Lithuanian Mathematical Journal, Vol. 53, No. 3, July, 2013, pp. 241–263

Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres Olga Aryasova a , Alessandro De Gregorio b , and Enzo Orsingher b a

Institute of Geophysics, National Academy of Sciences of Ukraine, Palladin Ave. 32, 03680 Kiev-142, Ukraine b Dipartimento di Scienze Statistiche, “Sapienza”, University of Rome, P.le Aldo Moro 5, 00185 Rome, Italy (e-mail: [email protected]; [email protected]; [email protected])

Received August 7, 2012; revised March 29, 2013 d Abstract. We consider diffusion processes (Xd (t))t0 moving inside spheres SR ⊂ Rd and reflecting orthogonally on their surfaces. We present stochastic differential equations governing the reflecting diffusions and explicitly derive their d kernels and distributions. Reflection is obtained by means of the inversion with respect to the sphere SR . The particular cases of Ornstein–Uhlenbeck process and Brownian motion are examined in detail. The hyperbolic Brownian motion on the Poincaré half-space Hd is examined in the last part of the paper, and its reflecting counterpart within hyperbolic spheres is studied. Finally, a section is devoted to reflecting hyperbolic Brownian motion in the Poincaré disc D within spheres concentric with D.

MSC: 60H10, 60J35 Keywords: Bessel process, circular inversion, hyperbolic distance, Meyer–Itô formula, Ornstein–Uhlenbeck process, Poincaré half-space

1

Introduction

d of radius R, We consider diffusion processes (Xd (t))t0 , d  2, moving inside d-dimensional spheres SR d d . A relstarting from the origin of R (that is, Xd (0) = 0d ), and reflecting orthogonally on the surface of SR evant process related to Xd (t) is the radial process Bd (t) = Xd (t), t  0, representing the distance of d . The Brownian motion is rotationally the randomly moving point Xd (t) from the center of the sphere SR invariant, and its distribution can be obtained by multiplying the distribution of Bd (t) by the uniform law on the sphere S1d . We consider the reflecting process (X+ d (t))t0 , where reflection is performed by spherical inversion. We recall the definition of circular inversion. Let C be a circle with radius r and center O. The map I that associates to the point P the point P  on the ray OP such that OP · OP  = r2 is called the inversion in the circle C . The sample paths of X+ d (t) are obtained from those of Xd (t), when Xd (t) > R, by mapping them pointwise inside the sphere according to the spherical inversion rule introduced above (see also Fig. 1 in the planar case). Therefore, in the description of the reflecting process, a major role is played by the radial + process Bd+ (t) = X+ d (t), t  0, where Bd (t) is related to the free radial process Bd (t) by means of the

c 2013 Springer Science+Business Media New York 0363-1672/13/5303-0241 

241

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O. Aryasova, A. De Gregorio, and E. Orsingher

relationship     R2 Bd+ (t) = Bd (t)1(0,R) Bd (t) + 1[R,∞) Bd (t) . Bd (t)

(1.1)

Hence, in this paper, we will focus our attention on the radial process Bd+ (t). For a d-dimensional diffusion Xd (t) = (X1 (t), . . . , Xd (t)), t  0, the (independent) components Xi (t) are governed by the stochastic differential equations     dXi (t) = bi Xi (t) dt + σi Xi (t) dWi (t), Xi (0) = 0, (1.2) where Wi (t) are independent standard Brownian motions, while bi and σi represent the drift and diffusion coefficients, respectively. The probability distribution pd (xd , t) of (Xd (t))t0 satisfies the partial differential equation  d     ∂  ∂ 1 ∂2  2 σ (xi )pd (xd , t) − bi (xi )pd (xd , t) pd (xd , t) = ∂t 2 ∂x2i i ∂xi i=1

