# Notes on hyperbolic branching Brownian motion

Wolfgang Woess Institut für Diskrete Mathematik, Technische Universität Graz,
Steyrergasse 30, A-8010 Graz, Austria<EMAIL_ADDRESS>
(Date: May 12, 2024)

###### Abstract.

Euclidean branching Brownian motion (BBM) has been intensively studied during
many decades by renowned researchers. BBM on hyperbolic space has received
less attention. A profound study of Lalley and Sellke (1997) provided insight
on the recurrent, resp. transient regimes of BBM on the Poincare’ disk. In
particular, they determined the Hausdorff dimension of the limit set on the
boundary circle in dependance on the fission rate of the branching particles.
In the present notes, some further features are exhibited, such as the rate of
the maximal hyperbolic distance to the starting point and the behaviour of the
empiricial distributions of the branching population, as time goes to
infinity.

###### Key words and phrases:

Hyperbolic disk, hyperbolic Laplacian, branching Brownian motion, maximal
distance, empirical distributions, boundary convergence

##  1\. Introduction

_This is ongoing work; these notes present its state of mid 2024._

Euclidean branching Brownian motion on $\mathbb{R}$ has been an intensively
studied topic. A good reference for the earlier developments is the
comprehensive monograph by Athreya and Ney [1]. In short, a particle performs
Brownian motion for an exponentially distributed time and then fissions in two
new particles, each of which continues its own Brownian motion independently
of the other for exponential time, then in turn fissions in two, and so on.
Primary object of studies has been the evolution of the random population at
time $t$, as $t\to\infty$. One of the many interesting features concerns the
position of the rightmost particle at time $t$. Famous work of McKean [24]
showed that the distribution of the rightmost particle is directly linked with
the travelling wave equation studied in the famous work of Kolmogorov,
Petrovski and Piscounov [21].

Similarly, there is the study of branching random walk (BRW) in discrete time,
where particles evolve according to a Galton-Watson process: at the points of
fission, the new particles perform independent steps according to a given
random walk. Branching random walk on $\mathbb{R}$ appeared, for example, in
the monograph by Harris [15, §III.16]. Both models have evolved significantly,
and both are examples of tree-indexed Markov processes in the sense of
Benjamini and Peres [3], where the linear time is replaced by a - possibly
random - tree (discrete or continuous).

Here, the focus is on branching Brownian motion (BBM) on the hyperbolic disk.
That is, the motion of the particles follows the continuous time Markov
process whose infinitesimal generator is $\frac{1}{2}\mathfrak{L}$, where
$\mathfrak{L}$ is the Laplace-Beltrami operator of the unit disk equipped with
the Poincaré metric (or equivalently, the upper half plane model). It has not
been as widely considered as several other branching random processes. A very
significant in-depth study is due to Lalley and Sellke [23]. They considered
hyperbolic BBM where the random times between successive fissions are
exponentially distributed with parameter $\lambda>0$. There is a phase
transition: for $\lambda\leq 1/8$ (transient phase), each compact subset of
$\mathbb{D}$ is no more visited by the branching population from some random
time onwards, while for $\lambda>1/8$ (recurrent phase) each open subset of
$\mathbb{D}$ continues to be visited by the population. The main results of
[23] concern the random _limit set_ $\Lambda$, that is, the set of
accumulation points of the trace of the population on the boundary
$\partial\mathbb{D}$, the unit circle. It is the full circle when
$\lambda>1/8$, while it is a perfect set with _Hausdorff dimension_
$(1-\sqrt{1-8\lambda}\,)/2$ when $\lambda\leq 1/8$.

An analogous phenomenon was exhibited for branching random walk on regular
trees, resp. free groups, by Liggett [22] and Hueter and Lalley [16]. This was
extended to BRW on free products of groups by Candellero, Gilch and Müller
[8], and very recently to BRW on hyperbolic groups by Sidoravicius, Wang and
Xiang [26] and subsequently to BRW on relatively hyperbolic groups by
Dussaule, Wang and Yang [12]. For BRW on finitely generated groups, resp.
transitive graphs, this study has been accompanied by the investigation of the
_trace,_ that is, the subgraph spanned by all points visited by the BRW. See
Benjamini and Müller [2], Candellero and Roberts, [10], Gilch and Müller [13],
Hutchcroft [17] as well as [26] and [12].

Returning to hyperbolic BBM, in the present notes some results are added to
the profound study of [23]. For the _maximal hyperbolic distance_ from the
origin of a particle at time $t$, we show that its rate is the same as for the
comparison process which is Euclidean BBM where the underlying one-dimensional
BM has drift $1/2$. Then we consider the _empirical distributions_ of
hyperbolic BBM: these are the normalised occupation measures of the population
at the times $t>0$.

Following a suggestion by V. A. Kaimanovich, the asymptotic behaviour of these
random distributions was recently studied for BRW on graphs in parallel work
of Kaimanovich and Woess [18] and Candellero and Hutchcroft [9]. In the
Euclidean setting of BRW and BBM, they had been studied since the mid 1960ies,
see e.g. Ney [25], Stam [27], Kaplan and Asmussen [19], Uchiyama [28], Biggins
[4], while [18] and [9] are more relevant in a “non-amenable” setting.

Here, variants of CLT-type asymptotics of the empricial distributions of
hyperbolic BBM are derived once more via the Euclidean comparison process.
Then then the average rate of escape, that is, the rate of the average
distance from the origin of the particles at time $t$, is shown to coincide
with the one of (non-branching) hyperbolic BM. It is proved that the empirical
distributions converge weakly (on the closed disk) to a random limit
distribution on the boundary, that is, the unit circle. Some properties of the
limit distribution are derived, and various open questions are posed.

In these notes, the author has chosen to present the initial parts in a rather
broad way. This concerns, in paricular, the construction of the underlying
time tree (Yule tree) and the chosen notation, which differs a bit from the
previous mainstream.

Acknowledgements. The author acknowledges helpful email exchange with John
Biggins, Maury Bramson, Nicola Kistler, Steve Lalley, Sebastian Müller, Enzo
Orsingher, Yuichi Shiozawa, Anton Wakolbinger (personal contact) and Ofer
Zeitouni.

##  2\. The Yule tree

The introduction of the continuous random population tree described in this
section goes back to Yule [29].

Consider the binary tree $\\{\mathfrak{l},\mathfrak{r}\\}^{*}$ consisting of
all binary sequences (words)
$\mathfrak{v}=\mathfrak{s}_{1}\cdots\mathfrak{s}_{n}$ with $n\geq 0$ and
$\mathfrak{s}_{k}\in\\{\mathfrak{l},\mathfrak{r}\\}$.111To avoid confusion
with numbers, we avoid the binary symbols $0,1$. The symbols $\mathfrak{l}$,
$\mathfrak{r}$ and $\mathfrak{s}$ stand for “left”, “right” and “side”,
respectively. For $n=0$, this is the empty sequence $\epsilon$. If
$\mathfrak{v}$ has the form $\mathfrak{v}=\mathfrak{u}\mathfrak{s}$ with
$\mathfrak{s}\in\\{\mathfrak{l},\mathfrak{r}\\}$, then the _predecessor_ of
$\mathfrak{v}$ is $\mathfrak{v}^{\prime}=\mathfrak{u}$, and $\mathfrak{v}$ is
one of the two _successors_ of $\mathfrak{u}$. The edges of the tree are all
$[\mathfrak{v}^{\prime},\mathfrak{v}]$, where $\mathfrak{v}\neq\epsilon$. We
augment this tree by an additional vertex $\alpha$ which is only connected to
$\epsilon$, so that $\epsilon^{\prime}=\alpha$. Now consider a sequence of
i.i.d. random variables $\ell_{\mathfrak{v}}\,$,
$\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$ having exponential
distribution with parameter $\lambda>0$. For the moment, we can realise it for
example on the product space

$\bigl{(}\Omega^{\text{Yule}}\,,\mathcal{A}^{\text{Yule}}\,,\mathbb{P}^{\text{Yule}}\bigr{)}=\bigotimes_{\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}}\bigl{(}\mathbb{R}_{+},\mathcal{B}_{\mathbb{R}^{+}},\textit{Exp}_{\lambda}\bigr{)}_{\mathfrak{v}}\,,$

where each factor is a copy of the probability space consisting of the
positive real half-line with the Borel sigma-algebra and the exponential
distribution with paramteter $\lambda$, so that $\ell_{\mathfrak{v}}$ is the
projection on the $v$-coordinate. Later on, it will be embedded into a bigger
probability space.

We then get a _random tree_ $\mathcal{T}$, a $1$-complex where each edge
$[\mathfrak{v}^{\prime},\mathfrak{v}]$ is an interval with the random length
$\ell_{\mathfrak{v}}\,$, and we write the edge as

(1) $[\mathfrak{v}^{\prime},\mathfrak{v}]=\\{\tau=\mathfrak{v}_{s}:0\leq
s\leq\ell_{\mathfrak{v}}\,\\},$

where $v_{s}$ is the point in the interval at distance $s$ from
$\mathfrak{v}^{\prime}$. Then the length (= distance from $\alpha$) of a
vertex $\mathfrak{v}$ is defined recursively by $|\alpha|=0$ and
$|\mathfrak{v}|=|\mathfrak{v}^{\prime}|+\ell_{\mathfrak{v}}$, and the length
of $\tau=v_{s}\in[v^{-},v]$ is $|\tau|=|v^{-}|+s$.

This is the Yule tree. It is interpreted as a _genealogical tree_ , where
$\alpha$ is the “ancestor” at time $0$, an thinking of it as a particle, after
time $\ell_{\epsilon}$ it fissions in two particles. The timelines of the new
particles are the edges $[\epsilon,\mathfrak{l}]$ and
$[\epsilon,\mathfrak{r}]$, respectively, and after times
$\ell_{\mathfrak{l}}\,$, resp. $\ell_{\mathfrak{r}}\,$, each of them fissions
again in $2$ particles. Recursively, a particle at the end of its timeline
$[\mathfrak{v}^{\prime},\mathfrak{v}]$ fissions in two, whose new timelines
are the edges $[\mathfrak{v},\mathfrak{v}\mathfrak{l}]$ and
$[\mathfrak{v},\mathfrak{v}\mathfrak{r}]$, respectively.

The _population_ at time $t\geq 0$ is the set
$\mathcal{T}(t)=\mathcal{T}_{\epsilon}(t)$ of all particles ($\equiv$ vertices
or interior points on the edges) at distance $t$ from $\alpha$. We write

$N(t)=N_{\,\epsilon}(t)=|\mathcal{T}(t)|$

for their number. Note that by continuity of the exponential distribution, at
any time $t\geq 0$, with probability $1$ there is at most one vertex $v$ of
$\mathcal{T}$ with $|v|=t$. That is, no two fissions take place
simultaneously.

For any $\mathfrak{u}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$ let
$\mathcal{T}_{\mathfrak{u}}$ be the subtree of $\mathcal{T}$ in which
$\mathfrak{u}$ has the role of $\epsilon$ in the above description and
$\mathfrak{u}^{\prime}$ the role of $\alpha$. In other words, it is spanned by
the vertex set

$\\{\mathfrak{u}^{\prime}\\}\cup\bigl{\\{}\mathfrak{u}\mathfrak{v}:\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}\bigr{\\}}.$

We write $\mathcal{T}_{\mathfrak{u}}(t)$ for the associated part of the
population, that is, the set of elements of $\mathcal{T}_{\mathfrak{u}}$ at
distance $t$ from $u^{-}$, and $N_{\,u}(t)$ for their number.

For any subset $U\subset\\{\mathfrak{l},\mathfrak{r}\\}^{*}$ of vertices which
is prefix-free (that is, no element of $U$ is an initial part of another
element in $U$), then the trees
$\mathcal{T}_{\mathfrak{u}}\,,\;\mathfrak{u}\in U$ are i.i.d. In particular,
all the generation sizes $N_{\,u}(t)$,
$\mathfrak{u}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$ have the same
distribution on $\mathbb{N}$. The following is well-known (even for more
general models); we provide a short proof.

