma 4.1]). The conclusion now follows from Proposition 6.1. ∎

To prepare the proof for Proposition 6.1, we first need a few geometric
lemmas. Denote as usual by $\mathbb{P}=\\{gP:g\in\Gamma,P\in\mathbb{P}_{0}\\}$
the collection of all parabolic cosets, and by $\ell(\gamma)$ the length of a
path $\gamma$. Fix a geodesic $[x,z]$ and $\epsilon_{1},\epsilon_{2}\in[0,1]$.
An $[\epsilon_{1},\epsilon_{2}]$-percentage of $[x,z]$ consists of points
$w\in[x,z]$ such that $\epsilon_{1}\leq d(x,w)/d(x,z)\leq\epsilon_{2}$.

######  Lemma 6.3.

Let $\gamma$ be a geodesic segment such that
$[\epsilon,1-\epsilon]$-percentage of $\gamma$ contains no
$(\eta,L)$-transition point. Then there exists a unique $P\in\mathbb{P}$ such
that the entry and exit points of $\gamma$ in $N_{\eta}(P)$ have distance at
most $\epsilon\ell(\gamma)$ to the corresponding endpoints of $\gamma$.

###### Proof.

By assumption, the middle point $m\in\gamma$ is $(\eta,L)$-deep in a unique
$P\in\mathbb{P}$. Let $u,v$ be the corresponding entry and exit point of
$\gamma$ in $N_{\eta}(P)$. By Lemma 2.7, $u,v$ are $(\eta,L)$-transition
points, so $d(x,u)\leq\epsilon d(x,y)$ and $d(v,y)\leq\epsilon d(x,y)$, which
concludes the proof. ∎

Fix $C>0$ and $x\in S_{n}$. The $C$-partial cone $\Omega(x,C)$ consists of
points $z\in G$ such that $[e,z]$ contains an $(\eta,L)$-transition point
$C$-close to $x$.

Let $B([x,z])$ be the ball centered at the middle point of $[x,z]$ of radius
${d(x,z)/2}$. Define $U(x)$ to be the union of the partial cone $\Omega(x,C)$
and the balls $B([x,z])$ for all geodesics $[x,z]$ between $x$ and
$z\in\Omega(x,C)$. That is,

$U(x):=\Omega(x,C)\cup\left(\bigcup\left\\{B([x,z]):\forall[x,z],\forall
z\in\Omega(x,C)\right\\}\right).$

Note that any ball of centered at $w\in[x,z]$ of radius
$\min\\{d(w,x),d(w,z)\\}$ is contained in $B([x,z])$.

In what follows, let $C>0$ be given by Lemma 2.9.

######  Lemma 6.4.

Let $\alpha$ be a path starting from $e$ and first entering at a point $z\in
U(x)$. Let $w\in[x,z]$ be a transition point. Set
$S:=\min\\{d(w,x),d(w,z)\\}$. Then $B(w,S-2C)$ is contained in $U(x)$ so
$\alpha$ has distance at least $S-2C$ to the point $w$.

###### Proof.

We first consider the case $z\in\Omega(x,C)$. By definition of $U(x)$, any
ball centered at $w\in[x,z]$ of radius $S=\min\\{d(x,w),d(z,w)\\}$ is
contained in $U(x)$. The statement follows immediately.

Assume now that $z$ lies in a ball $B([x,\hat{z}])$ where $[x,\hat{z}]$ is a
geodesic between $x$ and some $\hat{z}$ in $\Omega(x,C)$. Let
$\hat{w}\in[x,\hat{z}]$ be the middle point.

Consider the triangle with vertices $x,\hat{w},z$. As $w$ is a transition
point on $[x,z]$, Lemma 2.9 shows that $d(w,w^{\prime})\leq C$ for some
$w^{\prime}\in[\hat{w},x]\cup[\hat{w},z]$. Thus, $B(w,\kappa-2C)\subset
B(w^{\prime},\kappa-C)$. As $w^{\prime}$ is on the radius $[\hat{w},x]$ or
$[\hat{w},z]$ of the ball $B([x,\hat{z}])$, we have the ball
$B(w^{\prime},S-C)$ is contained in $B(\hat{w},\eta)\subseteq U(x)$. ∎

######  Lemma 6.5.

Let $\xi\in\partial_{\mathcal{B}}\Gamma$ be a conical point and consider a
sequence of points $z_{n}\to\xi$. Let $x\in[e,\xi]$ be an
$(\eta,L)$-transition point. Then for for all but finitely many $z_{n}$, there
exists an $(\eta,L)$-transition point $x_{n}$ on $[e,z_{n}]$ such that
$d(x_{n},x)\leq C$. In particular, $z_{n}\in\Omega(x,C)$ for large enough $n$.

###### Proof.

If $z_{n}\to\xi$ then $\delta_{x}(z_{n},\xi)<\lambda^{C}\delta$ for all large
enough $n$, where $\delta=\delta(\eta,L)$ is given by Lemma 2.8. Applying
Lemma 2.9 to the triangle with vertices $e$, $\xi$, $z_{n}$, there is an
$(\eta,L)$-transition point $x_{n}$ on $[z_{n},\xi]$ or $[e,z_{n}]$ such that
$d(x,x_{n})\leq C$. It suffices to prove that $x_{n}\in[e,z_{n}]$. Let $y$ be
any transition point on $[z_{n},\xi]$. Then by Lemma 2.8,
$\delta_{y}(z_{n},\xi)>\delta$. It follows from (1) that
$\lambda^{C}\delta>\delta_{x}(z_{n},\xi)\geq\lambda^{d(x,y)}\delta_{y}(z_{n},\xi)>\lambda^{d(x,y)}\delta$
and hence $d(x,y)>C$. The conclusion follows. ∎

Fix $\epsilon\in(0,1/2)$. Let $U_{\epsilon}(x)$ be the set of points $z\in
U(x)$ such that $[x,z]$ contains a transition point $w$ being at distance at
least $\epsilon d(x,z)$ to one of the endpoints:

$\max\\{d(w,x),d(w,z)\\}\geq\epsilon d(x,z).$

By Lemma 2.7, the set $U(x)\setminus U_{\epsilon}(x)$ consists of points $z\in
U(x)$ such that the $[\epsilon,1-\epsilon]$-percentage of $[x,z]$ is contained
in the $\eta$-neighborhood of a unique peripheral coset $P\in\mathbb{P}$.
Explicitly, there exists a subsegment of $[x,z]$ with length at least
$(1-2\epsilon)d(x,z)$ contained in $N_{\eta}(P)$.

######  Lemma 6.6.

Let $\alpha$ be a path starting from $e$ and first entering $U(x)$ at a point
$z\in U_{\epsilon}(x)$. Assume that $\epsilon d(x,z)>10C$. Then there exists a
transition point $y$ on $[e,z]$ such that $\alpha$ lies outside the ball
around $y$ with radius $\epsilon d(x,z)-3C$.

###### Proof.

By definition of $z\in U_{\epsilon}(x)$, $[x,z]$ contains a transition point
$w$ such that

$\epsilon d(x,z)\leq\max\\{d(w,x),d(w,z)\\}.$

Setting $S=\epsilon d(x,z)$, Lemma 6.4 implies that $B(w,\kappa-2C)\subset
U(x)$, so $\alpha$ does not intersect $B(w,S-2C)$. To conclude the proof, it
remains to find a $(\eta,L)$-transition point $y\in[e,z]$ such that
$d(w,y)\leq C$. In particular, $\alpha$ does not intersect $B(y,S-3C)$,
completing the proof.

Indeed, as in the proof of Lemma 6.4, $z$ lies on the ball $B([x,\hat{z}])$
centered at the middle point $\hat{w}$ of $[x,\hat{z}]$ for some
$\hat{z}\in\Omega(x,C)$. By assumption, $w$ is an $(\eta,L)$-transition point
on $[x,z]$. Lemma 2.9 applied for the triangle with vertices $x,\hat{w},z$
shows that $d(w,w^{\prime})\leq C$ for some
$w^{\prime}\in[\hat{w},x]\cup[\hat{w},z]$. A schematic figure is shown below.

Similarly for the triangle with vertices $e,x,z$, Lemma 2.9 implies that
either $[e,z]$ or $[e,x]$ contains an $(\eta,L)$-transition point $y$ such
that $d(w,y)\leq C$ and then $d(y,w^{\prime})\leq 2C$. As
$\hat{z}\in\Omega(x,C)$, $d(x,[e,\hat{z}])\leq C$ holds, so the triangle
inequality shows

$d(e,x)+d(x,\hat{z})-2C\leq d(e,\hat{z}).$

In a different term, this implies that the path $p:=[e,x][x,\hat{z}]$ is
$(2C)$-taut: $\ell(p)\leq d(p_{-},p_{+})+2C$. It follows from triangle
inequality that any subpath of a $(2C)$-taut path $p$ is $(2C)$-taut.

We first claim that either $w^{\prime}\notin[x,\hat{w}]$ or $y\notin[e,x]$.
Otherwise, we have $w^{\prime}\in[x,\hat{w}]$ and $y\in[e,x]$. Since
$[y,x][x,w^{\prime}]$ is a $(2C)$-taut subpath of $p$, we obtain that
$d(y,x)+d(x,w^{\prime})\leq d(y,w^{\prime})+2C\leq 4C$, so $d(x,w)\leq
d(x,w^{\prime})+d(w^{\prime},w)\leq 5C$. This is a contradiction because
$d(x,w)\geq\epsilon d(x,z)>5C$.

Let us assume now $y\in[e,x]$ to derive a contradiction. The above claim shows
$w^{\prime}\notin[x,\hat{w}]$ and then $w^{\prime}\in[\hat{w},z]$. Using the
$(2C)$-taut path $[y,x][x,\hat{w}]$, we have

$d(y,\hat{w})+2C\geq d(y,x)+d(x,\hat{w}).$

which combined with $d(x,\hat{w})=d(z,\hat{w})\geq d(w^{\prime},\hat{w})$
gives

$d(y,\hat{w})+2C\geq d(y,x)+d(w^{\prime},\hat{w}).$

On the other hand, $|d(y,\hat{w})-d(w^{\prime},\hat{w})|\leq
d(y,w^{\prime})\leq 2C$. These together show that $d(y,x)\leq 4C$ and then
$d(w^{\prime},x)\leq d(w^{\prime},y)+d(y,x)\leq 6C$. Thus,
$|d(x,\hat{w})-d(w^{\prime},\hat{w})|\leq 6C$. Since
$d(x,\hat{w})=d(z,\hat{w})$ and $w^{\prime}$ is on $[\hat{w},z]$, we have
$d(w^{\prime},z)\leq 6C$, so $d(x,z)\leq 12C$. This contradicts the assumption
$\epsilon d(x,z)>10C$, hence we proved that $y\in[e,z]$ is impossible, so $y$
is the desired transition point on $[e,z]$. ∎

For any $m\geq 1$, let $U_{\epsilon}(x,m)$ be the set of elements $z\in
U_{\epsilon}(x)$ such that $d(x,z)\geq m$.