(1.3)

with initial condition pd (xd , 0) = δ(x1 )δ(x2 ) · · · δ(xd ), where δ(·) is the Dirac delta function. We prove that Bd (t) satisfies the stochastic differential equation (2.2) below. For the process (Bd (t))t0 , we define the kernel function qd (r, t) as the fundamental solution to the partial differential equation ∂ (1.4) qd (r, t) = Aqd (r, t), ∂t where A is the infinitesimal generator of Bd (t), while the density function pd (r, t) of (Bd (t))t0 satisfies the following equation: ∂ pd (r, t) = A∗ pd (r, t), ∂t where A∗ represents the adjoint operator of A. For the classical Bessel process, that is, when Xi (t) are independent standard Brownian motions, the partial differential equation governing the distribution gd (r, t), r > 0, has the form      ∂ 1 ∂2 d−1 ∂ d−1 − gd (r, t), gd (r, t) = + ∂t 2 ∂r2 2r ∂r 2r2

and gd (r, t) = rd−1 fd (r, t),

where fd (r, t) =

1 2 e−r /(2t) 2d/2−1 Γ(d/2)td/2

is the heat kernel. It is well known that ∂fd 1 ∂ 2 fd d − 1 ∂fd + = . ∂t 2 ∂r2 2r ∂r

Our main interest here is to study the explicit distribution for the process Bd+ (t). By means of the spherical inversion we will show that p+ d (r, t) =

P{Bd+ (t) ∈ dr} dr

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Reflecting diffusion processes

has the form p+ d (r, t)

=r

d−1

 2  R2d R qd (r, t) + d+1 qd ,t , r r

where 0 < r  R. Furthermore, we have that  qd+ (r, t)

= qd (r, t) + qd

 R2 ,t , r

where qd+ (r, t) is the reflecting kernel function. We also provide a stochastic differential equation governing (Bd+ (t))t0 , by using the Meyer–Itô formula, which is a generalization of the Tanaka formula. For the Ornstein–Uhlenbeck process with components t Xi (t) = σ

e−b(t−s) dWi (s),

0

by means of the spherical inversion, we explicitly obtain the distribution of the radial component of the reflecting Ornstein–Uhlenbeck as  2  R2d R + d−1 zd (r, t) = r wd (r, t) + d+1 wd ,t , r r where wd (r, t) =

1 2 e−r /(2λ) , d/2−1 d/2 2 Γ(d/2)λ

r > 0, λ =

e2bt − 1 . 2b

By specializing the results on the reflecting Ornstein–Uhlenbeck process, we obtain the explicit law of a reflecting Brownian motion within a sphere (treated also in [9]). The d-dimensional hyperbolic Brownian motion with d  2 is a diffusion process defined on the Poincaré upper half-space Hd = {xd : xd−1 ∈ Rd−1 , xd > 0} with metric 2

ds =

d

2 i=1 dxi . x2d

The hyperbolic distance η from the origin Od = (0d−1 , 1) of xd ∈ Hd is expressed by means of the following equality:

d x2 + 1 cosh η = i=1 i . (1.5) 2xd The transition density pd (xd , t) of the hyperbolic Brownian motion satisfies the following heat-type equation:  d 1 2  ∂2 ∂ ∂ x − (d − 2)xd pd (xd , t) pd (xd , t) = ∂t 2 d ∂xd ∂x2i i=1

subject to the initial condition pd (xd , 0) = δ(x1 )δ(x2 ) · · · δ(xd−1 )δ(xd − 1). Lith. Math. J., 53(3):241–263, 2013.

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Therefore, we can define the hyperbolic Brownian motion on Hd as a diffusion process with generator given by  d 1 2  ∂2 ∂ x − (d − 2)xd . 2 d ∂xd ∂x2i i=1 On this process, the reader can consult, for instance, [4, 6, 7, 8, 11] and [12]. The hyperbolic Brownian motion starting from the origin of the Poincaré upper half-space has rotationally invariant distribution, which can be written as the product of the distribution of the radial component (ηd (t))t0 and the uniform law on the hyperbolic sphere (see [1]). For this reason, we concentrate on the reflecting radial component (ηd+ (t))t0 . Since the explicit laws of ηd (t), d = 2, 3, are known, we give the following formula:  

+ 

 S2 P ηd (t) > η = P ηd (t) > η − P ηd (t) > , d = 2, 3. η We conclude this paper by analyzing the reflecting hyperbolic Brownian motion in the Poincaré disc D = {w = reiθ : |r| < 1, θ ∈ [0, 2π]}.