######  Lemma 2.1.

For any $t\geq 0$, the population size at time $t$ has geometric distribution:

$\mathbb{P}[N(t)=n]=e^{-\lambda t}\,(1-e^{-\lambda t})^{n-1}\,,\quad
n\in\mathbb{N}.$

In particular, the expected population size is
$\;\mathbb{E}\bigl{(}N(t)\bigr{)}=e^{\lambda t}$.

###### Proof.

We have

$N_{\,\epsilon}(t)=\begin{cases}1\,,&\text{if }\;\ell_{\epsilon}\geq t\\\
N_{\mathfrak{l}}(t-\ell_{\epsilon})+N_{\mathfrak{r}}(t-\ell_{\epsilon})\,,&\text{if
}\;\ell_{\epsilon}<t.\end{cases}$

Therefore the characteristic function (in the variable $x$) is

$\displaystyle\varphi_{N(t)}(x)=\mathbb{E}\Bigl{(}\exp\bigl{(}\mathfrak{i}\,x\,N_{\epsilon}(t)\bigr{)}\Bigr{)}$
$\displaystyle=e^{\mathfrak{i}\,x}\,\mathbb{P}[\ell_{\epsilon}\geq
t]+\mathbb{E}\Bigl{(}\exp\bigl{(}\mathfrak{i}\,x\,N_{\mathfrak{l}}(t-\ell_{\epsilon})+\mathfrak{i}\,x\,N_{\mathfrak{r}}(t-\ell_{\epsilon})\bigr{)}\,\mathbf{1}_{[\ell_{\epsilon}<t]}\Bigr{)}$
$\displaystyle=e^{\mathfrak{i}\,x-\lambda
t}+\int_{0}^{t}\varphi_{N(t-s)}(x)^{2}\,\lambda e^{-\lambda s}\,ds\,.$

Just for the purpose of this proof, set $f(t)=e^{\lambda
t}\,\varphi_{N(t)}(x)$. Then the above equation transforms into the integral
equation

$f(t)=e^{\mathfrak{i}\,x}+\lambda\int_{0}^{t}f(s)^{2}\,e^{-\lambda s}\,ds.$

Thus, $f(t)=1/(e^{-\mathfrak{i}\,x}+e^{-\lambda t}-1)$, and

$\varphi_{N(t)}(x)=\frac{e^{\mathfrak{i}\,x}\,e^{-\lambda
t}}{1-e^{\mathfrak{i}\,x}(1-e^{-\lambda t})},$

which we recognise as the characteristic function of the geometric
distribution with parameter (success probability) $p=e^{-\lambda t}$. ∎

The population $\mathcal{T}(t)$ at time $t$ can be viewed as a _cross-section_
of $\mathcal{T}$ at level $t$. For the tree viewed as a spatial generalisation
of time, it can be interpreted to have a role analogous to the one of a
stopping time. If we cut the tree $\mathcal{T}$ along that cross-scetion, we
obtain the subtree $\mathcal{T}(\prec\\!t)$ of all elements $\tau$ with
$|\tau|\leq t$. We let
$V\bigl{(}\mathcal{T}(\prec\\!t)\bigr{)}=\\{\mathfrak{l},\mathfrak{r}\\}^{*}\cap\mathcal{T}(\prec\\!t)$
be the random set consisting of those vertices of the original binary tree
which are part of $\mathcal{T}(\prec\\!t)\,$. By Lemma 2.1,
$V\bigl{(}\mathcal{T}(\prec\\!t)\bigr{)}$ is finite with probability $1$.

The sigma-algebra $\mathcal{F}_{t}^{\text{Yule}}$ comprising the information
inherent to the Yule tree up to and including time $t$ is generated by all
random variables $\ell_{\mathfrak{v}}$ which intervene in the construction of
$\mathcal{T}(\prec\\!t)$. The following is well known.

######  Proposition 2.2.

The process $\;\bigl{(}N(t)\,e^{-\lambda t}\bigr{)}_{t\geq 0}$ is a martingale
with respect to the filtration $(\mathcal{F}_{t}^{\text{Yule}})_{t\geq 0}\,$,
and there exists an almost surely positive random variable $W$ such that

$\lim_{t\to\infty}N(t)\,e^{-\lambda t}=W\quad\text{almost surely and in
}L^{1}.$

##  3\. Hyperbolic disk and Brownian motion

A standard model of two-dimensional hyperbolic space is the _Poincaré disk_
with the hyperbolic length element and resulting metric

(2) $ds=\frac{2\sqrt{dx^{2}+dy^{2}}}{1-|z|^{2}}\quad\text{and}\quad
d(z,w)=\log\frac{|1-z\bar{w}|+|z-w|}{|1-z\bar{w}|-|z-w|}.$

Its orientation preserving isometry group consists of all Möbius
transformations of the form

(3) $gz=\frac{az+c}{\bar{c}z+\bar{a}}\,,\quad
a,c\in\mathbb{C},\;|a|^{2}-|c|^{2}=1\,.$

Together with the reflection $z\mapsto-\bar{z}$, it generates the full
isometry group. Of course, these transformations also act on the boundary of
$\mathbb{D}$, the unit circle $\partial\mathbb{D}$. We shall use the group
$\mathcal{A}$ of all transformations as in (3) which fix the boundary point
$1$, that is, $a+c\in\mathbb{R}$. It acts simply transitively on
$\mathbb{D}\,$. In particular, For each $z_{0}\in\mathbb{D}$, there is a
unique element $g_{z_{0}}\in\mathcal{A}$ which maps $0$ to $z_{0}$. It is
given by

(4)
$g_{z_{0}}z=\frac{(1-z_{0})z+(z_{0}-|z_{0}|^{2})}{(\bar{z}_{0}-|z_{0}|^{2})z+(1-\bar{z}_{0})}\,.$

The hyperbolic Laplace (or Laplace-Beltrami) operator in the variable
$z=x+\mathfrak{i}\,y$ is

(5)
$\mathfrak{L}=\frac{(1-|z|^{2})^{2}}{4}\,\Bigl{(}\partial_{x}^{2}+\partial_{y}^{2}\Bigr{)}\,.$

It is self-adjoint on $L^{2}(\mathbb{D},\mathsf{m})$, where
$\mathsf{m}=\mathsf{m}_{\mathbb{D}}$ is the hyperbolic measure,

(6) $d\mathsf{m}(z)=\frac{4dz}{(1-|z|^{2})^{2}}\,.$

The infinitesimal generator of _hyperbolic Brownian motion_ $(B_{t})_{t\geq
0}$ is $\frac{1}{2}\mathfrak{L}$. (We remark that the factor $\frac{1}{2}$ is
used because in the analogous Euclidean setting we want that Brownian motion
at time $t=1$ has standard normal distribution.)

One can see the latter as a version of the standard two-dimensional Euclidean
Brownian motion slowed down as it gets close to the unit circle. Responsible
for slowing down is the factor $\frac{(1-|z|^{2})^{2}}{4}$ which only depends
on the Euclidean distance of the current position $z$ from the circle.

Hyperbolic Brownian motion (BM) has independent increments and continuous
trajectories (almost surely). Since the hyperbolic Laplacian commutes with all
hyperbolic isometries, hyperbolic BM is invariant under the latter: if $g$ is
as in (3) and $(B_{t})$ is (a version of) hyperbolic BM starting at
$z_{0}\in\mathbb{D}$ then $(gB_{t})$ is (a version of) hyperbolic BM starting
at $gz_{0}\,$. Equivalently, the heat kernel $p_{t}(\cdot,\cdot)$ with respect
to $\mathsf{m}$ associated with $\frac{1}{2}\mathfrak{L}$ is invariant under
the diagonal actions of every hyperbolic isometry. We write $P_{t}$ for the
associated transition operator of hyperbolic BM: for any measurable set
$K\subset\mathbb{D}$ and $z\in\mathbb{D}$, resp., measurable function
$f:\mathbb{D}\to\mathbb{R}$,

(7) $P_{t}(z,K)=\int_{K}p_{t}(z,w)\,d\mathsf{m}(w)\quad\text{and}\quad
P_{t}f(z)=\int_{\mathbb{D}}p_{t}(z,w)f(w)\,d\mathsf{m}(w)\,,$

whenever that integral is well defined.

Now let $(\Omega_{b}\,,\mathcal{A}_{b}\,,\mathbb{P}_{b})$ be a suitabe
probability space on which hyperbolic BM starting at the origin is defined.
With starting point $0$, as $t\to\infty$, the process converges almost surely
to a $\partial\mathbb{D}$-valued random variable $B_{\infty}\,$. The
distribution of $B_{\infty}\,$, being rotation invariant, is equidistribution
on the unit circle: $d\xi=\frac{1}{2\pi}e^{i\phi}\,d\phi$, where $d\phi$ is
the Lebesgue measure on $[-\pi\,,\,\pi]$. (The elements of
$\partial\mathbb{D}$ are denoted $\xi,\eta$, etc.)

When the starting point is $z_{0}$ then, working on the same probability
space, (a model of) the limit random variable is $g_{z_{0}}B_{\infty}\,$. The
density of its distribution $\nu_{z_{0}}$ with respect to $d\xi$ is the
_Poisson kernel_

(8) $\Pi(z_{0},\xi)=\frac{1-|z_{0}|^{2}}{|\xi-
z_{0}|^{2}}\qquad(z_{0}\in\mathbb{D}\,,\;\xi=e^{\mathfrak{i}\,\phi}\in\partial\mathbb{D}).$

Since $d(B_{t},o)\to\infty$ almost surely, we are also interested in the
speed. It is linear, and to understand it, it may be useful to pass to two
other models of hyperbolic space.

The second one, besides the disk, is the _upper half plane model_
$\mathbb{H}=\\{u+\mathfrak{i}\,v:u,v\in\mathbb{R},v>0\\}$. The metric
$d=d_{\mathbb{D}}$ of (2) is transported to the metric $d_{\mathbb{H}}$ via
the Möbius map from $\mathbb{D}$ to $\mathbb{H}$

(9) $z\mapsto\mathfrak{i}\,\frac{1+z}{1-z}.$

It maps $0$ to $\mathfrak{i}\,$, and the boundary points $-1$ to $0$ and $1$
to $\mathfrak{i}\,\infty$, and

(10) $d_{\mathbb{H}}(z,w)=\log\frac{|z-\bar{w}|+|z-w|}{|z-\bar{w}|-|z-w|}.$

We shall often switch back and forth between $\mathbb{D}$ and $\mathbb{H}$,
and will mostly use unified notation $o$ for our origin, that is, $o=0$ in
$\mathbb{D}$ and $o=\mathfrak{i}\,$ in $\mathbb{H}$. We also remark here that
in the upper half plane model, the group $\mathfrak{A}$ of all transformations
that fix the boundary point $1$ in $\mathbb{D}$ (that is, $a+c\in\mathbb{R}$
for $a,c$ as in (3)) becomes, via conjugation with the map (9), the _affine
group_ af all transformations

$z\mapsto az+b\,,\quad a>0\,,\;b\in\mathbb{R}\quad(z\in\mathbb{H}).$

(The $a$ here is not the same as in (3).)

In the coordinates $(u,v)\in\mathbb{H}$, the hyperbolic Laplacian becomes
$v^{2}(\partial^{2}u+\partial^{2}v)$. Then we make one more change of
variables, setting $w=\log v$ to obtain the _logarithmic model,_ where now
$(u,w)\in\mathbb{R}^{2}$ and the hyperbolic Laplacian becomes

(11) $\mathfrak{L}=e^{2w}\partial^{2}_{u}+\partial^{2}_{w}-\partial_{w}\,.$

Its projection on the vertical coordinate $w$ is
$\partial^{2}_{w}-\partial_{w}$, so that
$\frac{1}{2}(\partial^{2}_{w}-\partial_{w})$ is the infinitesimal generator of
one-dimensional Euclidean Brownian motion with drift $-1/2$. Tracing this back
to the upper half plane and disk models, writing $B_{t}^{\mathbb{H}}$ and
$B_{t}^{\mathbb{D}}$ for hyperbolic Brownian motion in the respective
coordinates, when $B_{0}^{\mathbb{H}}=\mathfrak{i}\,$, resp.
$B_{0}^{\mathbb{D}}=0$,

(12) $\log\bigl{(}\Im
B_{t}^{\mathbb{H}}\bigr{)}=\log\frac{1-|B_{t}^{\mathbb{D}}|^{2}}{|1-B_{t}^{\mathbb{D}}|^{2}}\sim
N\bigl{(}-\tfrac{1}{2}t,t\bigr{)},$

where (as usual) $\Im$ dentotes the imaginary part, $\sim$ means “has
distribution” and $N(a,s^{2})$ is the normal distribution with mean $a$ and
variance $s^{2}$. From this we get the following, where we can omit the
superscript referring to the respective model.