######  Lemma 6.7.

For any $\epsilon\in(0,1/2)$, there exists $\kappa>0$ with the following
property. Almost surely, there exists $n_{0}>0$ such that for all $n>n_{0}$
and all $x\in S_{n}$: if $\mathrm{BRW}(\Gamma,\nu,\mu)$ first enters $U(x)$ at
a point $z$, then $z\in U(x)\setminus U_{\epsilon}(x,\kappa\log|x|)$.

###### Proof.

Let us freeze all particles of the branching random walk when they enter
$U(x)$ at the first time. Denote by $\mathcal{Z}(x,m)$ the collection of
frozen particles $z\in U_{\varepsilon}(x,m)$. Then for $z\in\mathcal{Z}(x,m)$
we have

  1. (1)

$d(x,z)\geq m$,

  2. (2)

$\max\\{d(y,x),d(y,z)\\}>\epsilon d(x,z)$ where $y$ is an
$(\eta,L)$-transition point on $[e,z]$ given by Lemma 6.6.

Note that the genealogy path from $e$ to $z$ does not intersect $B(y,\epsilon
d(x,z))$.

Let $\delta=\delta(\eta,L)$ be given by Lemma 2.11. Then the expected number
of particles frozenat $z\in U_{\epsilon}(x)$ is upper bounded by

$G_{r}(e,z;[U_{\epsilon}(x)]^{c})\leq G_{r}(e,z;[B(y,\epsilon
d(x,z))]^{c})\leq e^{-e^{\delta\epsilon d(x,z)}}.$

By Lemma 2.1 there exists $c>0$ such that $\sharp S_{n}\leq ce^{vn}$ for any
$n\geq 1$. Thus, there exist $\epsilon_{1}=\epsilon_{1}(\epsilon,\delta,v)$
and $m_{0}>0$ such that for any $m>m_{0}$, we have

$\mathbf{E}[\sharp\mathcal{Z}({x,m})]\leq\sum_{z\in
U_{\epsilon}(x,m)}G_{r}\bigl{(}e,z;[B(y,\epsilon
d(x,z)]^{c})\bigr{)}\leq\sum_{k=m}^{\infty}ce^{kv}e^{-e^{\delta\epsilon
k}}\leq e^{-e^{\epsilon_{1}m}}.$

Choose $\kappa$ so that $\kappa\epsilon_{1}>1$ and let $m=\kappa\log n$.
Consider the event

$A_{n}=\left\\{\mathcal{Z}(x,m)\geq 1\,\text{for some }x\in S_{n}\right\\},$

i.e. $A_{n}$ is the event such that if the branching random walk visits
$U_{x}$ for some $x\in S_{n}$, then the first frozen particle is in
$U_{\epsilon}(x,m)$. Then

$\mathbf{P}(A_{n})\leq\sum_{x\in S_{n}}\mathbf{P}(\sharp\mathcal{Z}({x,m})\geq
1)\leq\sum_{x\in S_{n}}\mathbf{E}[\sharp\mathcal{Z}({x,m})]\leq
ce^{vn-e^{\epsilon_{1}\kappa\log n}}\leq ce^{vn-n^{\epsilon_{1}\kappa}}.$

Therefore, $\sum_{n=1}^{\infty}\mathbf{P}(A_{n})<\infty$ and so the conclusion
follows from the Borel-Cantelli Lemma. ∎

Similarly, we prove the following.

######  Lemma 6.8.

For every $K\geq 0$ and $C\geq 0$, there exists $\kappa>0$ such that the
following holds. Almost surely, for all but finitely many $x$, if the
branching random walk ever visits a point $y$ with $d(e,y)\leq Kd(e,x)$ and
such that $x$ is within $C$ of a transition point on $[e,y]$, then it first
enters $B(x,\kappa\log|x|)$.

###### Proof.

Given $x\in S_{n}$, we consider the set $U(x)$ of $y\in\Gamma$ so that
$|y|\leq K|x|$ and $x$ is within $C$ of a transition point on $[e,y]$. We
freeze particles when they first reach a point $y\in U(x)$, without entering
$B(x,\kappa\log|x|)$. We denote by $\mathcal{Z}_{n}$ the set of frozen
particles for all $x\in S_{n}$. By Lemma 2.11, for $n$ large enough, the
expected number of particles frozen at $y\in U(x)$ is upper bounded by

$G_{r}(e,y;[U(x)]^{c})\leq G_{r}(e,y;[B(x,\kappa\log|x|)]^{c})\leq
e^{-n^{\delta\kappa}}.$

By Lemma 2.1, there exist $c>0$ such that $\sharp S_{n}\leq ce^{vn}$ and
$\sharp U(x)\leq ce^{vKn}$ for any $n\geq 1$ and $x\in\Gamma$. Thus, we have

$\mathbf{E}[\sharp\mathcal{Z}_{n}]\leq\sum_{x\in S_{n}}\sum_{y\in
U(x)}G_{r}(e,y;[U(x)]^{c})\leq ce^{v(K+1)n}e^{-n^{\delta\kappa}}.$

If $\kappa$ is chosen large enough, then
$\sum_{n=1}^{\infty}\mathbf{E}[\sharp\mathcal{Z}_{n}]<\infty$, so the proof
follows from the Borel-Cantelli Lemma. ∎

Recall that if $P\in\mathbb{P}$ is a parabolic coset, $\eta\geq 0$ and
$x\in\Gamma$, we denote by

$\pi_{N_{\eta}(P)}(x):=\\{y\in N_{\eta}(P):d(x,y)=d(x,N_{\eta}(P))\\}$

the set of its shortest projections on the $\eta$-neighborhood $N_{\eta}(P)$
of $P$. Also, for $x,y\in\Gamma$, we denote by

$d_{N_{\eta}(P)}(x,y):={\textbf{diam}\big{(}\pi_{N_{\eta}(P)}(x)\cup\pi_{N_{\eta}(P)}(y)\big{)}}$

and [Hru10, Corollary 8.2] the shortest projection is coarsely Lipschitz,

$d_{N_{\eta}(P)}(x,y)\leq kd(x,y)+k$

for a fixed $k\geq 1$ depending only on $\eta$. Thus, $\pi_{N_{\eta}(P)}(x)$
has bounded diameter.

######  Lemma 6.9.

[Sis13, Lemma 1.15] For every large enough $\eta$, there exists $C=C(\eta)>0$
such that for every $x\in\Gamma$, $P\in\mathbb{P}$ and for every geodesic
$\gamma$ starting at $x$ and entering $N_{\eta}(P)$, we have

$\pi_{N_{\eta}(P)}(x)\subset B(y,C)$

where $y$ is the entrance point of $\gamma$ in $N_{\eta}(P)$.

######  Lemma 6.10.

There exists $\eta_{0}\geq 0$ such that for $\eta\geq\eta_{0}$, the following
holds. Almost surely, for all but finitely many parabolic cosets
$P\in\mathbb{P}$, if the branching random walk ever visits a point $z$
satisfying both that $d_{N_{\eta}(P)}(e,z)\geq d(e,N_{\eta}(P))$ and
$d(z,N_{\eta}(P))\leq d_{N_{\eta}(P)}(e,z)$, then it first needs to enter
$N_{\eta}(P)$ at a point $w$ such that $d_{N_{\eta}(P)}(e,w)\leq
d(e,N_{\eta}(P))$.

In particular, if the branching random walk ever enters $N_{\eta}(P)$, then
the first entrance point must be within $d(e,N_{\eta}(P))$ of the projection
of $e$ on $N_{\eta}(P)$.

###### Proof.

Given $P\in\mathbb{P}$, we freeze particles when they first visit some point
$z\in\Gamma$ with

$d_{N_{\eta}(P)}(e,z)\geq d(e,N_{\eta}(P))$

and

$d(z,N_{\eta}(P))\leq d_{N_{\eta}(P)}(e,z),$

without having entered $\\{w\in N_{\eta}(P):d_{N_{\eta}(P)}(e,w)\leq
d(e,N_{\eta}(P))\\}$. Denote by $\mathbb{P}_{k}$ the collection of parabolic
cosets $P$ with $d(e,N_{\eta}(P))=k$. We denote by $\mathcal{Z}_{k}$ the set
of such frozen particles for those $P\in\mathbb{P}_{k}$. Then,

$\mathbf{E}[\sharp\mathcal{Z}_{k}]\leq\sum_{P\in\mathbb{P}_{k}}\sum_{\underset{d_{N_{\eta}(P)}(x,e)\geq
k}{x\in N_{\eta}(P)}}\sum_{\underset{d(x,y)\leq
k}{y\colon\pi_{N_{\eta}(P)}(y)=x}}G_{R}(e,y;N_{\eta}(P)^{c}).$

As $x\in\pi_{N_{\eta}(P)}(z)$, there exists a trajectory for the $\mu$-random
walk from $z$ to $x$ that stays outside $N_{\eta}(P)$ of linear length in
$d(x,z)$, since the support of $\mu$ is finite. In particular,

$G_{R}(z,x;N_{\eta}(P)^{c})\geq\mathrm{e}^{-\alpha
d(x,z)}\geq\mathrm{e}^{-\alpha k}$

for some positive $\alpha$. Note that by Lemma 2.1, for a fixed $x$, the set
of such elements $z$ being contained in a ball of radius $k$ grows as an
exponential function in $k$. Thus,

(24)
$\mathbf{E}[\sharp\mathcal{Z}_{k}]\leq\sum_{P\in\mathbb{P}_{k}}\sum_{\underset{d(x,\pi_{N_{\eta}(P)}(e))\geq
k}{x\in N_{\eta}(P)}}G_{R}(e,x;N_{\eta}(P)^{c})\mathrm{e}^{\beta k},$

where $\beta$ depends both on $\alpha$ and on the growth rate $v$ of the word
distance. By Lemma 2.12, for every $M\geq 0$, there exists $\eta_{0}$ such
that for all $\eta\geq\eta_{0}$,

$G_{R}(e,x;N_{\eta}(P)^{c})\leq C\mathrm{e}^{-Md(x,\pi_{\eta,P}(e))}.$

Choosing $M>v+\beta$, where $v$ is the growth rate of the word distance, we
have by Lemma 2.1 and by (24) that for $\eta\geq\eta_{0}$,

$\mathbf{E}[\sharp\mathcal{Z}_{k}]\leq C\mathrm{e}^{-k(M-v-\beta)}.$

By the choice of $M$, the sum $\sum_{k\geq
0}\mathbf{E}[\sharp\mathcal{Z}_{k}]$is finite. The result again follows from
the Borel-Cantelli lemma. ∎

######  Lemma 6.11.

There exists $\kappa$ such that almost surely, for any conical limit point
$\xi$ in $\Lambda$ and for all but finitely many transition point $x$ on
$[e,\xi]$, $\mathcal{P}$ intersects $B(x,\kappa\log|x|)$.