2

Notation and preliminary results

For i = 1, 2, . . . , d and d  2, let bi and σi be bounded measurable functions on R satisfying the following conditions: (A1) There exists μ > 0 such that σi (x) > μ for all x ∈ R. (A2) For all x, y ∈ R, |σi (x) − σi (y)|  L|x − y|, where L is a positive constant. For i = 1, 2, . . . , d, we consider the stochastic differential equations     dXi (t) = bi Xi (t) dt + σi Xi (t) dWi (t), Xi (0) = 0,

(2.1)

where (Wi (t))t0 are standard one-dimensional Wiener processes. For each i, there exists a unique strong solution of Eq. (2.1) (see [15]). The process Xi (t), t  0, possesses a density function pi (xi , t), xi ∈ R, t  0. Let Xd (t) = (X1 (t), X2 (t), . . . , Xd (t)), t  0, be a d-dimensional diffusion process with independent coordinates Xi (t). Therefore, the probability density function of (Xd (t))t0 has the form pd (xd , t) =

d 

pi (xi , t),

xi ∈ R, i = 1, 2, . . . , d, t  0,

i=1

where xd = (x1 , x2 , . . . , xd ), and pi (xi , t) is the probability density of the process Xi (t) governed by (1.3). Let  ·  be the Euclidean distance in Rd , and Bd (t) = Xd (t), t  0, be the radial process related to Xd (t). Bd (t) is a generalization of the classical Bessel process and represents the main object of interest in this section. Now, we take up the study of the stochastic differential equation satisfied by Bd (t). Letting f (xd ) = xd , we observe that δij xi xj xi ∂f (xd ) ∂ 2 f (xd ) = = , , − ∂xi xd  ∂xi ∂xj xd  xd 3 where δij is the Kronecker delta. Therefore, f ∈ C 2 (Rd \ {0}). In other words, f is not differentiable at the origin, and we cannot apply the Itô formula to f . For this reason, we carry out the stochastic analysis of the above process by means of arguments similar to those used for the classical Bessel process (see [10, p. 159]).

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Reflecting diffusion processes

Theorem 1. The process (Bd (t))t0 satisfies the following stochastic differential equation: dBd (t) =

d   Xi (t)  σi Xi (t) dWi (t) Bd (t) i=1

      X 2 (t) 2  1 − i2 σi Xi (t) + 2Xi (t)bi Xi (t) dt. Bd (t) i=1

1  2Bd (t) d

+

Proof.

Let us define

 2 Yd (t) := Bd2 (t) = Xd (t) .

By applying the Itô formula we obtain that

Yd (t) = 2

d 

t

Xi (s) dXi (s) +

i=1 0

which is of class C 2 and such that limε→0 gε (y) = 

t

  σi2 Xi (s) ds.

i=1 0

Now, for ε > 0, we consider the following function: 3√ 3 √ 8 ε + 4 εy − gε (y) = √ y,

 ∂  gε xd 2 = ∂xi

d 

3 √ x 2 ε i xi xd  ,

√

−

1√ 2 y , 8ε ε

y < ε, y > ε,

y for all y > 0. Since

1√ xd 2 xi , 2ε ε

xd 2 < ε, xd 2 > ε,

and ⎧ 3 − 2ε1√ε xd 2 − ⎨ 2√  ε ∂2  2 gε xd  = ⎩ 1 − x2i , ∂x2i x  x 3 d

d

1 √ x2 , ε ε i

xd 2 < ε, xd 2 > ε,

the Itô rule provides the following equality: d d     gε Yd (t) = Ai (ε) + Bi (ε) + C(ε), i=1

i=1

where t  Ai (ε) := 0

      1 1 3 √ − √ Yd (s) 1(0,ε) Yd (s) + Yd (s) Xi (s) dXi (s), 1 Bd (s) [ε,∞) 2 ε 2ε ε

1 Bi (ε) := 2

t  0

Lith. Math. J., 53(3):241–263, 2013.

    1 Xi2 (s) 2  σi Xi (s) 1[ε,∞) Yd (s) ds, − 3 Bd (s) Bd (s)

(2.2)

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O. Aryasova, A. De Gregorio, and E. Orsingher

    Y (s) 1 √ 3d − (d + 2) d 1(0,ε) Yd (s) ds ε 4 ε

t C(ε) := 0

(for simplicity, we omit the dependence on time t). The Lebesgue measure of {0  s  t: Bd (s) = 0} is zero a.s., and then t  0

  1 Xi2 (s) 2  σi Xi (s) ds < ∞ − 3 Bd (s) Bd (s)

a.s.