######  Lemma 3.1.

The following central limit theorem and rate of escape results hold, as
$t\to\infty\,$:

$\frac{d(B_{t},B_{0})-\frac{1}{2}t}{\sqrt{t}}\to\mathcal{N}(0,1)\quad\text{in
law, and}\;\quad\frac{d(B_{t},B_{0})}{t}\to\frac{1}{2}\quad\text{almost
surely.}$

###### Proof.

As stated above (8), in terms of the disk model, it is well known that
$B_{t}^{\mathbb{D}}$ converges almost surely to a limit random variable
$B_{\infty}^{\mathbb{D}}$ with continuous distribution. Taking this to the
upper half plane model, we get $B_{\infty}^{\mathbb{H}}\in\mathbb{R}$ and $\Im
B_{t}^{\mathbb{H}}\to 0$ almost surely (since
$\mathbb{P}[B_{\infty}^{\mathbb{H}}=\mathfrak{i}\,\infty]=0$). Now, if
$z=x+\mathfrak{i}\,y\in\mathbb{H}$ then an easy computation with the
hyperbolic metric in $\mathbb{H}$ as in (10) shows that

(13) $d_{\mathbb{H}}(x+\mathfrak{i}\,y,\mathfrak{i}\,)+\log
y=\log\frac{1}{2}\Bigl{(}1+|z|^{2}+\sqrt{(1+|z|^{2})^{2}-4y^{2}}\Bigr{)}$

We get that

$d_{\mathbb{H}}(B_{t},\mathfrak{i}\,)+\log(\Im
B_{t}^{\mathbb{H}})\to\log\bigl{(}1+(B_{\infty}^{\mathbb{H}})^{2}\bigr{)}\quad\text{almost
surely, as }\;t\to\infty,$

and the limit is almost surely finite. Combining this with (12) completes the
proof. ∎

We shall need estimates of the tail behaviour of the random variables
$d(B_{t},o)$, $t>0$. The heat kernel $p_{t}(o,z)$, $z\in\mathbb{D}$ (resp.,
$\in\mathbb{H}$) associated with $\frac{1}{2}\mathfrak{L}$ only depends on
$R=d(z,o)$, and it has a uniform estimate in space and time, see Davies and
Mandouvalos [11, Thm. 3.1] 222Concerning the heat kernel, different
normalisations of the Laplacian are an ongoing source of small confusion.
Analyists typically use the heat kernel for $\mathfrak{L}$ as given in (5).
Some of them also omit the factor $1/4$. In probability, we want that
Euclidean BM at time $t=1$ has variance $1$, so that we use the heat kernel
for $\frac{1}{2}\mathfrak{L}$. Although not explicitly stated, [11] uses the
first of these three options, so that here we had to replace their $t$ by
$t/2$.. We state here the upper bound in a way which is suited for our
purpose:

(14) $\displaystyle p_{t}(z,w)$
$\displaystyle\leq\frac{\textsl{Const}}{\sqrt{1+R}}\,\,\Psi\biggl{(}\frac{1+R}{t}\biggr{)}\,\exp\biggl{(}-\frac{(R+\tfrac{t}{2})^{2}}{2t}\biggr{)}\quad\text{for
}\;R,t>0\,,\quad\text{where}$ $\displaystyle R$
$\displaystyle=d(z,w)\quad\text{and}\quad\Psi(x)=\begin{cases}x^{3/2}\,,&0\leq
x\leq 1\,,\\\ x^{1/2}\,,&x\geq 1\,.\end{cases}$

For the following proposition, the author acknowledges a suggestion by Yuichi
Shiozawa.

######  Proposition 3.2.

Let $\;\mathcal{M}=\max\\{d(B_{t}\,,B_{0}):t\leq 1\\}.$ Then there is
$\textsl{K}>0$ such that for any $c>0$

$\mathbb{P}[\mathcal{M}\geq c]\leq 2\,\mathbb{P}[d(B_{1}\,,B_{0})\geq
c]\leq\textsl{K}\,\exp\biggl{(}-\frac{(c-\tfrac{1}{2})^{2}}{2}\biggr{)}\,.$

###### Proof.

We work in the upper half plane model, and we may suppose without loss of
generality that $B_{0}=B_{0}^{\mathbb{H}}=o\,(=\mathfrak{i}\,)$. Here, we
shall omit the sub- and superscripts $\mathbb{H}$. Recall from (14) that in
any model, $(B_{t})_{t\geq 0}$ is istropic, that is, its transition kernel
only depends on time and hyperbolic distance:
$\bigl{(}d(B_{t}\,,o)\bigr{)}_{t\geq 0}$ is a Markov process on the state
space $[0\,,\,\infty)$ with continuous trajectories. In $\mathbb{H}$, the
point $e^{-c}$ is at hyperbolic distance $c$ from $o=\mathfrak{i}\,$. Thus,
using (13), for $s,t>0$,

(15) $\displaystyle\mathbb{P}[d(B_{s+t},o)\geq c\mid d(B_{s},o)=c]$
$\displaystyle=\mathbb{P}[d(B_{t},o)\geq c\mid B_{0}=e^{-c}]$
$\displaystyle\geq\mathbb{P}[-\log(\Im B_{t})\geq c\mid-\log(\Im B_{0})=c]\geq
1/2,$

because by (12), if $\;-\log(\Im B_{0})=c\;$ then $\;-\log(\Im B_{t})\sim
N\bigl{(}c+\tfrac{1}{2}t,t\bigr{)}\,$.

Now consider the a.s. finite stopping time

$T_{c}=\inf\\{t>0:d(B_{t}\,,o)=c\\}\,.$

Then, using the strong Markov property, the fact that $d(B_{T_{c}}\,,o)=c$,
and isotropy of hyperbolic BM,

$\displaystyle\mathbb{P}[\mathcal{M}\geq c]$
$\displaystyle=\mathbb{P}_{0}[T_{c}\leq 1]$
$\displaystyle=\mathbb{P}[d(B_{1}\,,o)\geq c\,,T_{c}\leq
1]+\mathbb{P}[d(B_{1}\,,o)<c\,,T_{c}\leq 1]$
$\displaystyle=\mathbb{P}[d(B_{1}\,,o)\geq
c]+\int_{0}^{1}\mathbb{P}[d(B_{1}\,,o)<c\mid
d(B_{s}\,,o)=c]\,\,d\,\mathbb{P}[T_{c}=s]$ $\displaystyle\noindent
nowapplying\eqref{eq:gec}$ $\displaystyle\leq\mathbb{P}[d(B_{1}\,,o)\geq
c]+\int_{0}^{1}\mathbb{P}[d(B_{1}\,,o)\geq c\mid
d(B_{s}\,,o)=c]\,\,d\,\mathbb{P}[T_{c}=s]$
$\displaystyle=2\,\mathbb{P}[d(B_{1}\,,o)\geq c].$

This proves the first of the two inequalities. The second is going to be
derived from (14). Note that for $t=1$, we have to use $\Psi(x)=x^{1/2}$ in
that heat kernel estimate.

We return to the disk model. We can express the hyperbolic measure of (6)
first in terms of Euclidean polar coordinates $(r,\varphi)$ with $r<1$ and
then replace $r=|z|$ by $R=d(z,o)=\log\frac{1+r}{1-r}$. In the coordinates
$(R,\varphi)$, we get

(16) $d\mathsf{m}(z)=\sinh R\,\,dR\,d\varphi\quad\text{in }\;\mathbb{D}\,.$

Then

$\displaystyle\mathbb{P}[d(B_{1},o)\geq c]$
$\displaystyle=\int_{\\{z\in\mathbb{D}\,:\,d(z,o)\geq
c\\}}p_{1}(0,z)\,d\mathsf{m}(z)$
$\displaystyle\leq\textsl{Const}\int_{-\pi}^{\pi}\int_{c}^{\infty}\,\exp\biggl{(}-\frac{(R+\tfrac{1}{2})^{2}}{2}\biggr{)}\,\frac{e^{R}-e^{-R}}{2}\,dR\,d\varphi$
$\displaystyle\leq\textsl{Const}^{\prime}\int_{c}^{\infty}\exp\biggl{(}-\frac{(R-\tfrac{1}{2})^{2}}{2}\biggr{)}\,dR$
$\displaystyle\leq\textsl{Const}^{\prime\prime}\exp\biggl{(}-\frac{(c-\tfrac{1}{2})^{2}}{2}\biggr{)}$

for a suitable constant $\textsl{Const}^{\prime\prime}$. ∎

##  4\. Hyperbolic branching Brownian motion

We now construct hyperbolic branching Brownian motion. Whenever it is
suitable, we can switch between the different models of hyperbolic plane, but
primarily we have the disk model in mind.

We need a probability space $(\Omega,\mathcal{A},\mathbb{P})$ on which one can
realise countably many i.i.d. random variables $\ell_{\mathfrak{v}}\,$,
$\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$ and, independently of the
latter, countably many independent hyperbolic Brownian motions
$(B_{t}^{\mathfrak{v}})\,$,
$\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$, each one starting at
$0$. If $(\Omega_{b}\,,\mathcal{B}_{b}\,,\mathbb{P}_{b})$ is one probability
space on which hyperbolic BM can be defined, then we can take the product
space

$\begin{gathered}\bigl{(}\Omega\,,\mathcal{A}\,,\mathbb{P}\bigr{)}=\bigl{(}\Omega^{\text{Yule}}\,,\mathcal{A}^{\text{Yule}}\,,\mathbb{P}^{\text{Yule}}\bigr{)}\otimes\bigl{(}\Omega_{\textit{{B}}}\,,\mathcal{A}_{\textit{{B}}}\,,\mathbb{P}_{\textit{{B}}}\bigr{)}\,,\quad\text{where}\\\
\bigl{(}\Omega_{\textit{{B}}}\,,\mathcal{A}_{\textit{{B}}}\,,\mathbb{P}_{\textit{{B}}}\bigr{)}=\bigotimes_{\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}}\bigl{(}\Omega_{b},,\mathcal{B}_{b}\,,\mathbb{P}_{b}\bigr{)}_{\mathfrak{v}}\,,\end{gathered}$

along with the corresponding projections. (It may also be realised on the
Lebesgue space = the unit interval with the Lebesgue measure).

With each fission point $\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$,
we associate a random element $\mathbf{g}_{\mathfrak{v}}\in\mathfrak{A}$,
where $\mathfrak{A}$ is the affine group, defined below (3) for the disk
model: if $z_{\mathfrak{v}}=B_{\ell_{\mathfrak{v}}}^{\mathfrak{v}}o$ then
$\mathbf{g}_{\mathfrak{v}}=g_{z_{\mathfrak{v}}}$ as defined by (4). The
$\mathbf{g}_{\mathfrak{v}}\,$,
$\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$, are i.i.d.

Furthermore, we take a starting point $z_{0}\in\mathbb{D}$. We shall often
write $\mathbb{P}_{\\!z_{0}}$ for the probability measure and
$\mathbb{E}_{z_{0}}$ for the corresponding expectation in order to remember
the starting point.