###### Proof.

Let $\xi\in\Lambda$ be a conical point. Then $[e,\xi]$ contains infinitely
many $(\eta,L)$-transition points (see [Yan22, Lemma 2.20]). Consider any
$(\eta,L)$-transition point $x\in[e,\xi]$ so that $|x|>n_{0}$. According to
Lemma 6.5, for $\Omega(x,C)\subset U(x)$, the branching random walk must enter
$U(x)$.

Set $\epsilon\in(0,1/2)$ so that $K=\epsilon^{-1}\geq 4$. Let $\kappa$ be
given by Lemmas 6.7 and 6.8. Up to enlarging $\eta$, we may assume that it is
big enough to apply Lemma 6.10.

Let $z\in U_{\epsilon}(x)\setminus U_{\epsilon}(x,\kappa\log|x|)$ be the first
entrance point by Lemma 6.7, so one of the following statements is true:

  1. (1)

$d(z,x)\leq\kappa\log|x|$,

  2. (2)

the $[\epsilon,1-\epsilon]$-percentage of $[x,z]$ does not contain any
$(\eta,L)$-transition point.

If the case (1) happens, then we are done. We now assume
$d(x,z)>\kappa\log|x|$.

By Lemma 6.3, there exist a unique coset $P\in\mathbb{P}$ such that if
$y_{1},y_{2}$ are the entrance and exit points of $[x,z]$ in $N_{\eta}(P)$,
then

$\max\\{d(x,y_{1}),d(y_{2},z)\\}\leq\epsilon d(x,z),$

so

$d(y_{1},y_{2})\geq(1-2\epsilon)d(x,z)\geq(1-2\epsilon)\kappa\log n_{0}.$

By definition, $z$ is contained in a ball $B([x,\hat{z}])$ centered at
$\hat{w}$ for some $\hat{z}\in\Omega(x,C)$.

By definition of $\Omega(x,C)$, there exists a transition point $\hat{x}$ on
$[e,\hat{z}]$ such that $d(x,\hat{x})\leq C$. By Lemma 2.9 for the triangle
with vertices $e,z,\hat{z}$, we see that $\hat{x}$ is within $C$ and so $x$ is
within $2C$ of a transition point on $[e,z]$. According to Lemma 6.8, if the
branching random walk does not enter $B(x,\kappa\log|x|)$, then we have
$|z|>K|x|=\epsilon^{-1}|x|$. Noting as above that $x$ is within $2C$ of a
transition point on $[e,z]$, Lemma 6.9 implies that $\pi_{N_{\eta}(P)}(e)$ is
within a bounded distance of the entry point $y_{1}$ of $[x,z]$ into
$N_{\eta}(P)$, which implies that $d(y_{1},y_{2})\leq
d_{N_{\eta}(P)}(e,z)+C^{\prime}$ for some constant $C^{\prime}$ depending on
$C$. Moreover, if $n_{0}$ is large enough, then $d(y_{1},y_{2})\geq
C^{\prime}/2$, so we get $d(y_{1},y_{2})\leq 2d_{N_{\eta}(P)}(e,z)$, hence

$d(z,N_{\eta}(P))\leq\frac{2\epsilon}{1-2\epsilon}d_{N_{\eta}(P)}(e,z).$

Furthermore, $d(e,N_{\eta}(P))\leq|x|+d(x,y_{1})$. Since $|x|\leq\epsilon|z|$
and $d(x,y_{1})\leq\epsilon d(x,z)$, we get

$d(e,N_{\eta}(P))\leq\left(\frac{\epsilon}{1-\epsilon}+\epsilon\right)d(x,z)\leq\left(\frac{\epsilon}{1-\epsilon}+\epsilon\right)(1-2\epsilon)2d_{N_{\eta}(P)}(e,z).$

Thus, if $\epsilon$ is small enough, the conditions of Lemma 6.10 holds, hence
the branching random walk first enters $N_{\eta}(P)$ at a point $w$ such that
$d(\pi_{N_{\eta}(P)}(e),w)\leq d(e,N_{\eta}(P))$. Since $x$ is within $2C$ of
a transition point on $[e,w]$, $d(e,N_{\eta}(P))\geq|x|+d(x,N_{\eta}(P))-2C$
and so

$d(x,w)\leq d(x,N_{\eta}(P))+d_{N_{\eta}(P)}(e,w)\leq
2d(e,{N_{\eta}(P)})+2C\leq 2d(e,w)+2C.$

Thus, for $n_{0}$ large enough, we have $d(e,w)\geq d(e,x)-2C\geq C$ so
$d(x,w)\leq 4d(e,w)$. Applying Lemma 6.8 with $K\geq 4$ again, we see that the
branching random walk necessarily enters $B(x,\kappa\log|x|)$. This concludes
the proof. ∎

We can now end the proof of the upper-bound.

###### Proof of Proposition 6.1.

Let $r\leq\rho^{-1}$ and fix $h$ such that

$h>\frac{\omega_{\Gamma}(r)}{-\log\lambda}.$

Let $0<\epsilon<1/2$. Let $\xi\in\Lambda^{con}_{\mathcal{F}}(r)$ and let $x$
be a transition point on $[e,\xi]$. By Lemma 6.11, almost surely, there exists
$n_{0}$ such that if $|x|\geq n_{0}$, we can find $z\in\mathcal{P}$ such that
$z\in B(x,\kappa\log|x|)$. In particular, $|x|\leq|z|+\kappa\log|x|$ and if
$|x|$ is large enough, then $|z|\geq(1-\epsilon)|x|$, hence

$x\in B\left(z,\kappa\log\frac{|z|}{1-\epsilon}\right).$

Consequently, for every $m$,

$\Lambda^{con}_{\mathcal{F}}(r)\subset\bigcup_{n\geq
m}\bigcup_{z\in\mathcal{P}_{n}}\Pi\left(z,\kappa\log\frac{|z|}{1-\epsilon}\right).$

By Lemma 2.5, the diameter of
$\Pi\left(z,\kappa\log\frac{|z|}{1-\epsilon}\right)$ is bounded by
$C\lambda^{|z|}|z|^{\alpha}\log|z|$. Thus,

$\sum_{n\geq
m}\sum_{z\in\mathcal{P}_{n}}\mathbf{diam}\left(\Pi\left(z,\kappa\log\frac{|z|}{1-\epsilon}\right)\right)^{h}\leq
C\sum_{n\geq m}M_{n}\lambda^{hn}n^{h\alpha}(\log n)^{h}.$

Since $\limsup\frac{1}{n}\log M_{n}\leq\omega_{\Gamma}(r)$, by the choice of
$h$, this last quantity converges to 0 as $m$ tends to infinity. This
concludes the proof. ∎

We deduce the following result. We denote by $\Lambda_{\mathcal{B}}(r)$ the
limit set of the branching random walk inside the Bowditch boundary, endowed
with the shortcut distance $\overline{\delta}_{e}$.

######  Corollary 6.12.

Let $r\leq\rho^{-1}$. Almost surely,

${\rm
Hdim}(\Lambda_{\mathcal{B}}(r),\overline{\delta}_{e})\leq\frac{-1}{\log\lambda}\omega_{\Gamma}(r).$

###### Proof.

Combining Proposition 6.1 and (2), we get that

${\rm
Hdim}(\Lambda_{\mathcal{B}}(r)\cap\partial^{con}_{\mathcal{B}}\Gamma,\overline{\delta}_{e})\leq\frac{-1}{\log\lambda}\omega_{\Gamma}(r).$

The complement of the set of conical limit points in the Bowditch boundary is
the set of parabolic limit points, which is countable. This yields the desired
upper-bound. ∎

Theorem 1.2 is a consequence of Proposition 5.1 and Corollary 6.12. Recall
that if $\Gamma$ is hyperbolic, then it is also relatively hyperbolic and its
Bowditch, Floyd and Gromov boundaries coincide. Moreover, the shortcut
distance and the visual distance are bi-Lipschitz by [PY19, Proposition 6.1].
Thus, Corollary 1.5 follows from Theorem 1.2.

##  Appendix A Convergence of random finite measures

###  A.1. Random finite measures

Let $(X,\mathcal{B})$ be a Polish space endowed with its Borelian
$\sigma$-algebra. Let $(\Omega,\mathcal{F},\mathbf{P})$ be a probability
space.

######  Definition A.1.

A random finite measure on $X$ is a map

$\mu:(\omega,B)\in\Omega\times\mathcal{B}\mapsto\mu_{\omega}(B)\in\mathbb{R}$

such that

  1. (a)

for every $B\in\mathcal{B}$, the map $\omega\mapsto\mu_{\omega}(B)$ is
measurable,

  2. (b)

for $\mathbf{P}$-almost every $\omega$, $\omega\mapsto\mu_{\omega}$ is a
finite Borelian measure on $X$,

  3. (c)

the expectation $\mathbf{E}[\mu_{\omega}(X)]$ is finite,

We identify $\mu$ with the family of maps
$\mu_{\omega}:\mathcal{B}\to\mathbb{R}$ and we write
$\mu=(\mu_{\omega})_{\omega}$.

We denote by $\mathcal{M}_{\Omega}(X)$ the set of random finite measure on $X$
and by $\mathcal{M}(X)$ the set of finite measures on $X$. Given a random
finite measure $\mu=(\mu_{\omega})_{\omega}$, we define the measure
$\pi_{X}(\mu)$ on $X$ by

$\pi_{X}(\mu)(B)=\mathbf{E}\left[\mu_{\omega}(B)\right]$

for every Borelian set $B\in\mathcal{B}$. We also define the measure
$\pi_{\Omega}(\mu)$ on $\Omega$ by

$\pi_{\Omega}(\mu)(A)=\mathbf{E}[\mathbf{1}_{A}\mu_{\omega}(X)].$

We call $\pi_{X}(\mu)$, respectively $\pi_{\Omega}(\mu)$, the $X$-marginal,
respectively the $\Omega$-marginal of $\mu$.

If $\mu=(\mu_{\omega})_{\omega}$ is a random finite measure, then one can
define a finite measure $\tilde{\mu}$ on $X\times\Omega$ by setting for every
measurable set $A$ of $X\times\Omega$

$\mu(A)=\mathbf{E}\left[\int\mathbf{1}_{A}(x,\omega)d\mu_{\omega}(x)\right].$

Then, $\pi_{X}(\mu)$ and $\pi_{\Omega}(\mu)$ are the push-forward measures of
$\mu$ by the canonical projections $\pi_{X}:X\times\Omega\to X$ and
$\pi_{\Omega}:X\times\Omega\to\Omega$.