Hence, by the dominated convergence theorem we obtain that 1 lim Bi (ε) = ε→0 2

t  0

1 = 2

t  0

    1 X 2 (s) 2  σi Xi (s) 1[0,∞) Yd (s) ds − i3 Bd (s) Bd (s)   1 Xi2 (s) 2  σi Xi (s) ds − 3 Bd (s) Bd (s)

a.s.

Furthermore, under conditions (A1) and (A2), pi (xi , t) admits the upper Gaussian bound Kt−1/2 e−xi /(2κt) , where K and κ are positive constants (see, for example, [13, Chap. 2]). Then we have that 2

3d 0  EC(ε)  √ 4 ε

t

3d P Yd (s) < ε ds  √ 4 ε

0

3d K √ 4 ε

√

t 0 √

3d =K √ 2 ε





ds s



t 0

−ρ2 /(2κs)

3d dρ = K √ 4 ε

0



t

ε

ρ dρ 0

∞

ρ dρ √ ρ/ κt

0

√

ε

ρe

ε

 P X12 (s) + X22 (s) < ε ds

e−w /2 dw w 2



e−ρ

2

/(2κs)

s

ds

0



ρ w=√ , κs

and by means of l’Hôpital’s rule we can conclude that limε→0 EC(ε) = 0. Letting Zi (t) = Bd (s) dXi (s), we get that

E Zi (t) − Ai (ε)

2

 t =E













1(0,ε) Yd (s) 0

 t =E

1(0,ε) Yd (s) 0

 t =E

1(0,ε) 0







1 − Bd (s) 1 − Bd (s)

1 Yd (s) 1 − 2







1 3 √ − √ Yd (s) 2 ε 2ε ε 1 3 √ − √ Yd (s) 2 ε 2ε ε

t 0

Xi (s)/

2

 Xi (s) dXi (s)

2

 2





Xi2 (s)σi Xi (s) ds

  2    Yd (s) Xi (s) 2 2  1 3 − Yd (s) σi Xi (s) ds ε ε Bd (s)

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Reflecting diffusion processes

 t E



1(0,ε) Yd (s)



σi2





Xi (s) ds



t H

0

 P Yd (s) < ε ds,

0

which tends to zero as ε goes to zero (where supx σi2 (x) < H since σi (x) is a bounded function).   For σi (Xi (t)) = σ and bi (Xi (t)) = bXi (t), the process (Xi (t))t0 becomes the Ornstein–Uhlenbeck process, namely, t Xi (t) = σ

e−b(t−s) dWi (s).

(2.3)

0

Clearly, in this case, bi (x) = bx is not a bounded function in R. Nevertheless, the statement in Theorem 1 still holds for the Ornstein–Uhlenbeck process because it is a Gaussian process with variance (e2bt − 1)/(2b). Then we can mimic the proof of Theorem 1, and Eq. (2.2) reduces to  d d   1 Xi (t) dBd (t) = σ (d − 1)σ 2 + 2b Xi2 (t) dt. dWi (t) + Bd (t) 2Bd (t) i=1

(2.4)

i=1

t

We observe that di=1 0 Xi (s)/Bd (s) dWi (s) := di=1 Ui (t) is a standard Brownian motion. Indeed, we have that the cross-variation process [Ui , Uj ]t (see, for instance, [10, p. 31]) becomes t [Ui , Uj ]t = 0

1 Xi (s)Xj (s) d[Wi , Wj ]s = δij 2 Bd (s)

t 0

1

Xi (s)Xj (s) ds, Bd2 (s)

2 and then di=1 [Ui ]t = t. Furthermore, Ui is a square-integrable martingale because

d we have that E(Ui (t)) < ∞. Therefore, by means of Lévy’s characterization theorem we conclude that i=1 Ui (t) is a standard Brownian motion that we denote by W (t). Therefore, we get that 

 (d − 1)σ 2 dBd (t) = σ dW (t) + + bBd (t) dt. 2Bd (t)

(2.5)

From (2.5) by setting b = 0 and σ = 1 we obtain the stochastic differential equation for the classical Bessel process.