Then hyperbolic branching Brownian motion assigns a random variable
$\textit{{B}}_{\tau}$ to every element $\tau$ of the Yule tree $\mathcal{T}$
(vertex or interior element of some edge) as follows, using the notation of
(1):

  * (A)

If the path from $\alpha$ to
$\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$ in our tree has the
vertices
$\alpha=\mathfrak{v}_{0}\,,\mathfrak{v}_{1}\,,\dots,\mathfrak{v}_{k}=\mathfrak{v}$
(with $\mathfrak{v}_{j}^{-}=\mathfrak{v}_{j-1}$) then for
$\tau=\mathfrak{v}_{s}\in[\mathfrak{v}^{\prime},\mathfrak{v}]$,

$\textit{{B}}_{\tau}=g_{z_{0}}\mathbf{g}_{\mathfrak{v}_{1}}\cdots\mathbf{g}_{\mathfrak{v}_{k-1}}B_{s}^{v}\,.$

This can also be described by the following recursive construction:

  * (B1)

At the “ancestor” $\alpha$,

$\textit{{B}}_{\alpha}=z_{0}\,.$

  * (B2)

If for a vertex $\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$, we
already have $\textit{{B}}_{\mathfrak{v}^{\prime}}=z\in\mathbb{D}$, then on
the edge $[\mathfrak{v}^{\prime},\mathfrak{v}]$, we continue with

$\textit{{B}}_{\mathfrak{v}_{s}}=g_{z}B_{s}^{\mathfrak{v}}\,,\quad 0\leq
s\leq\ell_{\mathfrak{v}}\,,$

where the random element $g_{z}\in\mathfrak{A}$ is given by (4), with $z$ in
the place of $z_{0}\,$.

In particular, in our construction, hyperbolic BBM starting at $z_{0}$ is the
image under $g_{z_{0}}$ of hyperbolic BBM starting at $0$.

######  Remarks 4.1.

(a) In principle, we could choose any initial family of Möbius transformations
such that for each $z_{0}\in\mathbb{D}$ there is precisely one $g_{z_{0}}$
mapping $o$ to $z_{0}$. For example (suggestion by Steve Lalley) one could
take the map

$z\mapsto\frac{z+z_{0}}{\bar{z}_{0}z+1}\,,$

which is as in (3) with $a=1/(1-|z_{0}|^{2})^{1/2}$ and
$c=z_{0}/(1-|z_{0}|^{2})^{1/2}$. When $z_{0}=0$ it is the identity map, while
otherwise it fixes the diameter segment of the unit disk through the origin
and $z_{0}\,$.
(b) However, the equivalence between the above two constructions (A) and
(B1)+(B2) relies on the fact that the group $\mathfrak{A}$ acts simply
transitively, while this is not the case for the semigroup generated by the
mappings of (a).
A feature of the construction (A) is that in this way, hyperbolic BBM is
governed by a branching random walk on the group $\mathfrak{A}$. The
underlying Galton-Watson tree is not random, but the binary tree
$\\{\mathfrak{l},\mathfrak{r}\\}^{*}$. That is, each member of the
corresponding population fissions in precisely 2 children. The branching
random walk starting at $g_{z_{0}}$ is then
$(g_{z_{0}}\mathbf{G}_{\mathfrak{v}})_{\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}}$,
where

(17)
$\mathbf{G}_{\mathfrak{v}}=\mathbf{g}_{\mathfrak{v}_{1}}\cdots\mathbf{g}_{\mathfrak{v}_{k}}$

when the path from $\epsilon$ to $\mathfrak{v}$ is
$[\epsilon=\mathfrak{v}_{1}\,,\mathfrak{v}_{2}\,,\dots,\mathfrak{v}_{k}=\mathfrak{v}]$.

An infinite ray $\pi$ in the Yule tree $\mathcal{T}$ is a line isometric with
$[0\,,\,\infty)$ which is spanned by a sequence of vertices
$[\alpha,\epsilon,...]$ where every vertex $\mathfrak{v}\in\pi$ has precisely
one successor in $\pi$. Along $\pi$, the process
$(\textit{{B}}_{\tau})_{\tau\in\pi}$ is a hyperbolic BM starting at $z_{0}$
whose element at time $t$ is $\textit{{B}}_{\tau}$ with $\tau=v_{s}$, where
$v\in\pi$ is a vertex and $|v_{s}|=t$ as defined below (1). If we have two
distinct rays $\pi$ and $\pi^{\prime}$ then their _confluent_
$\pi\wedge\pi^{\prime}$ is the furthest vertex from $\alpha$ shared by the
two. Then the two hyperbolic Brownian motions along $\pi$ and $\pi^{\prime}$
coincide up to $\pi\wedge\pi^{\prime}$ and thereafter continue independently.

##  5\. The maximal distance

We first consider an issue which has been intensively studied for Euclidean
BBM since the pioneering work of Kolmogorov, Petrovski and Piscounov [21],
namely, the behaviour of the maximal hyperbolic distance from the starting
point of the population at time $t$. We only consider the basic result on the
rate of escape. For details in the Euclidean case, see e.g. Bramson [7] and,
in higher dimension, Kim, Lubetzky and Zeitouni [20].

######  Theorem 5.1.

Set
$\;\textsf{Max}_{t}=\max\bigl{\\{}d(\textit{{B}}_{\tau},o):\tau\in\mathcal{T}(t)\bigr{\\}}\,$.
Then

$\frac{\textsf{Max}_{t}}{t}\to r^{*}=\frac{1}{2}+\sqrt{2\lambda}$

almost surely, as $t\to\infty\,$.

###### Proof.

In the upper half plane model, the family
$(-\log\Im\textit{{B}}_{\tau}^{\mathbb{H}})_{\tau\in\mathcal{T}}$ is one-
dimensional Euclidean branching BM with drift $1/2$. It is well known that

(18)
$\frac{1}{t}\max\bigl{\\{}-\log\Im\textit{{B}}_{\tau}:\tau\in\mathcal{T}(t)\bigr{\\}}\to
r^{*}\quad\text{almost surely, as }\;t\to\infty$

for $r^{*}$ as given above, see e.g. [24] and Remark 5.2 below.

Furthermore, again in the upper half plane model, $d(z,o)\geq-\log\Im z$ by
(13), which we have used already. This and (18) imply that

$\liminf_{t\to\infty}\frac{\textsf{Max}_{t}}{t}\geq r^{*}\quad\text{almost
surely.}$

For what follows, it is preferable to return to the disk model. Note that the
particles’ positions at a given time $t$, that is, the random variables

$\textit{{B}}_{\tau}\,,\;\tau\in\mathcal{T}(t)$

have the same distribution (but are not independent) as the ordinary
hyperbolic random variable $B_{t}$. Let us assume, without loss of generality,
that the starting point is $o$.

We now choose a constant $C>r^{*}$. If $N(t)=n$ then write
$\mathcal{T}(t)=\\{\tau_{1}\,,\dots,\tau_{n}\\}$. Then

(19) $\displaystyle\mathbb{P}[\textsf{Max}_{t}\geq Ct\,,\;N(t)=n]$
$\displaystyle=\mathbb{P}\Bigl{(}\bigcup_{j=1}^{n}[d(\textit{{B}}_{\tau_{j}}\,,o)\geq
Ct\,,\;N(t)=n]\Bigr{)}$
$\displaystyle\leq\sum_{j=1}^{n}\mathbb{P}[d(\textit{{B}}_{\tau_{j}}\,,o)\geq
Ct\,,\;N(t)=n]$ $\displaystyle=n\,\mathbb{P}[d(B_{t}\,,o)\geq
Ct]\,\mathbb{P}[N(t)=n]\,,$

since the position of $\textit{{B}}_{\tau_{j}}$ does not depend on the value
of $N(t)$.

Therefore

$\displaystyle\mathbb{P}[\textsf{Max}_{t}\geq Ct]$
$\displaystyle=\sum_{n=1}^{\infty}\mathbb{P}[\textsf{Max}_{t}\geq
Ct\,,\;N(t)=n]$ $\displaystyle=\sum_{n=1}^{\infty}n\mathbb{P}[d(B_{t},o)\geq
Ct]\,\mathbb{P}[N(t)=n]$ $\displaystyle=\mathbb{P}[d(B_{t},o)\geq
Ct]\,\mathbb{E}\bigl{(}N(t)\bigr{)}=\mathbb{P}[d(B_{t},o)\geq Ct]\,e^{\lambda
t}.$

We need to estimate $\mathbb{P}[d(B_{t},o)\geq Ct]$. This works similarly as
in the last part of the proof of Proposition 3.2. We use once more (14). Up to
a change of the leading constant, we can also use $\Psi(x)=x^{1/2}$ for $x\geq
1/2$ (instead of $x\geq 1$). This applies to the range which we are now
considering, namely $R\geq Ct>r^{*}t\geq t/2$. We perform the following
estimate for (non-branching) hyperbolic BM in $\mathbb{D}$ starting at $o$.

$\displaystyle\mathbb{P}[d(B_{t},o)\geq C\,t]$
$\displaystyle=\int_{\\{z:d(z,o)\geq Ct\\}}p_{t}(o,z)\,d\mathsf{m}(z)$
$\displaystyle\leq\textsl{Const}\int_{-\pi}^{\pi}\int_{Ct}^{\infty}\,\frac{\sqrt{1+R}}{t}\,\exp\biggl{(}-\frac{(R+\tfrac{t}{2})^{2}}{2t}\biggr{)}\,\frac{e^{R}-e^{-R}}{2}\,dR\,d\varphi$
$\displaystyle\leq\textsl{Const}^{\prime}\int_{Ct}^{\infty}\frac{\sqrt{1+R}}{t}\,\exp\biggl{(}-\frac{(R-\tfrac{t}{2})^{2}}{2t}\biggr{)}\,dR$
$\displaystyle\noindent substituting$s = (R-\tfrac{t}{2})/\sqrt{t}$$
$\displaystyle=\textsl{Const}^{\prime}\int_{(C-\frac{1}{2})\sqrt{t}}^{\infty}\sqrt{\frac{1}{t}+\frac{1}{2}+\frac{s}{\sqrt{t}}}\,\exp\biggl{(}-\frac{s^{2}}{2}\biggr{)}\,ds$
$\displaystyle\noindent since$\tfrac{s}{\sqrt{t}} \geq C - \tfrac{1}{2} >
\sqrt{2\lambda}$$
$\displaystyle\leq\textsl{Const}^{\prime\prime}\int_{(C-\frac{1}{2})\sqrt{t}}^{\infty}\frac{s}{\sqrt{t}}\exp\biggl{(}-\frac{s^{2}}{2}\biggr{)}\,ds=\textsl{Const}^{\prime\prime}\frac{1}{\sqrt{t}}\,\exp\biggl{(}-\frac{(C-\tfrac{1}{2})^{2}t}{2}\biggr{)},$

(The constants are not assumed to be the same as in the proof of Proposition
3.2. Also, here $\textsl{Const}^{\prime\prime}$ depends on $\lambda$.)

Combining our computations, we get

$\mathbb{P}[\textsf{Max}_{t}\geq
Ct]\leq\textsl{Const}^{\prime\prime}\frac{1}{\sqrt{t}}\,\exp\biggl{(}-\frac{\bigl{(}(C-\tfrac{1}{2})^{2}-2\lambda\bigr{)}t}{2}\biggr{)}\,.$

Since $C>r^{*}$, we have $(C-\tfrac{1}{2})^{2}-2\lambda>0$. Consequently, at
integer times

$\sum_{n=1}^{\infty}\mathbb{P}[\textsf{Max}_{n}\geq Cn]<\infty\,,$

and by the Borel-Cantelli Lemma,

$\limsup_{n\to\infty}\frac{\textsf{Max}_{n}}{n}\leq C\quad\text{almost
surely.}$

This holds for every $C>r^{*}$, whence in view of the lower bound

$\lim_{n\to\infty}\frac{\textsf{Max}_{n}}{n}=r^{*}\quad\text{almost surely.}$

We now need to fill in the “gaps” for real $t\in(n-1\,,\,n)$, where $n$ runs
through the positive integers. For this purpose, we first define

$\overline{\\!\mathcal{M}}=\max\\{d(\textit{{B}}_{\tau}\,,o):\tau\in\mathcal{T}(\prec\\!1)\\}$

_Claim._ For any $c>0$,

$\mathbb{P}[\,\overline{\\!\mathcal{M}}>c]\leq
e^{\lambda}\,\mathbb{P}[\mathcal{M}>c]\,,$

where $\mathcal{M}$ is as in Proposition 3.2, that is, the maximal distance of
ordinary hyperbolic BM from the starting point within the time interval
$[0\,,\,1]$.
_Proof of the Claim._ Suppose that $N(1)=k$, and just within this proof, let
$\mathcal{T}(1)=\\{\tau_{1}\,,\dots,\tau_{k}\\}$. For each $j$, consider the
geodesic path in $\mathcal{T}$ from the root $\alpha$ to $\tau_{j}\,$. Along
this timeline, we see an ordinary hyperbolic BM starting at $0$, runnig up to
time $1$. Let $\mathcal{M}_{j}$ be the maximal distance from $0$ of this BM.
The random variables $\mathcal{M}_{1}\,,\dots,\mathcal{M}_{k}$ are not
independent, but they all have the same distribution as $\mathcal{M}$.
Furthermore,

$\overline{\\!\mathcal{M}}=\max\\{\mathcal{M}_{1}\,,\dots,\mathcal{M}_{k}\\}.$

The proof of the Claim is now completed in the same way as in (19).