######  Definition A.2.

A random finite measure $\mu$ is called $\mathbf{P}$-regular if $\mathbf{P}$
and $\pi_{\Omega}(\mu)$ are absolutely continuous with respect to each other.

######  Lemma A.3.

Let $\mu=(\mu_{\omega})_{\omega}$ be a random finite measure. Then
$\pi_{\Omega}(\mu)$ is absolutely continuous with respect to $\mathbf{P}$.
Moreover, $\mu$ is $\mathbf{P}$-regular if and only if for $\mathbf{P}$-almost
every $\omega$, $\mu_{\omega}$ is not the null measure.

###### Proof.

If $A$ is such that $\mathbf{P}(A)=0$, then $\mathbf{P}$-almost surely,
$\mathbf{1}_{A}\mu_{\omega}(X)=0$, so
$\pi_{\Omega}(\mu)(A)=\mathbf{E}[\mathbf{1}_{A}\mu_{\omega}(X)]=0$. This
concludes the first part of the lemma.

For the second part, assume that the event $A=\\{\omega,\mu_{\omega}(X)=0\\}$
has positive probability. Then, $\mathbf{E}[\mathbf{1}_{A}\mu_{\omega}(X)]=0$.
So $\pi_{\Omega}(\mu)(A)=0$, but $\mathbf{P}(A)>0$. Conversely, assume that
$\mathbf{P}$-almost surely, $\mu_{\omega}(X)>0$ and let $A$ be such that
$\pi_{\Omega}(\mu)(A)=0$, hence $\mathbf{P}$-almost surely,
$\mathbf{1}_{A}\mu_{\omega}(X)=0$. Then, we necessarily have
$\mathbf{1}_{A}=0$ $\mathbf{P}$-almost surely, i.e. $\mathbf{P}(A)=0$. ∎

######  Remark A.4.

In fact, by definition of $\pi_{\Omega}(\mu)$, we have that the Radon-Nikodym
derivative of $\pi_{\Omega}(\mu)$ with respect to $\mathbf{P}$ is given by

$\frac{d\pi_{\Omega}(\mu)}{d\mathbf{P}}(\omega)=\mu_{\omega}(X).$

Thus, $\mathbf{P}$ is absolutely continuous with respect to
$\pi_{\Omega}(\mu)$ if and only if this Radon-Nikodym derivative is almost
surely positive and then,

$\frac{d\mathbf{P}}{d\pi_{\Omega}(\mu)}(\omega)=\frac{1}{\mu_{\omega}(X)}$

$\mathbf{P}$-almost surely.

If $\mu=(\mu_{\omega})_{\omega}$ is a random probability measure, i.e.
$\mathbf{P}$-almost surely, $\mu_{\omega}$ is a probability measure on $X$,
then the $\Omega$-marginal of $\mu$ is $\mathbf{P}$. Thus,
$\mathbf{P}$-regularity is automatic in this context. However, it is easy to
construct an example where $\mathbf{P}$-regularity fails, since one only needs
that $\mathbf{P}(\mu_{\omega}(X)=0)>0$. Indeed, let $\mu$ be any random finite
measure and let $A$ be an event such that $\mathbf{P}(A)\leq 1/2$, then
$\mathbf{1}_{A}\mu$ is also a random finite measure and
$\mathbf{P}(\mathbf{1}_{A}\mu(X)=0)\geq 1/2$. Restricting our attention to
$\mathbf{P}$-regular measures will be important in the following, mainly
because of the following result.

######  Proposition A.5.

Every finite measure $\mu$ on $X\times\Omega$ such that $\pi_{\Omega}(\mu)$
and $\mathbf{P}$ are absolutely continuous with respect to each other is
defined by a random finite measure.

###### Proof.

Let $\mu$ be a finite measure on $X\times\Omega$ and denote by
$\|\mu\|=\mu(X\times\Omega)$ its mass. Then, $\nu=\mu/\|\mu\|$ is a
probability measure on $X\times\Omega$ with marginal
$\mathbf{Q}=\pi_{\Omega}(\mu)/\|\mu\|$. By the disintegration theorem [Cra02,
Proposition 3.6], there exists a map

$(B,\omega)\in\mathcal{B}\times\Omega\mapsto\nu_{\omega}(B)$

such that $\omega\mapsto\nu_{\omega}$ is $\mathbf{Q}$-almost surely a
probability measure on $X$ and for every Borelian $B\in\mathcal{B}$,
$\omega\mapsto\nu_{\omega}(B)$ is measurable.

By assumption, the probability measure $\mathbf{Q}$ is absolutely continuous
with respect to $\mathbf{P}$. Let $f=d\mathbf{Q}/d\mathbf{P}$ be the
corresponding Radon-Nikodym derivative. Define then
$\mu_{\omega}=\|\mu\|f(\omega)\nu_{\omega}$. Then, $\mu_{\omega}$ is
$\mathbf{Q}$-almost surely, hence $\mathbf{P}$-almost surely, a finite measure
on $X$, for every Borelian set $B$, the map $\omega\mapsto\mu_{\omega}(B)$ is
measurable and for every non-negative measurable function $\phi$ on
$X\times\Omega$, we have

$\displaystyle\mu(\phi)$ $\displaystyle=\|\mu\|\nu(\phi)$
$\displaystyle=\|\mu\|\int\int\phi(x,\omega)d\nu_{\omega}(x)d\mathbf{Q}(\omega)$
$\displaystyle=\int\int\phi(x,\omega)\|\mu\|d\nu_{\omega}(x)f(\omega)d\mathbf{P}(\omega)$
$\displaystyle=\mathbf{E}\left[\int\phi(x,\omega)d\mu_{\omega}(x)\right].$

Thus, $\mu$ is defined by the random finite measure $(\mu_{\omega})_{\omega}$.
∎

In general, the disintegration theorem allows one to decompose a finite
measure on $X\times\Omega$ along its $\Omega$-marginal, so without assuming
$\mathbf{P}$-regularity, it cannot be seen as a random finite measure in the
sense of Definition A.1.

We now introduce some definitions based on [Cra02].

######  Definition A.6.

A random closed set is a map $\omega\in\Omega\mapsto C(\omega)\in 2^{X}$
taking values in closed subsets of $X$ and such that the map $\omega\mapsto
d(x,C(\omega))$ is measurable for every $x\in X$. A random open set is a map
$\omega\mapsto U(\omega)$ such that the complement map $\omega\mapsto
U^{c}(\omega)$ is a random closed set.

Let $C=\omega\mapsto C(\omega)$ be a random closed set, $U=\omega\mapsto
U(\omega)$ a random open set and $\mu=(\mu_{\omega})_{\omega}$ a random finite
measure. We set

$\mu(C)=\mathbf{E}\left[\mu_{\omega}(C(\omega))\right]$

and

$\mu(U)=\mathbf{E}\left[\mu_{\omega}(U(\omega))\right].$

######  Definition A.7.

A random bounded continuous function is a map $f:X\times\Omega\to\mathbb{R}$
such that

  1. (a)

for every $x\in X$, $\omega\mapsto f(x,\omega)$ is measurable,

  2. (b)

for every $\omega\in\Omega$, $x\mapsto f(x,\omega)$ is continuous and bounded,

  3. (c)

There exists $C\geq 0$ such that for $\mathbf{P}$-almost every $\omega$,
$\|f(\cdot,\omega)\|_{\infty}\leq C$.

######  Remark A.8.

The third condition can be reformulated as $\|f(\cdot,\omega)\|_{\infty}$ is
in $L_{\infty}(\Omega,\mathbf{P})$. In [Cra02], the author introduces several
spaces of random functions, replacing the third condition by
$\|f(\cdot,\omega)\|_{\infty}\in L_{p}(\Omega,\mathbf{P})$. For $p=1$ the
corresponding space of function is called the space of random continuous
functions there.

We denote by $C_{\Omega,b}(X)$ the space of random bounded continuous
functions and endow $C_{\Omega,b}(X)$ with the $L_{\infty}\times
L_{\infty}$-norm $\|\cdot\|_{\infty}$, defined by

$\|f\|_{\infty}=\inf\left\\{C\geq
0,\mathbf{P}\big{(}\omega,\|f(\cdot,\omega)\|_{\infty}>C\big{)}=0\right\\},$

for every $f\in C_{\Omega,b}(X)$. If $f$ is a random bounded continuous
function and $\mu$ is a random finite measure, then the integral

$\mu(f)=\mathbf{E}\left[\int f(x,\omega)d\mu_{\omega}(x)\right]$

is well defined.

Recall that a Lipschitz function on $X$ is a function $f:X\to\mathbb{R}$ such
that

$\|f\|_{L}=\sup_{x,y\in X}\frac{|f(x)-f(y)|}{d(x,y)}$

is finite. We then set

$\|f\|_{BL}=\sup\\{\|f\|_{\infty},\|f\|_{L}\\}$

and say that $f$ is bounded Lipschitz if $\|f\|_{BL}$ is finite. We denote by
$BL(X)$ the set of bounded Lipschitz functions on $X$.

######  Definition A.9.

A random bounded Lipschitz function is a random bounded continuous function
$f$ such that there exists $C\geq 0$ such that for $\mathbf{P}$-almost every
$\omega$, the map $x\mapsto f(x,\omega)$ is bounded Lipschitz and
$\|f(\cdot,\omega)\|_{BL}\leq C$.

######  Remark A.10.

In [Cra02], random bounded Lipschitz functions are called random Lipschitz
functions. We changed the terminology to insist on the fact that random
functions we are considering here are $\mathbf{P}$-essentially bounded.

We denote by $BL_{\Omega}(X)$ the set of random bounded Lipschitz functions on
$X$.

######  Lemma A.11.

If two random finite measures $(\mu_{\omega})_{\omega}$ and
$(\nu_{\omega})_{\omega}$ coincide on random bounded Lipschitz functions, i.e.
for every $f\in BL_{\Omega}(X)$, $\mu(f)=\nu(f)$, then for $\mathbf{P}$-almost
every $\omega$, $\mu_{\omega}=\nu_{\omega}$.

