Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

43 Hyperbolic Brownian Motion
3.2.3 Hyperbolic Brownian bridge
Let {Bz1 (t), t ≥ 0} be a Euclidean Brownian motion in Rn starting from z1 ∈ R n , n ≥ 2, then the
Euclidean Brownian bridge {W (t), t ∈ [0, 1]} between z1 and z2 ∈ R n is given by
W (t) = Bz1 (t) + t[z2 − Bz1 (t)]
for every t ∈ [0, 1]. For the Brownian bridge in the Euclidean space we have that the probability
that it stays uniformly close to the line joining z1 and z2 does non depend on the distance between
z1 and z2.
Let U(t) be the hyperbolic Brownian bridge in H n , n ≥ 2, between the origin O and z ′ ∈ H n ,
Eberle (see (13)) proved that on the hyperbolic space H n the sample paths of the hyperbolic
Brownian bridge actually concentrate, according to some exponential rate, around the geodesic
through O and z ′ when z ′ tends to infinity in finite time. Simon in (39) has found the exact
exponential rate of concentration in the case of the hyperbolic Brownian bridge on H 2 .
Let lz ′ be the unique geodesic segment joining O to z′ and define the following function on H 2
d O z ′(·) = inf{η(·, z), z ∈ lz ′}.
For every a > 0, it is possible to prove the following
lim
η(O,z ′ )→∞ −
1
η(O, z ′ ) log
P sup d
0≤t≤1
O
z ′(U(t)) > a = 2 log cosh a.
The proof of the above result is based on the comparison between the law of the hyperbolic bridge
and that of the hyperbolic Brownian motion conditioned to tend toward infinity in the direction
of z ′ .
3.2.4 Fractional hyperbolic Brownian motion
A generalization of the classical hyperbolic Brownian motion in H 2 is given in Lao and Orsingher
(26) where it is considered the time-fractional equation
with α ∈ (0, 1], subject to the initial condition
∂α ∂tα uα = 1
sinh η
sinh η
∂
uα, (3.32)
∂η
uα(η, 0) = δ(η).
We assume that the fractional derivative appearing in (3.32) is understand in the sense of Dzerbashyan-
Caputo that is, for f ∈ C m ,
dαf =
dtα t
1
Γ(m − α) 0
f (m) (s)
ds, m − 1 < α ≤ m.
(t − s) α+1−m