Note that the event $[\,|v|\notin\mathbb{N}\;\text{for
all}\;\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}]$ has probability
$1$. We work on that event. Now let $t>0$ and $n=\lfloor t\rfloor$. Each
$\tau\in\mathcal{T}(n)$ is the root of a single subtree
$\mathcal{T}_{\tau}=\mathcal{T}_{\tau}(\prec\\!1)$ of height $1$ within
$\mathcal{T}$, whose end-vertices belong to $\mathcal{T}(n+1)$. Let

$\overline{\\!\mathcal{M}}_{\tau}=\max\\{d(\textit{{B}}_{\theta}\,,\textit{{B}}_{\tau}):\theta\in\mathcal{T}_{\tau}\\}$

The random variables $\;\overline{\\!\mathcal{M}}_{\tau}\,$,
$\tau\in\mathcal{T}(n)$, are independent and (since the exponential
distribution is memoryless) have the same distribution as the RV
$\;\overline{\\!\mathcal{M}}$ of the Claim. We clearly have

$\textsf{Max}_{t}\leq\textsf{Max}_{n}+\max\\{\,\overline{\\!\mathcal{M}}_{\tau}:\tau\in\mathcal{T}(n)\\}.$

The proof of the theorem will be complete if we can show that

(20)
$\lim_{n\to\infty}\frac{1}{n}\max\\{\,\overline{\\!\mathcal{M}}_{\tau}:\tau\in\mathcal{T}(n)\\}=0\quad\text{almost
surely.}$

Let $c>0$. Then, once more in the same way as in (19), ad using the Claim,

$\displaystyle\mathbb{P}\bigl{[}\max\\{\,\overline{\\!\mathcal{M}}_{\tau}:\tau\in\mathcal{T}(n)\\}>cn]$
$\displaystyle\leq\mathbb{P}\bigl{[}\,\overline{\\!\mathcal{M}}>cn]\,e^{\lambda
n}$ $\displaystyle\leq\,\mathbb{P}[\mathcal{M}>cn]\,e^{\lambda(n+1)}$
$\displaystyle\leq\textsl{K}\,\exp\biggl{(}-\frac{(cn-\tfrac{1}{2})^{2}-2\lambda(n+1)}{2}\biggr{)}\,$

by Proposition 3.2. Summing over all $n$, the resulting series converges for
every $c>0$, and once more, the Borel-Cantelli Lemma yields (20). ∎

######  Remark 5.2.

Regarding the value of $r^{*}$, the literature deals with Euclidean BBM where
the underlying BM is standard with drift $0$. Also, usually only the case
$\lambda=1$ is considered for the fissioning rate (i.e., the parameter of the
exponential distribution of the edge lengths of the Yule tree). The
$\frac{1}{2}$ in our formula for $r^{*}$ comes from the drift $\frac{1}{2}$ of
our underlying Brownian motion. Then, in the drift-free case with $\lambda=1$,
on finds e.g. in [24] – see also Bovier [6, Lemma 3.4] – that the maximum at
time $t$ of Euclidean BBM has rate $\sqrt{2}$. Now let $\mathcal{T}$ be the
Yule tree with parameter $\lambda$ as in §2, and let
$(\boldsymbol{\beta}_{\tau})_{\tau\in\mathcal{T}}$ be the associated standard
Euclidean BBM (where the underlying BM is drift-free). Write
$\lambda\mathcal{T}$ for the random tree where all the edge lengths are
multiplied by $\lambda$. This is the Yule tree with fissioning rate $1$. Then
by the scale-invariance of BM,
$(\frac{1}{\sqrt{\lambda}}\,\boldsymbol{\beta}_{\tau})_{\tau\in\mathcal{T}}$
is standard Euclidean BBM. From here, one obtains the general formula. This is
of course well-known and clear to the experts, but the hints may be useful for
the novice.

##  6\. The empirical distributions

We can interpret hyperbolic BBM as a Markov process on the space of
_populations_ in $\mathbb{D}$, where a population is a finitely supported
measure of the form

$\mathbf{M}=\sum_{j=1}^{n}\delta_{z_{j}}\,,\quad
n\in\mathbb{N}\,,\;z_{j}\in\mathbb{D}\;\text{not necessarily distinct.}$

(Another interpretation is to see this as a multiset; see [18].)

######  Definition 6.1.

Starting with one particle at position $z_{0}\,$, the _occupation measure_ of
hyperbolic BBM at time $t\geq 0$ is the population

$\mathbf{M}_{t}=\mathbf{M}^{z_{0}}_{t}=\sum_{\tau\in\mathcal{T}:|\tau|=t}\delta_{\textit{{B}}_{\tau}}$

and the associated _empirical distribution_ is

$\mu_{t}=\mu_{t}^{z_{0}}=\frac{1}{N(t)}\mathbf{M}_{t}\,.$

Thus, $\mu_{t}$ is a finitely supported probability measure on the disk. Both
$\mathbf{M}_{t}$ and $\mu_{t}$ depend on the starting point $z_{0}\,$, and in
our construction, with $g_{z_{0}}$ as in (4),
$\mathbf{M}^{z_{0}}_{t}=g_{z_{0}}\mathbf{M}_{t}^{o}$ is the image of
$\mathbf{M}_{t}^{o}$ under the mapping $z\mapsto g_{z_{0}}z$. (That is, for a
measure $\mu$ on $\overline{\mathbb{D}}$ and an isometry $g$ of $\mathbb{D}$,
we have $g\mu(K)=\mu(g^{-1}K)$.) We shall often omit the starting point in the
notation.

Instead of starting BBM at a single point, we may start with an arbitrary
initial population $\mathbf{M}_{0}$. Then the population at time $t$ is

(21)
$\mathbf{M}_{t}=\sum_{j=1}^{n}\mathbf{M}^{z_{j}}_{t}\,,\quad\text{if}\quad\mathbf{M}_{0}=\sum_{j=1}^{n}\delta_{z_{j}}\,,$

where the occupation measures $\mathbf{M}^{z_{i}}_{t}$ are independent even
when some of the $z_{i}$ coincide.

We start with an obvious consequence of the continuity of the distributions of
the random variables $\textit{{B}}_{\tau}\,$.

######  Fact 6.2.

For distinct $\tau,\tau^{\prime}\in\mathcal{T}$, one has
$\mathbb{P}[\textit{{B}}_{\tau}=\textit{{B}}_{\tau^{\prime}}]=0$.

In particular, for any fixed $t\geq 0$, with probability $1$ the measure
$\mu_{t}$ is equidistribution on $N(t)$ distinct points.

Of course this does not imply that it never happens that $\mu_{t}$ is other
than equidistributed. However, the following is also quite clear.

######  Lemma 6.3.

Let $\pi$ and $\pi^{\prime}$ be two distinct rays in the Yule tree. Then, with
probability $1$, there is a random $t_{0}\geq 0$ such that
$\textit{{B}}_{\tau}\neq\textit{{B}}_{\tau^{\prime}}$ for all $\tau\in\pi$ and
$\tau^{\prime}\in\pi^{\prime}$ with $|\tau|,|\tau^{\prime}|>t_{0}\,$.

###### Proof.

Let $v=\pi\wedge\pi^{\prime}$. (Compare with the last lines of §4.)
Conditionally upon the location (value) $\textit{{B}}_{\mathfrak{v}}=z_{0}\,$,
beyond $v$ the two Brownian motions along $\pi$ and $\pi^{\prime}$ are
independent replicas of hyperbolic BM starting at $z_{0}\,$. Each of them has
an a.s. random limit $\xi$, resp. $\xi^{\prime}\in\partial\mathbb{D}$. Both
are distributed according to $\nu_{z_{0}}\,$; see (24) and the preceding
lines. Continuity of $\nu_{z_{0}}$ yields that $\xi\neq\xi^{\prime}$ almost
surely. This implies the statement of the lemma. ∎

Again, this does not necessarily mean that from some random time onwards, all
$\mu_{t}$ are equidistributed. But if we restrict to all $\mu_{n}\,$,
$n\in\mathbb{N}$, then with probability $1$ all of them _are_ equidistributed.

For a stronger result than Lemma 6.3 concerning branching random walks on
finitely generated groups, see Hutchcroft [17].

######  Question 6.4.

Can one prove a result analogous to [17] that in the regime $\lambda\leq 1/8$,
two independent copies of hyperbolic BBM meet at most finitely often? It
appears to be true at integer times for any $\lambda$.

In the place of $\mu_{t}\,$, it will also be useful to use the discrete random
measure

$\sigma_{t}=\sigma^{z_{0}}_{t}=\frac{1}{e^{\lambda
t}}\,\mathbf{M}^{z_{0}}_{t}\,,$

the image of $\sigma^{0}_{t}$ under $z\mapsto g_{z_{0}}z$. It is not a
probability measure, but its expectation is a deterministic probability
measure on $\mathbb{D}$, as the following Lemma shows.

######  Lemma 6.5.

Let $K\subset\mathbb{D}$ be compact. Then the expected number of particles
present within $K$ at time $t$ is

$\mathbb{E}_{z_{0}}\bigl{(}\mathbf{M}_{t}(K)\bigr{)}=e^{\lambda
t}\,\mathbb{P}_{z_{0}}[B_{t}\in K]\,,$

where $B_{t}$ is hyperbolic Brownian motion started at $z_{0}\,$. Thus,

$\mathbb{E}_{z_{0}}\left(\int_{0}^{\infty}\sigma_{t}(K)\,dt\right)<\infty\,,$

and therefore

$\lim_{t\to\infty}\mu_{t}(K)=0\quad\text{almost surely.}$

###### Proof.

Once more, if $N(t)=n$ then write
$\mathcal{T}(t)=\\{\tau_{1}\,,\dots,\tau_{n}\\}$. Then, similarly to (19),

$\displaystyle\mathbb{E}_{z_{0}}\bigl{(}\mathbf{M}_{t}(K)\bigr{)}$
$\displaystyle=\sum_{n=1}^{\infty}\mathbb{E}_{z_{0}}\Bigl{(}\sum_{j=1}^{n}\mathbf{1}_{K}(\textit{{B}}_{\tau_{j}}\mid
N(t)=n\Bigr{)}\,\mathbb{P}[N(t)=n]$
$\displaystyle=\sum_{n=1}^{\infty}n\,\mathbb{P}_{z_{0}}[B_{t}\in
K]\,\mathbb{P}[N(t)=n]=\mathbb{E}\bigl{(}N(t)\bigr{)}\,\mathbb{P}_{z_{0}}[B_{t}\in
K].$

This shows the first identity, and finiteness of
$\mathbb{E}_{z_{0}}\bigl{(}\int_{0}^{\infty}\sigma_{t}(K)\,dt\bigr{)}$ follows
from trancience of hyperbolic BM. It implies that the random variable
$\int_{0}^{\infty}\sigma_{t}(K)\,dt$ is a.s. finite, whence

$\lim_{t\to\infty}\mu_{t}(K)=\frac{1}{W}\lim_{t\to\infty}\sigma_{t}(K)=0\quad\text{almost
surely,}$

where $W$ is the martingale limit of Proposition 2.2. ∎

Thus, the mass of the empirical distributions $\mu_{t}$ “disappears at
infinity” in the hyperbolic metric, resp. topology. (More precisely, the
$\mu_{t}$ tend to 0 vaguely within the hyperbolic disk.)