###### Proof.

For every closed set $C\in X$, the sequence of functions

$f_{n}:x\mapsto 1-(1\wedge nd(x,C))$

is non-increasing and converges to $\mathbf{1}_{C}$. Moreover, for every $n$,
$f_{n}$ is bounded Lipschitz. Therefore, for every event $A$, the maps
$(x,\omega)\mapsto\mathbf{1}_{A}(\omega)f_{n}(x)$ are random bounded Lipschitz
functions. Thus by monotone convergence, it suffices to prove that if for all
closed set $C$, for all event $A$, $\mu(C\times A)=\nu(C\times A)$, then
$(\mu_{\omega})_{\omega}$ and $(\nu_{\omega})_{\omega}$ coincide
$\mathbf{P}$-almost surely. Since closed sets are closed under finite
intersection and the measures we are considering are finite, this follows from
the monotone class theorem [Bil86, Theorem 3.4]. ∎

###  A.2. The weak topology on random finite measures

Recall that the weak topology on $\mathcal{M}(X)$ is the smallest topology
such that for all bounded continuous function $f:X\to\mathbb{R}$, the map
$\mu\mapsto\mu(f)$ is continuous.

######  Definition A.12.

The weak topology on $\mathcal{M}_{\Omega}(X)$ is the topology generated by
the maps $\mu\mapsto\mu(f)$, for every $f\in C_{\Omega,b}(X)$, i.e. it is the
smallest topology on $\mathcal{M}_{\Omega}(X)$ such that for every $f\in
C_{\Omega,b}(X)$, the map
$\mu\in\mathcal{M}_{\Omega}(X)\mapsto\mu(f)\in\mathbb{R}$ is continuous.

######  Remark A.13.

In [Cra02], the author defines the narrow topology on the space of random
probability measures as the topology generated by the maps $\mu\mapsto\mu(f)$
for every random continuous function $f$. Recall that a random continuous
function as defined there is a function $f$ such that for all $\omega$,
$f(\cdot,\omega)$ is bounded continuous and $\|f(\cdot,\omega)\|_{\infty}$ is
in $L_{1}(\Omega,\mathbf{P})$.

  * •

First, we preferred to use the terminology weak topology which is more common,
although both exist in literature.

  * •

Second, by [Cra02, Lemma 3.16], for random probability measures, the induced
weak topology on the set of measures is the same when choosing either random
bounded continuous functions or random continuous functions. However, in our
context, the proof of this lemma does not apply and it seems that choosing
different spaces of functions can yield different notions of weak topologies.

######  Lemma A.14.

The weak topology is generated by the maps $\mu\mapsto\mu(f)$ for every $f\in
BL_{\Omega}(X)$, i.e. it is the smallest topology on $\mathcal{M}_{\Omega}(X)$
such that for every random bounded Lipschitz function $f$, $\mu\mapsto\mu(f)$
is continuous.

###### Proof.

This follows from the fact that bounded continuous functions can be
approximated by bounded Lipschitz functions, see [Cra02, Proposition 4.9] for
more details. ∎

We now prove the following generalization of the classical Portmanteau theorem
in terms of convergent nets. We refer to [AB06, Definition 2.11, Definition
2.15] for more details on nets.

######  Proposition A.15.

Let $\mu_{\alpha}=((\mu_{\alpha})_{\omega})_{\omega}$ be a net of random
finite measures and let $\mu=(\mu_{\omega})_{\omega}$ be a random finite
measure. Then, the following assertions are equivalent.

  1. (1)

The net $\mu_{\alpha}$ converges to $\mu$ for the weak topology.

  2. (2)

For all random closed set $C=\omega\mapsto C(\omega)$,
$\limsup_{\alpha}\mu_{\alpha}(C)\leq\mu(C)$ and $\mu_{\alpha}(X\times\Omega)$
converges to $\mu(X\times\Omega)$.

  3. (3)

For all random open set $U=\omega\mapsto U(\omega)$,
$\liminf_{\alpha}\mu_{n}(U)\geq\mu(U)$ and $\mu_{\alpha}(X\times\Omega)$
converges to $\mu(X\times\Omega)$.

###### Proof.

Taking complements, the second and third assertions are equivalent. Also, if
$\mu_{\alpha}$ converges to $\mu$, applying the definition of the weak
topology to the constant function $\mathbf{1}$, we see that
$\mu_{\alpha}(X\times\Omega)$ converges to $\mu(X\times\Omega)$.

Let us assume that $\mu_{\alpha}$ converges to $\mu$ and let $C$ be a closed
random set. For every $k\in\mathbb{N}$, set

$f_{k}(x,\omega)=1-(1\wedge kd(x,C(\omega))).$

Then, for every $k$, $f_{k}$ is a random bounded Lipschitz function and the
sequence $f_{k}$ is non-increasing and converges to
$\mathbf{1}_{C(\omega)}(x)$. Thus, for every $k$,

$\limsup_{\alpha}\mu_{\alpha}(C)\leq\lim_{\alpha}\mu_{\alpha}(f_{k})=\mu(f_{k}),$

so

$\limsup_{\alpha}\mu_{\alpha}(C)\leq\inf_{k}(\mu(f))=\mu(C).$

Consequently, the first assertion implies the second one.

Assume now that for all closed random set $C$,
$\limsup\mu_{\alpha}(C)\leq\mu(C)$ and that $\mu_{\alpha}(X\times\Omega)$
converges to $\mu(X\times\Omega)$. We show that for every non-negative random
bounded continuous function $f$,

(25) $\limsup\mu_{\alpha}(f)\leq\mu(f).$

Fix $m\in\mathbb{N}$ and set for every $0\leq k\leq m$

$C_{k}(\omega)=\left\\{x\in
X,f(x,\omega)\geq\frac{k}{m}\|f(\cdot,\omega)\|_{\infty}\right\\}.$

Then, $C_{k}=\omega\mapsto C_{k}(\omega)$ is a random closed set. By [Cra02,
Lemma 1.4], for every random finite measure $(\nu_{\omega})_{\omega}$,

$\frac{1}{m}\sum_{k=1}^{m}\nu(C_{k})\leq\nu(f)\leq\frac{1}{m}\sum_{k=0}^{m}\nu(C_{k}).$

Applying this both to $\mu_{\alpha}$ and $\mu$, we get

$\displaystyle\mu(f)\geq\frac{1}{m}\sum_{k=1}^{m}\mu(C_{k})$
$\displaystyle\geq\limsup_{\alpha}\frac{1}{m}\sum_{k=1}^{m}\mu_{\alpha}(C_{k})$
$\displaystyle=\limsup_{\alpha}\left(\frac{1}{m}\sum_{k=0}^{m}\mu_{\alpha}(C_{k})-\frac{\mu_{\alpha}(X\times\Omega)}{m}\right)$
$\displaystyle\geq\limsup_{\alpha}\mu_{\alpha}(f)-\frac{\mu(X\times\Omega)}{m}.$

Since $m$ is arbitrary, this proves (25). Using that
$\mu_{\alpha}(\mathbf{1})=\mu_{\alpha}(X\times\Omega)$ converges to
$\mu(\mathbf{1})=\mu(X\times\Omega)$ and applying (25) to the function
$\|f\|_{\infty}-f$, we get that $\mu_{\alpha}(f)$ converges to $\mu(f)$. This
is true for all non-negative random bounded continuous function, so
$\mu_{\alpha}$ converges to $\mu$ for the weak topology. ∎

######  Remark A.16.

By Lemma A.11, the weak topology is Hausdorff. We do not attempt to study
metrizability of the weak topology in here to avoid lengthily arguments, but
in [Cra02, Theorem 4.16], the author proves that the weak topology on the
space of random probability measures is metrizable, provided that the
probability space $(\Omega,\mathcal{F},\mathbf{P})$ is countably generated
(mod. $\mathbf{P}$). This might also holds in our situation.

###  A.3. A compactness criterion for random finite measures

In all this section, $\mathcal{M}(X)$ and $\mathcal{M}_{\Omega}(X)$ are
endowed with the weak topology. Let us recall the following definition.

######  Definition A.17.

A subset $M$ of $\mathcal{M}(X)$ is tight if for every $\epsilon>0$, there
exists a compact subset of $X$ such that for every $\mu\in M$, we have

$\mu(K^{c})\leq\epsilon.$

Following [Cra02], we define tightness for random finite measures as tightness
under the projection map $\pi_{X}$.

######  Definition A.18.

A subset $M_{\Omega}$ of $\mathcal{M}_{\Omega}(X)$ is tight if
$\pi_{X}(M_{\Omega})$ is tight, i.e. for every $\epsilon>0$, there exists a
compact subset of $X$ such that for all $(\mu_{\omega})_{\omega}\in
M_{\Omega}$, we have

$\mathbf{E}[\mu_{\omega}(K^{c})]\leq\epsilon.$

The classical Prokhorov theorem states that a set $M$ of probability measures
on $X$ is relatively compact if and only if it is tight. The following
generalizes this result to finite measures.

######  Theorem A.19.

Prokhorov Theorem for finite measures [Kal17, Lemma 4.4]. A subset $M$ of
$\mathcal{M}(X)$ is relatively compact if and only if $M$ is tight and
uniformly bounded, in the sense that $\sup_{\mu\in M}\mu(X)$ is finite.

Our goal is to generalize this to random finite measures. Unfortunately, we
will not get a necessary and sufficient condition for compactness as in the
Prokhorov Theorem but only a sufficient condition.

We follow the strategy of [Cra02, Theorem 4.4] and first prove the following
representation result. Recall that $BL_{\Omega}(X)$ denotes the set of random
bounded Lipschitz functions and $BL(X)$ denotes the set of bounded Lipschitz
functions on $X$. Then, if $f\in BL(X)$, $f$ can be viewed as an element of
$BL_{\Omega}(X)$ by setting $f(x,\omega)=f(x)$. Similarly, if $f$ is
$\mathbf{P}$ -essentially bounded, i.e. $f\in L_{\infty}(\Omega,\mathbf{P})$,
then $f$ can be viewed as an element of $BL_{\Omega}(X)$ by setting
$f(x,\omega)=f(\omega)$. If $L:BL_{\Omega}(X)\to\mathbb{R}$ is a function, we
denote by $\pi_{X}(L)$, respectively $\pi_{\Omega}(L)$ its restriction to
bounded Lipschitz functions, respectively to $\mathbf{P}$-essentially bounded
functions.

######  Lemma A.20.

Let $L:BL_{\Omega}(X)\to\mathbb{R}$. be a function. Assume that the following
conditions hold

  1. (a)

$L$ is linear,

  2. (b)

$L$ is non-negative, i.e. for every non-negative function $f$ in
$BL_{\Omega}(X)$, we have $L(f)\geq 0$,

  3. (c)

there exists $\kappa\in\mathcal{M}(X)$ such that for every $f\in BL(X)$, we
have

$\pi_{X}(L)(f)=\kappa(f)=\int fd\kappa,$

  4. (d)

there exists a constant $C>0$ such that for every $f\in
L_{\infty}(\Omega,\mathbf{P})$, we have

$\frac{1}{C}\pi_{\Omega}(L)(f)\leq\mathbf{E}[f]\leq C\pi_{\Omega}(L)(f).$

Then, there exists a random finite measure $\mu=(\mu_{\omega})_{\omega}$ such
that $L(f)=\mu(f)$ for every $f\in BL_{\Omega}(X)$.