3 3.1

Reflecting diffusion processes within an Euclidean sphere

General case

Let us consider a multidimensional diffusion process (Xd (t))t0 defined as in the previous section. When d with radius R and center at the a sample path of the process hits the surface of the d-dimensional sphere SR d origin 0d of R , the sample paths of the process are orthogonally reflected. This leads to a new process, namely, d the reflecting process (X+ d (t))t0 moving inside the d-dimensional sphere SR . We introduce the reflecting diffusion process by means of the circular inversion. If we consider a point x d , having polar coordinates equal to (r, θ), we can find another point y in the space Rd with polar inside SR coordinates given by (r , θ) (R < r ) such that rr = R2 . The point x is called the inverse point of y with d . Therefore, if we indicate by B (t) = X (t), t  0, the radial component of (X (t)) respect to SR t0 , we d d d Lith. Math. J., 53(3):241–263, 2013.

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O. Aryasova, A. De Gregorio, and E. Orsingher

Figure 1. In figure, two sample paths a and b are depicted. The trajectory wandering outd d side the circle SR (indicated by the dotted line) is reflected inside SR by circular inversion.

have that Bd+ (t) = X+ d (t) is equal to  Bd+ (t)

=

Bd (t)

if Bd (t) ∈ (0, R),

R2 Bd (t)

if Bd (t) ∈ [R, ∞),

(3.1)

or, equivalently,     R2 Bd+ (t) = Bd (t)1(0,R) Bd (t) + 1[R,∞) Bd (t) . Bd (t) The process (3.1) has two components: the first one in the interval (0, R) is related to the Bessel process without reflection, and the second component concerns the reflection of the trajectory crossing the surface of the sphere. In other words, the excursions of sample paths of (Xd (t))t0 outside the sphere are mapped inside d by using the circular inversion (see Fig. 1). When one sample path of the nonreflecting process tends to go SR far from the origin, the reflected trajectory is located near the origin. + Theorem 2. The probability density function p+ d (r, t) of Bd (t) has the form

p+ d (r, t)

=r

d−1

 2  R2d R qd (r, t) + d+1 qd ,t , r r

(3.2)

where 0 < r  R, and qd (r, t) satisfies Eq. (1.4). Furthermore, the kernel qd+ (r, t) defined as  qd+ (r, t)

= qd (r, t) + qd

 R2 ,t r

satisfies the Neumann boundary condition   ∂ + = 0. qd (r, t) ∂r r=R

(3.3)

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Reflecting diffusion processes

+ Proof. From (3.2) and (3.3) we see that the probability density p+ d (r, t) and the kernel qd (r, t) of the reflecting + d (the process Bd (t) are related to the contribution of the sample paths of the free process either lying inside SR first term) or reflected inside the sphere by circular inversion (the second term). The circular inversion principle implies that r = R2 /r and dr = R2 /r2 dr. Then, the density function of + (Bd (t))t0 is given by  2  R2d R + d−1 pd (r, t) = r qd (r, t) + d+1 qd ,t , r r

where 0 < r  R. Furthermore, we have that p+ d (r, t) integrates to 1: R

R p+ d (r, t) dr

=

0

R r

d−1

qd (r, t) dr + R

2d

0



1 rd+1

qd

 R2 , t dr r

0

∞ rd−1 qd (r, t) dr = 1.

= 0

It is easy to verify that qd+ (r, t) satisfies the Neumann boundary condition   ∂ + = 0. qd (r, t) ∂r r=R This result concludes the proof.   Remark 1. Since the kernel qd (r, t) of the diffusion is bounded by Kt−1/2 e−r /(2κt) , where K and κ are positive constants, the component related to the reflection, that is, qd (R2 /r, t), tends to zero as R → ∞. Then 2

qd+ (r, t) → rd−1 qd (r, t) d

or, equivalently, Bd+ (t) → Bd (t) as R → ∞. By means of Theorem 1 we are able to provide a stochastic differential equation governing the reflecting process (Bd+ (t))t0 . Furthermore, we recall the following result, also known as the Meyer–Itô formula giving a generalization of the Itô rule to the convex functions. Theorem 3. (See [14, p. 214].) Let f be the difference of two convex functions, let D− f be its left derivative, and let μ be the signed measure (when restricted to compacts), which is the second derivative of f in the generalized function sense. Then the following equation holds: 







t

f X(t) − f X(0) =



1 D− f X(s) dX(s) + 2

0

+∞ μ(da) Lt (a),



−∞

where (X(t))t0 is a continuous semimartingale, and Lt (a) is its local time at a. Corollary 1. (See [14, p. 216].) Let (X(t))t0 be a continuous semimartingale with local time Lt (a), a ∈ R. Let g be a bounded Borel-measurable function. Then +∞ t   Lt (a)g(a) da = g X(s) d[X, X]s −∞ Lith. Math. J., 53(3):241–263, 2013.