We can express the first formula of Lemma 2.1 in terms of the transition
operator (7) of (non-branching) hyperbolic BM: for a measurable set
$K\subset\mathbb{D}$ and starting point $z_{0}$,

(22) $\mathbb{E}_{z_{0}}\bigl{(}\sigma_{t}(K)\bigr{)}=P_{t}(z,K)\,.$

Thus, for a measurable function $f:\mathbb{D}\to\mathbb{R}$

(23)
$\mathbb{E}_{z_{0}}\left(\int_{\mathbb{D}}f\,d\mathbf{M}_{t}\right)=e^{\lambda
t}\,P_{t}f(z_{0})\quad\text{and}\quad\mathbb{E}_{z_{0}}\left(\int_{\mathbb{D}}f\,d\sigma_{t}\right)=P_{t}f(z_{0})\,,$

as long as the involved integral is well defined. A _harmonic function_ is a
$C^{2}$-function $h$ on $\mathbb{D}$ which satisfies $\mathfrak{L}h\equiv 0$.
It is well known that every harmonic function satisfies

$P_{t}h=h\quad\text{for every }\;t>0.$

Now let $\mathcal{F}_{t}=\mathcal{F}_{t}^{\text{BBM}}$ be the sigma-algebra
generated by the information of hyperbolic BBM up to and including time $t$.
(It projects naturally onto $\mathcal{F}_{t}^{\text{Yule}}$.)

######  Proposition 6.6.

If $h$ is a bounded or positive harmonic function, then the family of random
variables

$\int h\,d\sigma_{t}=e^{-\lambda
t}\sum_{\tau\in\mathcal{T}:|\tau|=t}h(\textit{{B}}_{\tau})\,,\quad t\geq 0$

is a martingale with respect to the filtration $(\mathcal{F}_{t})_{t\geq
0}\,$.

###### Proof.

For any bounded or positive measurable function $f:\mathbb{D}\to\mathbb{C}$,
consider its lift $\widetilde{f}$ to the space of populations:

$\widetilde{f}(\mathbf{M})=\int_{\mathbb{D}}f\,d\mathbf{M}=\sum_{z\in\mathbb{D}}f(z)\,\mathbf{M}(\\{z\\})\,,$

a finite sum. Let $(\widetilde{P}_{t})_{t\geq 0}$ be the family of transition
operators (i.e. the transition semigroup) of hyperbolic BBM, that is, of the
Markov process $(\mathbf{M}_{t})_{t\geq 0}$. By (21), for any population
$\mathbf{M}_{0}=\sum_{j=1}^{n}\delta_{z_{j}}$,

$\displaystyle\widetilde{P}_{t}\widetilde{f}(\mathbf{M}_{0})$
$\displaystyle=\mathbb{E}_{\mathbf{M}_{0}}\bigl{(}\widetilde{f}(\mathbf{M}_{t})\bigr{)}=\mathbb{E}_{\mathbf{M}_{0}}\left(\sum_{z}f(z)\,\mathbf{M}_{t}(\\{z\\})\right)$
$\displaystyle=\mathbb{E}_{\mathbf{M}_{0}}\left(\sum_{z}f(z)\sum_{j=1}^{n}\mathbf{M}^{z_{j}}_{t}(z)\right)=\sum_{j=1}^{n}\mathbf{M}_{0}(z_{j})\,\mathbb{E}_{z_{j}}\left(\sum_{z}f(z)\,\mathbf{M}^{z_{j}}_{t}(z)\right)$
$\displaystyle=\sum_{j=1}^{n}\mathbf{M}_{0}(z_{j})\,\mathbb{E}_{z_{j}}\left(\int
f\,d\mathbf{M}^{z_{j}}_{t}(z)\right)=\sum_{j=1}^{n}\mathbf{M}_{0}(z_{j})\,e^{\lambda
t}\,P_{t}f(z_{j})$ $\displaystyle=e^{\lambda
t}\,\widetilde{P_{t}f}(\mathbf{M}_{0})$

by (23). In particular, if $h$ is a bounded or positive harmonic function on
$\mathbb{D}$ then $\widetilde{P}_{t}\widetilde{h}=e^{\lambda
t}\,\widetilde{h}$ for every $t\geq 0$. This yields that $\int
h\,d\sigma_{t}=e^{-\lambda t}\,\int h\,d\mathbf{M}_{t}$ is a martingale. ∎

######  Theorem 6.7.

With probability 1, as $t\to\infty$, the measures $\sigma_{t}$ converge weakly
to a Borel measure $\sigma_{\infty}$ on $\partial\mathbb{D}$, and the
probability measures $\mu_{t}$ converge weakly to a probability measure
$\mu_{\infty}$ on $\partial\mathbb{D}$. With $W$ as in Proposition 2.2, we
have

$\sigma_{\infty}=W\mu_{\infty}\,,\quad\text{and}\quad\mathbb{E}_{z_{0}}(\sigma_{\infty})=\nu_{z_{0}}\,,$

the measure on $\partial\mathbb{D}$ whose density with respect to the
normalised Lebesgue measure on the circle is the Poisson kernel (8).

In particular, for every $\xi\in\partial\mathbb{D}$, we have
$\mu_{\infty}(\\{\xi\\})=0$ almost surely.

###### Proof.

We use a potential theoretic argument. Note that the harmonic functions for
$\mathfrak{L}$ are the same as the harmonic functions for the Euclidean
Laplacian on $\mathbb{D}$. The corresponding Dirichlet problem is solvable:
given a continuous function $\varphi$ on $\partial\mathbb{D}$, there is a
unique harmonic function $h_{\varphi}$ on $\mathbb{D}$ which provides a
continuous extension of $\varphi$ to the interior of the disk. Indeed,
$h_{\varphi}$ is the Poisson transform of $\varphi$,

(24) $h_{\varphi}(z)=\int_{\partial\mathbb{D}}\Pi(z,\xi)\varphi(\xi)\,d\xi\,.$

Now let $f$ be any continuous function on the closed disk
$\overline{\mathbb{D}}$. Considering $\sigma_{t}$ as a measure on
$\overline{\mathbb{D}}$, we have to show that
$\int_{\mathbb{D}}f\,d\sigma_{t}$ converges almost surely. Let
$\varphi=f|_{\partial\mathbb{D}}$ be the restriction of $f$ to the unit
circle, and let $h_{\varphi}$ be the associated solution of the Dirichlet
problem. This is a bounded harmonic function, so that

$\int_{\mathbb{D}}h_{\varphi}\,d\sigma_{t}$

converges almost surely by Proposition 6.6. On the other hand,

$\lim_{|z|\to 1}f(z)-h_{\varphi}(z)=0\quad\text{uniformly in $z$.}$

Given $\varepsilon>0$, take $r\in(0\,,\,1)$ such that
$|f(z)-h_{\varphi}(z)|<\varepsilon$ for $|z|\geq r$. Let
$K=\\{z\in\mathbb{D}:|z|\leq r\\}$. By Proposition 2.2 and Lemma 6.5,

$\int_{K}(f-h_{\varphi})\,d\mu_{t}=\frac{e^{\lambda
t}}{N(t)}\int_{K}(f-h_{\varphi})\,d\sigma_{t}\to 0\quad\text{almost surely,}$

and since $\mu_{t}$ is a probability measure,

$\left|\int_{\mathbb{D}\setminus
K}(f-h_{\varphi})\,d\mu_{t}\right|<\varepsilon\,.$

This shows weak convergence. The identity $\sigma_{\infty}=W\mu_{\infty}$ is
immediate from Proposition 2.2. Finally, if $f\in C(\overline{\mathbb{D}})$
then by (23) and dominated convergence

$\displaystyle\mathbb{E}_{z_{0}}\left(\int_{\mathbb{D}}f\,d\sigma_{\infty}\right)$
$\displaystyle=\lim_{t\to\infty}\mathbb{E}_{z_{0}}\left(\int_{\mathbb{D}}f\,d\sigma_{t}\right)=\lim_{t\to\infty}P_{t}f(z_{0})=\lim_{t\to\infty}\mathbb{E}_{z_{0}}\bigl{(}f(B_{t})\bigr{)}=\mathbb{E}_{z_{0}}\bigl{(}f(B_{\infty})\bigr{)}$
$\displaystyle=\int_{\partial\mathbb{D}}f\,d\nu_{z_{0}}\,.$

In particular, for any $\xi\in\partial\mathbb{D}$,
$\mathbb{E}_{z_{0}}\bigl{(}\sigma_{\infty}(\\{\xi\\})\bigr{)}=\nu_{z_{0}}(\\{\xi\\})$,
so that $\sigma_{\infty}(\\{\xi\\})=0$ almost surely. ∎

Note that the latter does not necessarily imply that almost surely, the limit
measure carries no point mass.

The _radial projection_ of $\mathbb{D}$ to the boundary circle
$\partial\mathbb{D}$ is the mapping

$\textsl{rad}(re^{\mathfrak{i}\,\psi})=e^{\mathfrak{i}\,\psi}\,,\;\text{if
}\;0<r<1\,,\quad\text{and}\quad\textsl{rad}(0)=1\,.$

(The value $\textsl{rad}(0)$ is of no specific importance and might be chosen
arbitrarily.) We can consider the radial projection of hyperbolic Brownian
motion, and the image of $\mu_{t}$ under the mapping rad.

######  Corollary 6.8.

With probability 1, the probability measures

$\mu_{t}^{\textsl{rad}}=\frac{1}{N(t)}\sum_{\tau\in\mathcal{T}:|\tau|=t}\delta_{\textsl{rad}(\textit{{B}}_{\tau})}$

on $\partial\mathbb{D}$ converge weakly to $\mu_{\infty}\,$, as
$t\to\infty\,$.

###### Proof.

Let $\varphi\in C(\mathbb{D})$, and let $h_{\varphi}$ be its Poisson transform
(24), providing the continuous extension of $\varphi$ to
$\overline{\mathbb{D}}$ which is harmonic on $\mathbb{D}$. To “hide” the
discontinuity of $\textsl{rad}(\cdot)$ at $0$, let

$f(z)=\begin{cases}\Bigl{(}h_{\varphi}(z)-\varphi\bigl{(}\textsl{rad}(z)\bigr{)}\Bigr{)}\,\min\\{2|z|,1\\}\,&\text{if
}\;z\in\mathbb{D},\\\ 0\,&\text{if }\;z\in\partial\mathbb{D}\end{cases}$

Then $f\in C(\overline{\mathbb{D}})$. We get that with probability 1,

$\lim_{t\to\infty}\int_{\overline{\mathbb{D}}}f\,d\mu_{t}=0$

On the other hand,

$\lim_{t\to\infty}\int_{\overline{\mathbb{D}}}h_{\varphi}\,d\mu_{t}=\int_{\partial\mathbb{D}}\varphi\,d\mu_{\infty}\,.$

This concludes the proof. ∎

In analogy with Corollary 6.8, we have the following for the real parts of
hyperbolic BBM in the upper half plane model. We omit the very similar proof.

######  Corollary 6.9.

Let

$\mu_{t}^{\Re}=\frac{1}{N(t)}\sum_{\tau\in\mathcal{T}:|\tau|=t}\delta_{\Re\textit{{B}}_{\tau}^{\mathbb{H}}}.$

Then, with probability one, $\mu_{t}^{\Re}$ converges weakly to
$\mu_{\infty}^{\mathbb{H}}\,$, as $t\to\infty\,$.

Another feature of the “disappearance” of the population at infinity is the
average rate of escape.

######  Definition 6.10.

The _(empirical) distance distribution_ of hyperbolic BBM at time $t\geq 0$ is
the finitely supported random measure $\mu_{t}^{d}$ on $\mathbb{R}_{+}$ which
is the image of $\mu_{t}$ under the mapping $\mathbb{D}\ni z\mapsto d(z,o)$,
that is,

$\mu_{t}^{d}=\frac{1}{N(t)}\sum_{\tau\in\mathcal{T}:|\tau|=t}\delta_{d(\textit{{B}}_{\tau},o)}.$

######  Theorem 6.11.

With probability one, $\mu_{t}^{d}$ is asymptotically normal with mean $t/2$
and variance $t$. That is, its distribution function satisfies

$\mu_{t}^{d}\bigl{(}-\infty\,,\tfrac{1}{2}t+x\sqrt{t}\,\bigr{]}\to\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-s^{2}/2}\,ds\quad\text{for
every $x\in\mathbb{R}$,}$

as $t\to\infty\,$.