###### Proof.

The proof relies on the general Stone-Daniell representation theorem. We claim
that if $f_{n}$ is a non-increasing sequence of functions of $BL_{\Omega}(X)$
converging to 0, then $L(f_{n})$ is non-increasing and converges to 0. By
[Cra02, Theorem 4.11], we deduce that there exists a measure $\mu$ on
$X\times\Omega$ such that $L(f)=\mu(f)$ for every $f\in BL_{\Omega}(X)$.
Conditions (c) and (d) ensure that $\mu$ is finite. Moreover, the
$\Omega$-marginal $\pi_{\Omega}(\mu)$ of $\mu$ is $\pi_{\Omega}(L)$ and so the
fourth condition shows $\pi_{\Omega}(\mu)$ and $\mathbf{P}$ are absolutely
continuous with respect to each other. Therefore, we can apply Proposition
A.5, so that $\mu$ is defined by a random finite measure.

We just need to prove the claim to conclude the proof. Let $f_{n}$ be a non-
increasing sequence of random bounded Lipschitz functions converging to 0.
Since $L$ is linear and non-negative, we get that $L(f_{n})$ is non-
inscreasing. We need to prove that $L(f_{n})$ converges to 0. Fix
$\epsilon>0$. Let $K$ be a compact such that $\kappa(K^{c})\leq\epsilon$. For
every $n$, there exists $C_{n}$ such that $\|f_{n}(\cdot,\omega)\|_{BL}\leq
C_{n}$, for $\mathbf{P}$-almost every $\omega$. Set
$\delta_{n}=\epsilon/C_{n}$ and consider the function

$\chi_{n}:x\mapsto\chi_{n}(x)=1-\big{(}1\wedge\delta_{n}^{-1}d(x,K)\big{)}.$

Then, $\chi_{n}$ vanishes outside the $\delta_{n}$-neighborhood of $K$.
Moreover, the function $g_{n}:(x,\omega)\mapsto f_{n}(x,\omega)\chi_{n}(x)$ is
in $BL_{\Omega}(X)$. Now, $f_{n}=g_{n}+f_{n}(1-\chi_{n})$ and so

(26) $L(f_{n})=L(g_{n})+L(f_{n}(1-\chi_{n})).$

We first deal with the second term in the right-hand side of (26). Since
$f_{n}$ is non-increasing, we have that
$f_{n}(x,\omega)\leq\|f_{1}(\cdot,\omega)\|_{\infty}$ Moreover, there exists
$M\geq 0$ such that the event $A=\\{\|f_{1}(\cdot,\omega)\|_{\infty}\leq M\\}$
satisfies $\mathbf{P}(A^{c})=0$. Note that we can write $f_{n}(1-\chi_{n})\leq
M(1-\chi_{n})\mathbf{1}_{A}+\|f_{1}(\cdot,\omega)\|\mathbf{1}_{A^{c}}$. By
Condition (d), $L(\|f_{1}(\cdot,\omega)\|\mathbf{1}_{A^{c}})=0$. Thus,

$L(f_{n}(1-\chi_{n}))\leq ML(1-\chi_{n})=M\int(1-\chi_{n})d\kappa.$

Note that $1-\chi_{n}$ vanishes in $K$, so

(27) $L(f_{n}(1-\chi_{n}))\leq M\kappa(K^{c})\leq M\epsilon.$

We now deal with the first term in the right-hand side of (26). For every
$\omega$ such that $\|f_{n}(\cdot,\omega)\|_{BL}\leq C_{n}$ and for every
$x,y\in X$, we have

$|g_{n}(x,\omega)-g_{n}(y,\omega)|\leq C_{n}d(x,y)+2f_{n}(y,\omega).$

If $x$ is in the $\delta_{n}$-neighborhood of $K$, choose $y\in K$ such that
$d(x,y)\leq\delta_{n}$. Then, by what precedes,

$g_{n}(x,\omega)\leq 3\sup_{y\in K}f_{n}(y,\omega)+C_{n}\delta_{n}\leq
3\sup_{y\in K}f_{n}(y,\omega)+\epsilon.$

Since $\chi_{n}$ vanishes outside the $\delta_{n}$-neighborhood of $K$, this
yields

$\sup_{x\in X}g_{n}(x,\omega)\leq 3\sup_{y\in K}f_{n}(y,\omega)+\epsilon.$

We write $h_{n}(\omega)=\sup_{y\in K}f_{n}(y,\omega)$. The event
$A_{n}=\\{\omega$, $\|f_{n}(\cdot,\omega)\|_{BL}>C_{n}\\}$ satisfies
$\mathbf{P}(A_{n})=0$, hence the same manipulation as above shows that

(28) $L(g_{n})\leq 3L(h_{n})+\epsilon L(\mathbf{1})\leq
3C\mathbf{E}(h_{n})+\kappa(X)\epsilon,$

using Conditions (c) and (d).

Combining (26), (27) and (28), we get

$L(f_{n})\leq 3C\mathbf{E}(h_{n})+(\kappa(X)+M)\epsilon.$

Using that $f_{n}$ is non-increasing and converges to 0, that for all
$\omega$, $f_{n}(\cdot,\omega)$ is continuous and that $K$ is compact, we get
that $h_{n}$ is non-increasing and converges to 0. By monotone convergence,
$\mathbf{E}(h_{n})$ converges to 0, so for large enough $n$,
$\mathbf{E}(h_{n})\leq\epsilon$. Thus, for large enough $n$,

$L(f_{n})\leq(3C+\kappa(X)+M)\epsilon.$

Since $\epsilon$ is arbitrary, this concludes the proof of the claim. ∎

The main ingredient in proving a compactness criterion for random finite
measures is the following proposition.

######  Proposition A.21.

Let $C>0$. If $K$ is compact in $\mathcal{M}(X)$, then

$\pi_{X}^{-1}(K)\cap\left\\{\mu\in\mathcal{M}_{\Omega}(X),\frac{1}{C}\pi_{\Omega}(\mu)\leq\mathbf{P}\leq
C\pi_{\Omega}(\mu)\right\\}$

is compact in $\mathcal{M}_{\Omega}(X)$.

###### Proof.

Let $f\in BL_{\Omega}(X)$ and set

$M_{f}^{+}=\sup_{\mu\in\pi_{X}^{-1}(K)}\mu(f),$
$M_{f}^{-}=\inf_{\mu\in\pi_{X}^{-1}(K)}\mu(f).$

Recall that if $f\in BL_{\Omega}(X)$, then in particular, $f\in
C_{\Omega,b}(X)$ and so for $\mathbf{P}$-almost every $\omega$,
$|f(\cdot,\omega)|$ is bounded and $\mathbf{E}[\|f(\cdot,\omega)\|_{\infty}]$
is bounded. Thus, $\pi_{X}(\mu)(f)$ is finite. Since $K$ is compact, this
proves that $M_{f}^{+}$ and $M_{f}^{-}$ are finite.

We set

$\mathbf{C}=\prod_{f\in BL_{\Omega}(X)}[M_{f}^{-},M_{f}^{+}]$

and we endow $\mathbf{C}$ with the product topology. Then $\mathbf{C}$ is
compact by the Tychonoff Theorem [AB06, Theorem 2.61]. We can identify
$\mathbf{C}$ with the set of all functions $L:BL_{\Omega}(X)\to\mathbb{R}$
such that $L(f)\in[M_{f}^{-},M_{f}^{+}]$. A neighborhood basis of an element
$L_{0}$ of $\mathbf{C}$ is given by

$U_{\delta}(f_{1},...f_{n})(L_{0})=\\{L\in\mathbf{C},|L(f_{k})-L_{0}(f_{k})|<\delta,1\leq
k\leq n\\},$

where $\delta>0$ and $f_{1},...,f_{n}\in BL_{\Omega}(X)$. We define the map

$I:\mu\in\pi_{X}^{-1}(K)\mapsto(\mu(f))_{f\in BL_{\Omega}(X)}\in\mathbf{C}.$

Then, $I$ is one-to-one and continuous. Moreover, by Lemma A.14, a
neighborhood basis of an element $\mu_{0}\in\pi_{X}^{-1}(K)$ for the weak
topology is given by

$V_{\delta}(f_{1},...,f_{n})(\mu_{0})=\\{\mu\in\pi_{X}^{-1}(K),|\mu(f_{k})-\mu_{0}(f_{k})|<\delta,1\leq
k\leq n\\},$

where $\delta>0$ and $f_{1},...,f_{n}\in BL_{\Omega}(X)$. Since

$I\big{(}V_{\delta}(f_{1},...,f_{n})(\mu_{0})\big{)}=U_{\delta}(f_{1},...,f_{n})(I(\mu_{0})),$

we see that $I$ is an open map and thus a homeomorphism onto its image in
$\mathbf{C}$.

We make a similar construction for non-random finite measures. Recall that
$BL(X)$ is the set of (non-random) bounded Lipschitz functions on $X$ and that
$BL(X)$ can be seen as a subset of $BL_{\Omega}(X)$ by setting
$f(x,\omega)=f(x)$ for any $f\in BL(X)$. We define

$\mathbf{c}=\prod_{f\in BL(X)}[M_{f}^{-},M_{f}^{+}]$

and

$i(\kappa)=(\kappa(f))_{f\in BL(X)}.$

Then, $i$ is a homeomorphism onto its image in $\mathbf{c}$. If
$L\in\mathbf{C}$, denote by $\pi(L)\in\mathbf{c}$ its restriction to non-
random bounded Lipschitz functions.

For every $f\in BL(X)$, seen as an element of $BL_{\Omega}(x)$ and for every
random finite measure $\mu$, we have

$\mu(f)=\mathbf{E}\left[\int f(x)d\mu_{\omega}(x)\right]=\pi_{X}(\mu)(f).$

Thus, $\pi\circ I=i\circ\pi_{X}$. Now, fix $C>0$. By Lemma A.20, we have

$\displaystyle
I\left(\pi_{X}^{-1}(K)\cap\left\\{\mu,\frac{1}{C}\leq\pi_{\Omega}(\mu)\leq\mathbf{P}\leq
C\pi_{\Omega}(\mu)\right\\}\right)$ $\displaystyle=$ $\displaystyle\hskip
5.69046pt\pi^{-1}(i(K))\cap\\{L\text{ linear}\\}\cap\\{L\text{ non-
negative}\\}$ $\displaystyle\hskip
8.5359pt\cap\left\\{L,\frac{1}{C}\pi_{\Omega}(L)\leq\mathbf{E}[\cdot]\leq
C\pi_{\Omega}(L)\right\\}.$

The four sets on the right-hand side are closed. Since $\mathbf{C}$ is compact
and $I$ is a homeomorphism onto its image, we get that

$\pi_{X}^{-1}(K)\cap\left\\{\mu,\frac{1}{C}\leq\pi_{\Omega}(\mu)\leq\mathbf{P}\leq
C\pi_{\Omega}(\mu)\right\\}$

is compact. ∎

######  Corollary A.22.

Consider a subset $M_{\Omega}$ of $\mathcal{M}_{\Omega}(X)$. Assume that
$M_{\Omega}$ is tight. Also assume that there exists $C\geq 0$ such that for
every $\mu\in M_{\Omega}$,

$\frac{1}{C}\pi_{\Omega}(\mu)\leq\mathbf{P}\leq C\pi_{\Omega}(\mu).$

Then, $M_{\Omega}$ is relatively compact.

