Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

31 Hyperbolic Brownian Motion
motion on H n may be obtained by solving the stochastic differential equations
⎧
⎨dXi(s)
= Y (s)dBi(s), i = 1, ..., n − 1,
⎩dY
(s) = Y (s)dBn(s) − n−2Y
(s)ds.
By Itô’s formula we have that the solution {Z(t) = (X(t), Y (t)), t ≥ 0} satisfying Z(0) = z = (x, y)
may be represented as follows:
2
⎧
⎨Xi(t)
= xi + y
⎩
t
0 exp{Bµ (s)} dBi(s), i = 1, ..., n − 1,
Y (t) = y exp{B µ (t)},
where µ = (n − 1)/2 and B µ (t) = Bn(t) − µt has constant drift −µ. Since µ is assumed to be
positive, Y (t) is the usual geometric Brownian motion with negative drift and then {Z(t), t ≥ 0}
converges almost surely to the boundary of H n as t → ∞.
3.2 Hyperbolic Brownian motion in H 2
3.2.1 Transition function
The 2-dimensional hyperbolic Brownian motion is a diffusion process on the upper half-plane
H 2 = {z = (x, y) : x ∈ R, y > 0}
associated with the Laplace operator ∆2 given by
(3.7)
∆2 = y 2
2 ∂ ∂2
+
∂x2 ∂y2
. (3.8)
The position of points in H 2 can be given either in terms of Cartesian coordinates (x, y) or by
means of the hyperbolic coordinates (η, α). The coordinate η represents the hyperbolic distance
of (x, y) from the origin O = (0, 1) of H 2 . The coordinate α is the slope of the tangent in O to
the geodesic through O and (x, y) with Euclidean center (x0, 0) and Euclidean radius r (see Figure
5.3).
The formulas which relates the hyperbolic coordinate to the cartesian coordinates are
x =
y =
sinh η cos α
cosh η−sinh η sin α ,
1
cosh η−sinh η sin α ,
with η > 0 and α ∈ [− π π
2 , 2 ]. See, for example, Rogers and Williams (37), page 213 or Terras (41)
formula (3.7). For a proof of the relations in (3.9) see Proposition 3.A.3.
In Proposition 3.A.4 it is showed that the Laplace operator (3.8) in hyperbolic coordinates takes
the form
1 ∂
sinh η
sinh η ∂η
∂
1
+
∂η sinh 2 ∂
η
2
.
∂α2 (3.10)
Since, for a fixed t, the transition function p2(z, z ′ , t) is a function of the hyperbolic distance η,
we can take into account only the radial part of the Laplace operator (3.10).
(3.9)