Before the proof, we make a detour to the upper half plane model and write
$(\textit{{B}}_{\tau}^{\mathbb{H}})_{\tau\in\mathcal{T}}$ for the resulting
model of hyperbolic BBM: it is the image of
$(\textit{{B}}_{\tau})_{\tau\in\mathcal{T}}$ as costructed on $\mathbb{D}$
under the mapping (9). Considering the logarithmic model, we see the
following.

######  Fact 6.12.

The vertical projection

$\textit{{B}}_{\tau}^{\text{vert}}=-\log\Im\textit{{B}}_{\tau}^{\mathbb{H}}\,,\quad\tau\in\mathcal{T},$

is one-dimensional branching Brownian motion on $\mathbb{R}$; the underlying
Euclidean Brownian motion at time $t$ has drift $\frac{1}{2}t$ and variance is
$t$; see (11).

The reason for choosing the minus sign is that this Euclidean BBM has the same
asymptotic drift and variance as the one proposed in Theorem 6.11, and we will
use comparison of the two processes.

For the following, see Ney [25, Thm. 2], Kaplan and Asmussen [19, Thm. 3] as
well as Biggins [5].

######  Proposition 6.13.

Let

$\mu_{t}^{\text{vert}}=\frac{1}{N(t)}\sum_{\tau\in\mathcal{T}:|\tau|=t}\delta_{\textit{{B}}_{\tau}^{\text{vert}}}.$

With probability 1, as $t\to\infty$, the family of discrete probability
distributions $\mu_{t}^{\text{vert}}$ on $\mathbb{R}$ is asymptotically normal
with mean $t/2$ and variance $t$.

[25] only has convergence in mean square, whence in probability. [19] has
almost sure convergence when the drift of the base Brownian motion is $0$ with
obvious extension to non-zero drift; see also the last section of [5].

###### Proof of Theorem 6.7.

We work with the upper half plane model and indicate this by the superscript
$\mathbb{H}$. Let $\Omega_{0}$ be the event on which $\mu_{t}^{\mathbb{H}}$
converges weakly and $\mu_{\infty}^{\mathbb{H}}(\mathfrak{i}\,\infty)=0$. On
$\Omega_{0}\,$, the latter measure assigns mass $0$ to the boundary point
$\mathfrak{i}\,\infty$, so that it lives on the lower boundary line
$\mathbb{R}$. Let $f\in C^{\infty}(\mathbb{R})$ be uniformly continuous and
set

$f_{t}(x)=f\biggl{(}\frac{x-\frac{1}{2}t}{\sqrt{t}}\biggr{)}.$

By the Portmanteau Theorem, we need to show that on $\Omega_{0}\,$,

$\lim_{t\to\infty}\int_{\mathbb{R}}f_{t}(x)\,d\mu_{t}^{d}(x)=\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}f(x)\,dx\,.$

Given $\varepsilon>0$ there is a random bound $M<\infty$ such that

$\mu_{\infty}^{\mathbb{H}}\bigl{(}[-M\,,\,M]\bigr{)}\geq 1-\varepsilon\,.$

Consider the closed set $U_{M}=\\{z\in\overline{\mathbb{H}}:|z|\leq M\,,\;\Im
z\leq 1\\}\subset\mathbb{H}$, where
$\overline{\mathbb{H}}=\mathbb{H}\cup\mathbb{R}\cup\\{\mathfrak{i}\,\infty\\}$
is the geometric compactification of the hyperbolic upper half plane. Then

(25) $\limsup_{t\to\infty}\mu_{t}^{\mathbb{H}}(\overline{\mathbb{H}}\setminus
U_{M})\leq\varepsilon\,.$

We decompose

$\int_{\mathbb{R}}f_{t}(x)\,d\mu_{t}^{d}(x)=\int_{\overline{\mathbb{H}}}f_{t}\bigl{(}d_{\mathbb{H}}(z,o)\bigr{)}\,d\mu_{t}^{\mathbb{H}}(z)$

into $\int_{U_{M}}+\int_{\overline{\mathbb{H}}\setminus U_{M}}$. Regarding the
second part, we have by (25)

$\limsup_{t\to\infty}\left|\int_{\overline{\mathbb{H}}\setminus
U_{M}}f_{t}\bigl{(}d_{\mathbb{H}}(z,o)\bigr{)}\,d\mu_{t}^{\mathbb{H}}(z)\right|\leq\varepsilon\,\|f\|_{\infty}\,.$

We now apply the same decomposition to

$\int_{\mathbb{R}}f_{t}(x)\,d\mu_{t}^{\text{vert}}(x)=\int_{\overline{\mathbb{H}}}f_{t}\bigl{(}-\log\Im
z\bigr{)}\,d\mu_{t}^{\mathbb{H}}(z)$

and get

$\limsup_{t\to\infty}\left|\int_{\overline{\mathbb{H}}\setminus
U_{M}}f_{t}\bigl{(}-\log\Im
z\bigr{)}\,d\mu_{t}^{\mathbb{H}}(z)\right|\leq\varepsilon\,\|f\|_{\infty}\,.$

We now consider the difference of the integrals over $U_{M}\,$, recalling that
for $\Im z\leq 1$ one has $d_{\mathbb{H}}(\Im z,o)=-\log\Im z\,$:

$\displaystyle\left|\int_{U_{M}}f_{t}\bigl{(}d_{\mathbb{H}}(z,o)\bigr{)}\,d\mu_{t}^{\mathbb{H}}(z)-\int_{U_{M}}f_{t}\bigl{(}-\log\Im
z\bigr{)}\,d\mu_{t}^{\mathbb{H}}(z)\right|$ $\displaystyle\hskip
128.0374pt\leq\int_{U_{M}}\Bigl{|}f_{t}\bigl{(}d_{\mathbb{H}}(z,o)\bigr{)}-f_{t}\bigl{(}d_{\mathbb{H}}(\Im
z,o)\bigr{)}\Bigr{|}\,d\mu_{t}^{\mathbb{H}}(z)\,.$

By (13), we have $|d_{\mathbb{H}}(z,o)-d_{\mathbb{H}}(\Im
z,o)|\leq\log(1+M^{2})$ on $U_{M}\,$. Therefore

$\left|\frac{d_{\mathbb{H}}(z,o)-\frac{1}{2}t}{\sqrt{t}}-\frac{d_{\mathbb{H}}(\Im
z,o)-\frac{1}{2}t}{\sqrt{t}}\right|\leq\frac{\log(1+M^{2})}{\sqrt{t}}\,,$

and by uniform continuity of $f$,

$\lim_{t\to\infty}\Bigl{|}f_{t}\bigl{(}d_{\mathbb{H}}(z,o)\bigr{)}-f_{t}\bigl{(}d_{\mathbb{H}}(\Im
z,o)\bigr{)}\Bigr{|}=0\quad\text{uniformly on }\;U_{M}\,.$

We infer that on $\Omega_{0}$

$\lim_{t\to\infty}\int_{\mathbb{R}}f_{t}(x)\,d\mu_{t}^{d}(x)=\lim_{t\to\infty}\int_{\mathbb{R}}f_{t}(x)\,d\mu_{t}^{\text{vert}}(x)\,,$

and Proposition 6.13 yields the proposed asymptotic behaviour. ∎

The _average displacement_ of hyperbolic BBM is

(26)
$\int_{\mathbb{D}}d(z,o)\,d\mu_{t}(z)=\frac{1}{N(t)}\sum_{\tau\in\mathcal{T}:|\tau|=t}d(\textit{{B}}_{\tau},o)$

######  Theorem 6.14.

With probability 1, the average displacement of hyperbolic BBM has rate of
escape

$\lim_{t\to\infty}\frac{1}{t}\int_{\mathbb{D}}d(z,o)\,d\mu_{t}(z)=\frac{1}{2}\,.$

###### Proof.

In the sample space $\Omega$ of hyperbolic BBM, let $\Omega_{1}$ be the set on
which $\mu_{t}^{d}$ is asymptotically normal as in Theorem 6.11, and

$\frac{1}{\textsf{Max}_{t}}\to r^{*}$

as in Theorem 5.1, where
$\textsf{Max}_{t}=\max\bigl{\\{}d(\textit{{B}}_{\tau},o):\tau\in\mathcal{T}\,,\;|\tau|=t\bigr{\\}}.$
We can consider this as providing a random environment for the family of
probability distributions $\mu_{t}^{d}$, and consider one quenched case, i.e.,
a fixed $\omega\in\Omega_{1}\,$. We can think of the associated deterministic
family $\mu_{t}^{d}$, $t>0$, as the distributions of a family of random
variables $X_{t}$. By asymptotic normality,

$\lim_{t\to\infty}\frac{X_{t}}{t}=\frac{1}{2}\quad\text{in probability.}$

On the other hand,

$\frac{|X_{t}|}{t}\leq\frac{\textsf{Max}_{t}}{t}\,,$

which is bounded. We can apply the Dominated Convergence Theorem, and denoting
by $\mathbb{E}_{\omega}$ the quenched expectation, we get

$\lim_{t\to\infty}\frac{\mathbb{E}_{\omega}(X_{t})}{t}=\frac{1}{2}.$

Now

$\mathbb{E}_{\omega}(X_{t})=\int_{\mathbb{R}}x\,d\mu_{t}^{d}(x)=\int_{\mathbb{D}}d(z,o)\,d\mu_{t}(z)\,,$

and this proves the claim. ∎

##  7\. Some properties of the random limit distributions

Here, we shall collect a few properties of the measures
$\mu_{\infty}^{z_{0}}\,$ plus related open questions. Before that, we start
with a general fact with a simple proof, comunicated to the author by V. A.
Kaimanovich during the work on the paper [18].

######  Proposition 7.1.

Let $\mathcal{X}$ be a separable compact space and $\mathcal{P}(\mathcal{X})$
be the space of Borel probability measures on $\mathcal{X}$. With the
weak*topology, it is again separable and compact. Now let $\mu$ and
$\mu^{\prime}$ be independent random probability measures on $\mathcal{X}$,
with distributions $\boldsymbol{\nu}$ and $\boldsymbol{\nu}^{\prime}$,
respectively. That is, the latter are probability measures on
$\mathcal{P}(\mathcal{X})$, and the distribution of $(\mu,\mu^{\prime})$ on
$\mathcal{X}^{2}$ is $\boldsymbol{\nu}\otimes\boldsymbol{\nu}^{\prime}$.

Suppose that the expectation of $\mu$ (barycentre of $\boldsymbol{\nu}$), that
is, the deterministic probability measure in $\mathcal{X}$ given by

$\overline{\mu}(K)=\int_{\mathcal{P}(\mathcal{X})}\mu(K)\,d\boldsymbol{\nu}(\mu)\quad(K\subset\mathcal{X}\;\text{compact})$

on $\mathcal{X}$ is purely non-atomic. Then, almost surely, $\mu^{\prime}$ and
$\mu$ share no atoms.

(“Almost surely” refers to the probability measure
$\boldsymbol{\nu}\otimes\boldsymbol{\nu}^{\prime}$.)