###### Proof.

If $M_{\Omega}$ satisfies the assumptions of the corollary, then
$\pi_{X}(M_{\Omega})\subset\mathcal{M}(X)$ is tight. Also, the condition
$\pi_{\Omega}(\mu)\leq C\mathbf{P}$ can be reformulated by

$\frac{d\pi_{\omega}(\mu)}{d\mathbf{P}}(\omega)=\mu_{\omega}(X)\leq C$

$\mathbf{P}$-almost surely. In particular,

$\pi_{X}\mu(X)=\mathbf{E}[\mu_{\omega}(X)]\leq C,$

so $\pi_{X}(\mu)(X)$ is uniformly bounded. By Theorem A.19,
$\pi_{X}(M_{\Omega})$ is relatively compact, i.e.
$\mathbf{cl}(\pi_{X}(M_{\Omega}))$ is compact. Moreover,

$M_{\Omega}\subset\pi_{X}^{-1}\big{(}\mathbf{cl}(\pi_{X}(M_{\Omega}))\big{)}\cap\left\\{\mu,\frac{1}{C}\pi_{\Omega}(\mu)\leq\mathbf{P}\leq
C\pi_{\Omega}(\mu)\right\\}.$

Thus, $\mathbf{cl}(M_{\Omega})$ is compact. ∎

This corollary does not give a necessary and sufficient condition for
compactness, but only a sufficient one. On the contrary, [Cra02, Theorem 4.4]
gives a necessary and sufficient condition for compactness in the context of
random probability measures : a set is relatively compact if and only if it is
tight. However, assuming that for $\mathbf{P}$-almost every $\omega$,
$\mu_{\omega}$ is a probability measure, we get that $\pi_{X}(\mu)$ is a
probability measures and that the $\Omega$-marginal of $\mu$ is exactly
$\mathbf{P}$. This ensures that the others conditions of Corollary A.22 are
automatically satisfied and so tightness is sufficient to get compactness.

Recall that the Radon-Nikodym derivative of $\pi_{\Omega}(\mu)$ with respect
to $\mathbf{P}$ is given by $\mu_{\omega}(X)$. In our context, on the one
hand, the assumption

$\frac{1}{C}\pi_{\Omega}(\mu)\leq\mathbf{P}\leq C\pi_{\Omega}(\mu)$

which is equivalent to

$\frac{1}{C}\leq\mu_{\omega}(X)\leq C$

for $\mathbf{P}$-almost every $\omega$ seems strong. Assuming only
$\mathbf{P}$-regularity, there is no reason for this Radon-Nikodym derivative
to be $\mathbf{P}$-essentially bounded from above and below. Even when
restricting our attention to random finite measures satisfying this property,
the maps

$\mu\mapsto\mathbf{P}-\textbf{ess}\sup\mu_{\omega}(X)$

and

$\mu\mapsto\mathbf{P}-\textbf{ess}\inf\mu_{\omega}(X)$

have no reason to be continuous. Thus, compactness for the weak topology does
not seem to imply the existence of a uniform $C$ such that the condition

$\frac{1}{C}\pi_{\Omega}(\mu)\leq\mathbf{P}\leq C\pi_{\Omega}(\mu)$

holds. On the other hand, it is not realistic to expect a sufficient condition
for compactness without assuming anything on the $\Omega$-marginals.

In the particular case where $X$ has an isolated point $x_{0}$, then the
following trick allows us to weaken a bit this assumption. For any subset of
$M_{\Omega}$ of $\mathcal{M}_{\Omega}(X)$, define

$\widetilde{M}_{\Omega}=\left\\{\nu\in\mathcal{M}_{\Omega}(X),\nu=\mathbf{D}(x_{0})+\mu,\mu\in
M_{\Omega}\right\\},$

where $\mathbf{D}$ is the Dirac measure on $x_{0}$. Assume that for every
$\mu\in M_{\Omega}$, we have $\mu_{\omega}(X)\leq C$ for $\mathbf{P}$-almost
every $\omega$, for some constant $C$. Then for every
$\nu\in\widetilde{M}_{\Omega}$,

(29) $1\leq\nu_{\omega}(X)\leq 1+C$

for $\mathbf{P}$-almost every $\omega$.

######  Corollary A.23.

Assume that $X$ has an isolated point $x_{0}$. Consider a subset $M_{\Omega}$
of $\mathcal{M}_{\Omega}(X)$. Assume that $M_{\Omega}$ is tight and that there
exists $C\geq 0$ such that for every $\mu\in M_{\Omega}$,
$\pi_{\Omega}(\mu)\leq C\mathbf{P}$. Then, $M_{\Omega}$ is relatively compact.

###### Proof.

The set $M_{\Omega}$ is tight, so there is a compact $K_{0}$ such that for
every $\mu\in M_{\Omega}$, $\pi_{X}(\mu)(K_{0}^{c})\leq\epsilon$. Then, the
set $K=K_{0}\cup\\{x_{0}\\}$ is also compact and for every
$\nu\in\widetilde{M}_{\Omega}$, $\pi_{X}(\nu)(K^{c})\leq\epsilon$. Thus,
$\widetilde{M}_{\Omega}$ is also tight. By (29), $\widetilde{M}_{\Omega}$
satisfies the assumptions of Corollary A.22, so it is relatively compact.

Let $\nu\in\mathbf{cl}(\widetilde{M}_{\Omega})$. By [AB06, Theorem 2.14],
there exists a net $(\nu_{\alpha})_{\alpha\in A}$ converging to $\nu$. Assume
by contradiction that $\mathbf{P}(\nu_{\omega}(\\{x_{0}\\})<1)>0$. Then, there
exists $c<1$ such that the event $A=\\{\nu_{\omega}(\\{x_{0}\\})\leq c\\}$
satisfies $\mathbf{P}(A)>0$. Since $x_{0}$ is isolated and $X$ is Hausdorff,
$\\{x_{0}\\}$ is both closed and open, hence the function $\mathbf{1}_{x_{0}}$
is bounded continuous and
$(x,\omega)\mapsto\mathbf{1}_{A}(\omega)\mathbf{1}_{x_{0}}(x)$ is a random
bounded continuous function. Since for every $\alpha$,
$\nu_{\alpha}(\\{x_{0}\\})\geq 1$, applying convergence to this function, we
get that

$\mathbf{P}(A)\leq\mathbf{E}[\mathbf{1}_{A}\nu_{\alpha}(\\{x_{0}\\})]\underset{\alpha\to\infty}{\longrightarrow}\mathbf{E}[\mathbf{1}_{A}\nu(\\{x_{0}\\})]\leq
c\mathbf{P}(A),$

which is a contradiction. Thus,

$\mu_{\omega}=\nu_{\omega}-\mathbf{D}(x_{0})$

is a well-defined random finite measure on $X$. In other words, the map

$F:\nu\in\mathbf{cl}(\widetilde{M}_{\Omega})\mapsto\nu-\mathbf{D}(x_{0})\in\mathcal{M}_{\Omega}(X)$

is well defined. Moreover, it is continuous, therefore
$F(\mathbf{cl}(\widetilde{M}_{\Omega}))$ is compact and by Lemma A.11, the
weak topology is Hausdorff, so $F(\mathbf{cl}(\widetilde{M}_{\Omega}))$ is
closed. Now, $M_{\Omega}\subset F(\mathbf{cl}(\widetilde{M}_{\Omega}))$ and so
$\mathbf{cl}(M_{\Omega})\subset F(\mathbf{cl}(\widetilde{M}_{\Omega}))$. Thus,
$\mathbf{cl}(M_{\Omega})$ is compact, i.e. $M_{\Omega}$ is relatively compact.
∎

## References

  * [AB06] Charalambos Aliprantis and Kim Border.  Infinite Dimensional Analysis: a hitchhiker’s guide.  Springer, 2006.
  * [AN04] Krishna Athreya and Peter Ney.  Branching processes.  Dover Publications, Inc., Mineola, NY, 2004.  Reprint of the 1972 original [Springer, New York; MR0373040].
  * [Anc88] Alano Ancona.  Positive harmonic functions and hyperbolicity.  In Potential theory-surveys and problems, pages 1–23. Lecture notes in mathematics, Springer, 1988.
  * [BB07] Sébastien Blachère and Sara Brofferio.  Internal diffusion limited aggregation on discrete groups having exponential growth.  Probab. Theory Relat. Fields, 137:323–343, 2007.
  * [Bil86] Patrick Billingsley.  Probability and Measure.  John Wiley and Sons, second edition, 1986.
  * [BM12] Itai Benjamini and Sebastian Müller.  On the trace of branching random walks.  Groups Geom. Dyn., 6:231–247, 2012.
  * [Bow12] Brian Bowditch.  Relatively hyperbolic group.  Int. J. Algebra Comput., 22:1–66, 2012.
  * [BP94] Itai Benjamini and Yuval Peres.  Markov chains indexed by trees.  Ann. Probab., 22(1):219–243, 1994.
  * [Car88] Donald Cartwright.  Some examples of random walks on free products of discrete groups.  Ann. di Mat. Pura ed Appl., 151:1–15, 1988.
  * [Car89] Donald Cartwright.  On the asymptotic behaviour of convolution powers of probabilities on discrete groups.  Monatsh. Math., 107:287–290, 1989.
  * [CG12] Elisabetta Candellero and Lorenz Gilch.  Phase transitions for random walk asymptotics on free products of groups.  Random Struct. Algorithms, 40:150–181, 2012.
  * [CGM12] Elisabetta Candellero, Lorenz Gilch, and Sebastian Müller.  Branching random walks on free products of groups.  Proc. London Math. Soc., 104:1085–1120, 2012.
  * [Cor17] Matthew Cordes.  Morse boundaries of proper geodesic metric spaces.  Groups Geom. Dyn., 11:1281–1306, 2017.
  * [Cra02] Hans Crauel.  Random probability measures on Polish spaces.  CRC Press, 2002.
  * [DFW19] François Dahmani, David Futer, and Daniel T. Wise.  Growth of quasiconvex subgroups.  Math. Proc. Cambridge Philos. Soc., 167(3):505–530, 2019.
  * [DG20] Matthieu Dussaule and Ilya Gekhtman.  Entropy and drift for word metrics on relatively hyperbolic groups.  Groups Geom. Dyn., 14:1455–1509, 2020.
  * [DG21] Matthieu Dussaule and Ilya Gekhtman.  Stability phenomena for Martin boundaries of relatively hyperbolic groups.  Probab. Theory Relat. Fields, 179:201–259, 2021.
  * [DOP00] Françoise Dal’bo, Jean-Pierre Otal, and Marc Peigné.  Séries de Poincaré des groupes géométriquement finis.  Isr. J. Math., 118:109–124, 2000.
  * [DPPS11] Françoise Dal’bo, Marc Peigné, Jean-Claude Picaud, and Andrea Sambusetti.  On the growth of quotients of Kleinian groups.  Ergod. Theory Dyn. Syst., 31:835–851, 2011.
  * [DPT22] Matthieu Dussaule, Marc Peigné, and Samuel Tapie.  A local limit theorem for convergent random walks on relatively hyperbolic groups.  arXiv:2202.11339, 2022.
  * [DS05] Cornelia Druţu and Mark Sapir.  Tree graded spaces and asymptotic cones of groups.  Topology, 44:959–1058, 2005.  with an Appendix by Denis Osin and Mark Sapir.
  * [Dus22a] Matthieu Dussaule.  Local limit theorems in relatively hyperbolic groups I : rough estimates.  Ergod. Theory Dyn. Syst., 42:1926 –1966, 2022.
  * [Dus22b] Matthieu Dussaule.  Local limit theorems in relatively hyperbolic groups II : the non spectrally degenerate case.  Compos. Math., 158:764–830, 2022.
  * [DY20] Matthieu Dussaule and Wenyuan Yang.  The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups.  arXiv:2010.07671, 2020.
  * [Far98] Benson Farb.  Relatively hyperbolic groups.  GAFA, 8:810–840, 1998.
  * [FK60] Harry Furstenberg and Harry Kesten.  Products of random matrices.  Ann. Math. Statist., 31:457–469, 1960.
  * [Flo80] William Floyd.  Group completions and limit sets of Kleinian groups.  Invent. Math., 57:205–2018, 1980.
  * [Fol99] Gerald Folland.  Real analysis : Modern techniques and their applications.  John Wiley and sons, 1999.
  * [Fre45] Hans Freudenthal.  Über die enden diskreter raüme und gruppen.  Comment. Math. Helvetici, 17:1–38, 1945.
  * [Ger12] Victor Gerasimov.  Floyd maps for relatively hyperbolic groups.  GAFA, 22:1361–1399, 2012.
  * [GGPY21] Ilya Gekhtman, Victor Gerasimov, Leonid Potyagailo, and Wenyuan Yang.  Martin boundary covers Floyd boundary.  Invent. Math., 223:759–809, 2021.
  * [GL13] Sébastien Gouëzel and Steven Lalley.  Random walks on co-compact Fuchsian groups.  Ann. Sci. Éc. Norm. Supér., 46:129–173, 2013.
  * [GM06] Nina Gantert and Sebastian Müller.  The critical branching Markov chain is transient.  Markov Process. Related Fields, 12:805–814, 2006.
  * [Gou14] Sébastien Gouëzel.  Local limit theorem for symmetric random walks in Gromov-hyperbolic groups.  JAMS, 27:893–928, 2014.
  * [GP13] Victor Gerasimov and Leonid Potyagailo.  Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups.  JEMS, 15:2115–2137, 2013.
  * [GP15] Victor Gerasimov and Leonid Potyagailo.  Non-finitely generated relatively hyperbolic groups and Floyd quasiconvexity.  Groups, Geom. Dyn., 9:369–434, 2015.
  * [Gui80] Yves Guivarc’h.  Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire.  In Conference on Random Walks, pages 47–98. Astérisque, vol. 74, Soc. Math. France, 1980.
  * [HL00] Irene Hueter and Steven Lalley.  Anisotropic branching random walks on homogeneoustrees.  Probab. Theory Relat. Fields, 116:57–88, 2000.
  * [Hru10] Geoffrey Hruska.  Relative hyperbolicity and relative quasiconvexity for countable groups.  Algebraic Geom. Topol., 10:1807–1856, 2010.
  * [Hut20] Tom Hutchcroft.  Non-intersection of transient branching random walks.  Probab. Theory Relat. Fields, 178:1–23, 2020.
  * [Kai87] Vadim Kaimanovich.  Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups.  Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 184:29–46, 1987.
  * [Kai00] Vadim Kaimanovich.  The Poisson formula for groups with hyperbolic properties.  Ann. Math., 152:659–692, 2000.
  * [Kal17] Olav Kallenberg.  Random measures, theory and applications, volume 77 of Probability Theory and Stochastic Modelling.  Springer, Cham, 2017.
  * [Kar03] Anders Karlsson.  Free subgroups of groups with nontrivial Floyd boundary.  Commun. Algebra, 31:5361–5376, 2003.
  * [KM99] Anders Karlsson and Gregory Margulis.  A multiplicative ergodic theorem and nonpositively curved spaces.  Commun. Math. Phys., 208:107–123, 1999.
  * [KW22] Vadim Kaimanovich and Wolfgang Woess.  Limit distributions of branching random walks.  arXiv:2205.13615, 2022.
  * [Led01] François Ledrappier.  Some asymptotic properties of random walks in free groups.  In Topics in Probability and Lie Groups: Boundary Theory, pages 117–152. CRM Proceedings and Lecture Notes, 2001.
  * [Lig96] Thomas M. Liggett.  Branching random walks and contact processes on homogeneous trees.  Probab. Theory Related Fields, 106(4):495–519, 1996.
  * [MT18] Joseph Maher and Giulio Tiozzo.  Random walks on weakly hyperbolic groups.  J. für die Reine und Angew. Math., 742:187–239, 2018.
  * [Osi06] Denis Osin.  Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems.  Mem. Am. Math. Soc., 20, 2006.
  * [Pei11] M. Peigné.  On some exotic Schottky groups.  Discrete Contin. Dyn. Syst., 118(31):559 – 579, 2011.
  * [PS18] Vincent Pit and Barbara Schapira.  Finitude des mesures de Gibbs sur les variétés non compactes à courbure négative pincée.  Ann. Institut Fourier, 68:457–510, 2018.
  * [PTV20] Marc Peigné, Samuel Tapie, and Pierre Vidotto.  Orbital counting for some convergent groups.  Ann. de l’Institut Fourier, 70:1307–1340, 2020.
  * [PW02] Panos Papasoglu and Kevin Whyte.  Quasi-isometries between groups with infinitely many ends.  Comment. Math. Helvetici, 77:133–144, 2002.
  * [PY19] Leonid Potyagailo and Wenyuan Yang.  Hausdorff dimension of boundaries of relatively hyperbolic groups.  Geom. Topol., 23:1779–1840, 2019.
  * [QRT20] Yulan Qing, Kasra Rafi, and Giulio Tiozzo.  Sublinearly Morse boundary II: Proper geodesic spaces.  arXiv:2011.03481, 2020.
  * [Rob03] Thomas Roblin.  Ergodicité et équidistribution en courbure négative.  Mém. Soc. Math. Fr., Nouv. Sér., 95:96, 2003.
  * [Sha09] Jay Shah.  Hausdorff dimension and its applications.  available at http://www.math.uchicago.edu/
may/VIGRE/VIGRE2009/REUPapers/Shah.pdf, 2009.

  * [Shi12] Zhan Shi.  Branching random walks.  In École d’été de Probabilités de Saint-Flour XLII, volume 2151. Lecture notes in mathematics, Springer, 2012.
  * [Sis12] Alessandro Sisto.  On metric relative hyperbolicity.  arXiv:1210.8081, 2012.
  * [Sis13] Alessandro Sisto.  Projections and relative hyperbolicity.  Enseign. Math., 59:165–181, 2013.
  * [ST21] Barbara Schapira and Samuel Tapie.  Regularity of entropy, geodesic currents and entropy at infinity.  Ann. Sci. Éc. Norm. Supér., 54:1–68, 2021.
  * [Sta71] John Stallings.  Group theory and three-dimensional manifolds.  Yale University Press, 1971.
  * [SW79] Peter Scott and Terry Wall.  Topological methods in group theory.  In London Math. Soc. Lecture Notes Ser. 36, pages 137–203. Cambridge University Press, 1979.
  * [SWX20] Vladas Sidoravicius, Longmin Wang, and Kainan Xiang.  Limit set of branching random walks on hyperbolic groups.  arXiv:2007.13267, 2020.
  * [Tio15] Giulio Tiozzo.  Sublinear deviation between geodesics and samples paths.  Duke Math. J., 164:511–539, 2015.
  * [Tra18] Hung Cong Tran.  Divergence spectra and Morse boundaries of relatively hyperbolic groups.  Geom. Dedicata, 194:99–129, 2018.
  * [Var86] Nicholas Theodoros Varapoulos.  Théorie du potentiel sur des groupes et des variétés.  C. R. Acad. Sci. Paris, 302:203–205, 1986.
  * [Vid19] Pierre Vidotto.  Ergodic properties of some negatively curved manifolds with infinite measure.  Mém. Soc. Math. Fr., Nouv. Sér., 160:1–32, 2019.
  * [Woe86a] Wolfgang Woess.  A description of the Martin boundary for nearest neighbour random walks on free products.  In Probability measures on groups VIII, pages 203–215. Lecture notes in mathematics, Springer, 1986.
  * [Woe86b] Wolfgang Woess.  Nearest neighbour random walks on free products of discrete groups.  Bolletino dell Unione Mat. Ital., 5-B:961–982, 1986.
  * [Woe89] Wolfgang Woess.  Boundaries of random walks on graphs and groups with infinitely many ends.  Isr. J. Math., 68:271–301, 1989.
  * [Woe00] Wolfgang Woess.  Random Walks on Infinite Graphs and Groups.  Cambridge Press University, 2000.
  * [Yan14] Wenyuan Yang.  Growth tightness for groups with contracting elements.  Math. Proc. Camb. Philos. Soc., 157:297–319, 2014.
  * [Yan19] Wenyuan Yang.  Statistically convex-cocompact actions of groups with contracting elements.  Int. Math. Res., 23:7259–7323, 2019.
  * [Yan22] Wenyuan Yang.  Patterson-Sullivan measures and growth of relatively hyperbolic groups.  Peking Math. J., 5:153–212, 2022.