###### Proof.

_Step 1._ Since $\overline{\mu}$ is non-atomic, for any $x\in\mathcal{X}$

$\mu(x)=0\quad\text{for }\;\boldsymbol{\nu}-\text{almost every
}\;\mu\in\mathcal{P}(\mathcal{X}).$

_Step 2._ Let $\mu^{\prime}\in\mathcal{P}(\mathcal{X})$ be fixed
(deterministic). Then $\boldsymbol{\nu}-\text{almost every
}\;\mu\in\mathcal{P}(\mathcal{X})$ has no common atoms with $\mu^{\prime}$,
that is,

$\int_{\mathcal{P}(\mathcal{X})}\mu(x)\mu^{\prime}(x)\,d\boldsymbol{\nu}(\mu)=0\quad\text{for
all }\;x\in\mathcal{X}\,.$

Proof. Let $x_{i}\,$, $i\in I$ be the finitely or countably many atoms of
$\mu^{\prime}$ (if any). Then the statement of Step 1 holds for each $x_{i}$.
_Conclusion._ Step 2 holds for every probability measure
$\boldsymbol{\nu}^{\prime}$ on $\mathcal{P}(\mathcal{X})$. Therefore

$\int\\!\\!\\!\\!\int_{\mathcal{P}(\mathcal{X})\times\mathcal{P}(\mathcal{X})}\mu(x)\mu^{\prime}(x)\,d\boldsymbol{\nu}(\mu)\,d\boldsymbol{\nu}^{\prime}(\mu^{\prime})=0\quad\text{for
all }\;x\in\mathcal{X}\,,$

which is the proposed result. ∎

For any vertex $\mathfrak{u}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$, consider
the subtree $\mathcal{T}_{\mathfrak{u}}$ as defined in §2. Its distribution is
the same as the one of $\mathcal{T}$. Recall that $u^{-}$ has the role of the
ancestor $\alpha$ in $\mathcal{T}_{\mathfrak{u}}\,$. For the associated
martingale according to Proposition 2.2, we write

(27) $W_{\mathfrak{u}}=\lim_{t\to\infty}N_{\mathfrak{u}}(t)e^{-\lambda t}$

for its almost surely existing and nondegenerate limit. If
$\textit{{B}}_{\mathfrak{u}^{\prime}}=z_{\mathfrak{u}^{\prime}}=z$ then
$\bigl{(}g_{z}^{-1}\textit{{B}}_{\tau}\bigr{)}_{\tau\in\mathcal{T}_{\mathfrak{u}}}$
is hyperbolic BBM starting at $o$, whose pieces along the edges come from the
same construction on our probability space as for the process starting with
the ancestor $\alpha$. We have $g_{z}=\mathbf{G}_{\mathfrak{u}^{\prime}}$ as
defined by (17). (Simple transitivity of the group $\mathfrak{A}$ is useful
here, since in this way we can avoid the need to handle uncountably many
different versions.) In particular, all the empirical distributions of
$\bigl{(}g_{z}^{-1}\textit{{B}}_{\tau}\bigr{)}_{\tau\in\mathcal{T}_{\mathfrak{u}}}$
converge almost surely, and by continuity of the action of $g_{z}$, we have
almost surely all the limits

(28)
$\mu_{\mathfrak{u},\infty}^{z}=\lim_{t\to\infty}\mu_{\mathfrak{u},t}^{z}\,,\quad\text{where
}\;z=z_{\mathfrak{u}^{\prime}}=\textit{{B}}_{\mathfrak{u}^{\prime}}\;\text{
and
}\;\mu_{\mathfrak{u},t}^{z}=\frac{1}{N_{\mathfrak{u}}(t)}\sum_{\tau\in\mathcal{T}_{\mathfrak{u}}\,:\,|\tau|=t+|u^{-}|}\textit{{B}}_{\tau}$

for for every $z$ and for each of the countably many vertices of
$\mathcal{T}$. Along with $\mu_{\mathfrak{u},t}^{z}$ and
$\mu_{\mathfrak{u},\infty}^{z}\,$, we also have

$\sigma_{\mathfrak{u},t}^{z}=N_{\mathfrak{u}}(t)e^{-\lambda
t}\,\mu_{\mathfrak{u},t}^{z}\quad\text{and}\quad\sigma_{\mathfrak{u},\infty}^{z}=W_{\mathfrak{u}}\,\mu_{\mathfrak{u},\infty}^{z}$

with $W_{\mathfrak{u}}$ as in (27).

######  Lemma 7.2.

Let $\mathfrak{u},\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$ be such
that none of the two is an ancestor (predecessor) of the other. Consider
hyperbolic BBM indexed by the time trees $\mathcal{T}_{\mathfrak{u}}$ and
$\mathcal{T}_{\mathfrak{v}}\,$, respectively, and starting at $o$. Then with
probability $1$, the limit distributions $\mu_{\mathfrak{u},\infty}^{o}$ and
$\mu_{\mathfrak{v},\infty}^{o}$ share no atoms.

###### Proof.

The two random measures are independent. Furthermore,

$\mu_{\mathfrak{u},\infty}^{o}=\frac{1}{W_{\mathfrak{u}}}\sigma_{\mathfrak{u},\infty}^{o}.$

Now we know from Theorem 6.7 that the expectation of the random measure
$\sigma_{\mathfrak{u},\infty}^{o}$ is $\nu_{o}\,$, the limit distribution on
$\mathbb{D}$ of hyperbolic Brownian motion starting at $o\,$. This is
normalised Lebesgue measure, whence it has no atoms. Since $W_{\mathfrak{u}}$
is almost surely finite and positive, also the expectation of
$\mu_{\mathfrak{u},\infty}^{o}$ has no atoms. The lemma now follows from
Proposition 7.1. ∎

Now we can deduce a first interesting property.

######  Theorem 7.3.

With probability $1$, the support of the random limit distribution
$\mu_{\infty}^{o}=\mu_{\epsilon,\infty}^{o}$ is infinite.

###### Proof.

For any measure $\mu$ on $\partial\mathbb{D}$, we shall write
$\mathsf{atoms}(\mu)$ for the set of atoms of $\mu$, and $\mathsf{supp}(\mu)$
for its support.

Let $\mathfrak{u}\in\\{\mathfrak{l},\mathfrak{r}\\}^{*}$ and consider
hyperbolic BBM according to our construction of §4 with time tree
$\mathcal{T}_{\mathfrak{u}}$ and starting point
$\textit{{B}}_{\mathfrak{u}^{\prime}}=o$. The two children of $u$ are
$u\mathfrak{l}$ and $u\mathfrak{r}$. For each of the two independent subtrees
$\mathcal{T}_{\mathfrak{u}\mathfrak{l}}$ and
$\mathcal{T}_{\mathfrak{u}\mathfrak{r}}\,$, the vertex $u$ has the role that
the ancestor $\alpha$ has in $\mathcal{T}$. Write
$z_{1}=\textit{{B}}_{\mathfrak{u}}\,$, so that
$g_{z_{1}}=\mathbf{g}_{\mathfrak{u}}\,$, the random element of $\mathfrak{A}$
as described in §4. Denote by
$\mathbf{M}_{\mathfrak{u}\mathfrak{l},t}^{z_{1}}$ and
$\mathbf{M}_{\mathfrak{u}\mathfrak{r},t}^{z_{1}}$ the occupation measures at
time $t$ (that is, distance $t$ from $u$ in $\mathcal{T}_{\mathfrak{l}}$ and
$\mathcal{T}_{\mathfrak{r}}$), respectively. Then

$\mathbf{M}_{\mathfrak{u},t}^{o}=\mathbf{M}_{\mathfrak{u}\mathfrak{l},t-\ell_{\mathfrak{u}}}^{z_{1}}+\mathbf{M}_{\mathfrak{u}\mathfrak{r},t-\ell_{\mathfrak{u}}}^{z_{1}}\,.$

Dividing by $N_{\epsilon}(t)$ and letting $t\to\infty\,$,

$\mu_{\mathfrak{u},\infty}^{o}=\mathbf{c}_{\mathfrak{u}\mathfrak{l}}\,\mu_{\mathfrak{u}\mathfrak{l},\infty}^{z_{1}}+\mathbf{c}_{\mathfrak{u}\mathfrak{r}}\,\mu_{\mathfrak{u}\mathfrak{r},\infty}^{z_{1}},\quad\text{where}\quad\mathbf{c}_{\mathfrak{u}\mathfrak{l}}=\frac{W_{\mathfrak{u}\mathfrak{l}}}{W_{\mathfrak{u}}}\,e^{-\lambda\ell_{\mathfrak{u}}}\;\text{
and
}\;\mathbf{c}_{\mathfrak{u}\mathfrak{r}}=\frac{W_{\mathfrak{u}\mathfrak{r}}}{W_{\mathfrak{u}}}\,e^{-\lambda\ell_{\mathfrak{u}}}\,,$

a non-trivial convex combination of two probability measures. We can also
rewrite this as

$\mathbf{g}_{\mathfrak{u}}^{-1}\mu_{\mathfrak{u},\infty}^{o}=\mathbf{c}_{\mathfrak{u}\mathfrak{l}}\,\mu_{\mathfrak{u}\mathfrak{l},\infty}^{o}+\mathbf{c}_{\mathfrak{u}\mathfrak{r}}\,\mu_{\mathfrak{u}\mathfrak{r},\infty}^{o}\,.$

From this identity and Lemma 7.2 we infer that

$\mathsf{atoms}(\mu_{\mathfrak{u},\infty}^{o})=\mathbf{g}_{\mathfrak{u}}\mathsf{atoms}(\mu_{\mathfrak{u}\mathfrak{l},\infty}^{o})\mathbin{\dot{\cup}}\mathbf{g}_{\mathfrak{u}}\mathsf{atoms}(\mu_{\mathfrak{u}\mathfrak{r},\infty}^{o})$

is almost surely a disjoint union.

Recursively, we get the following convex combination for each $n$:

(29)
$\mu_{\epsilon,\infty}^{o}=\sum_{\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{n}}\mathbf{c}^{(n)}_{\mathfrak{v}}\,\mathbf{G}_{\mathfrak{v}^{\prime}}\,\mu_{\mathfrak{v},\infty}^{o}$

with positive random constants $\mathbf{c}^{(n)}_{\mathfrak{v}}$ and
$\mathbf{G}_{v^{\prime}}$ as defined by (17), and

$\mathsf{atoms}(\mu_{\mathfrak{u},\infty}^{o})=\mathbin{\dot{\bigcup}}_{\mathfrak{v}\in\\{\mathfrak{l},\mathfrak{r}\\}^{n}}\mathbf{G}_{\mathfrak{v}^{\prime}}\,\mathsf{atoms}(\mu_{\mathfrak{v},\infty}^{o})\quad\text{almost
surely.}$

If $\mathsf{supp}(\mu_{\mathfrak{u},\infty}^{o})$ is finite then the support
of the measure coincides with the set of its atoms, and there must be
$\mathfrak{v}$ such that $\mu_{\mathfrak{v},\infty}^{o}$ has no atoms. But
along with $\mu_{\epsilon,\infty}^{o}\,$, by (29) all
$\mu_{\mathfrak{v},\infty}^{o}$ have finite support consisting only of atoms,
a contradiction. ∎

######  Definition 7.4.

The _limit set_ of hyperbolic BBM is the random set $\Lambda$ of accumulation
points of $(\textit{{B}}_{\tau})_{\tau\in\mathcal{T}}$ on the unit circle
$\partial\mathbb{D}$.

Lalley and Sellke [23] have proved the following significant result.

######  Theorem 7.5.

[23] In the unit circle with the arclength measure, there is the following
dichotomy.
_(i)_ For $\lambda\leq 1/8$, with probability $1$, $\Lambda$ is a Cantor set
(totally disconnected and perfect), and its Hausdorff dimension is
$(1-\sqrt{1-8\lambda}\,)/2$. Furthermore, $\Lambda$ is contained in a proper
sub-arc of $\mathbb{D}$.
_(ii)_ For $\lambda>1/8$, with probability $1$, $\Lambda=\partial\mathbb{D}$.

It is clear that

$\mathsf{supp}(\mu_{\infty}^{o})\subseteq\Lambda\,.$

At this point several interesting questions arise to which at present the
author does not know the anser.

######  Questions 7.6.

(a) Is $\mathsf{supp}(\mu_{\infty}^{o})\subset\Lambda$ properly ?
(b) What is the Hausdorff dimension of $\mu_{\infty}^{o}$ (i.e. the smallest
Hausdorff dimension of a set with full $\mu_{\infty}^{o}$-measure) ? Is it
strictly smaller than the Hausdorff dimension of $\Lambda\,$?
(c) Is $\mu_{\infty}^{o}$ purely non-atomic ? Is it absolutely continuous with
respect to Lebesgue (arc) measure in the supercritical regime $\lambda>1/8\,$?

All answers may well depend on the regime $\lambda\leq 1/8$, resp. $>1/8$.

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