0

a.s.

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Theorem 3 and Corollary 1 are useful in the proof of the next theorem. Theorem 4. The process (Bd+ (t))t0 solves the following stochastic differential equation: Bd+ (t)

=

d t  

1(0,R)



i=1 0

+

d 

t

i=1 0

   Xi (s)   R2 Bd (s) − 2 1[R,∞) Bd (s) σi Xi (s) dWi (s) Bd (s) Bd (s)

1 2Bd (s)







+ 2Xi (s)bi Xi (s)

1(0,R)



 +





  R2 Bd (s) − 2 1[R,∞) Bd (s) Bd (s)



   X 2 (s)  R2 i5 σi2 Xi (s) 1[R,∞) Bd (s) Bd (s)

  Xi2 (s) 2  1− 2 σi Xi (s) Bd (s)

 ds − Lt (R),

where Lt (R) is the local time of Bd+ (t) at the point R defined by 1 Lt (R) = lim ε→0 ε

t

  1[R,R+ε) Bd (s) ds.

0

Proof. We observe that the function  g(x) =

x ∈ (0, R),

x, 2

R x

,

x ∈ [R, ∞),

does not admit the second derivative at the point R. For this reason, we cannot apply the Itô formula, so we will use the Meyer–Itô formula (Theorem 3). First of all, we prove that the second generalized derivative of g is equal to g  (x) = 2

R2 1 (x) − 2δ(x − R). x3 [R,∞)

Let φ : (0, ∞) → R be a test function that is infinitely differentiable and identically zero outside some bounded interval of (0, ∞). We have that +∞ +∞ +∞ R R2     g (x)φ(x) dx = g(x)φ (x) dx = xφ (x) dx + φ (x) dx x 0

0 +∞

=

0

R



2R2 1 (x) − 2δ(x − R) φ(x) dx. x3 [R,∞)

0

Equation (2.2) implies that (Bd (t))t0 is a semimartingale, and then we can apply the Meyer–Itô formula to Bd+ (t) = g(Bd (s)) as follows: t Bd+ (t)

= 0





1 D− g Bd (s) dBd (s) + 2

+∞ Lt (a)g  (a) da −∞

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Reflecting diffusion processes

t  1(0,R)

=



0

   R2 Bd (s) − 2 1[R,∞) Bd (s) dBd (s) Bd (s) 

+∞  2  R2 R Lt (a) 3 1[R,∞) (a) − 2 δ(a − R) da + a a −∞

t  1(0,R)

=



0



  R2 Bd (s) − 2 1[R,∞) Bd (s) Bd (s)

  d i=1

 Xi (s)  σi Xi (s) dWi (s) Bd (s)

       Xi2 (s) 2  1− 2 σ Xi (s) + 2Xi (s)bi Xi (s) ds Bd (s) i i=1

 1 + 2Bd (s) d

+

d  i=1 0

t

    R2 2 2 (s)σ X (s) 1 B (s) ds − Lt (R), X i d [R,∞) i i Bd5 (s)

where, in the last step, we have used the cross-variation process d   Xi2 (s) 2  [Bd , Bd ]t = σi Xi (s) ds 2 Bd (s) i=1 t

0

and Corollary 1, which provides the following equality: +∞ t   R2 R2 Lt (a) 3 1[R,∞) (a) da = 1[R,∞) Bd (s) d[Bd , Bd ]s 3 a Bd (s)

−∞

0

=

d 

t

i=1 0

3.2

    R2 2 2 (s)σ X (s) 1 B (s) ds. X i d [R,∞) i i Bd5 (s)

 

Reflecting Ornstein–Uhlenbeck process

Let Xi (t) be the classical Ornstein–Uhlenbeck process (2.3), and let (Bd (t))t0 be the radial Ornstein– Uhlenbeck process, which is a solution of Eq. (2.5). The transition function for Bd (t) is given by  zd (r, r0 , t, s) =

d/2

r r0 eb(t−s)

     1  2 r0 eb(t−s) rr0 eb(t−s) 2 b(t−s) r + r0 e Id/2−1 exp − λ 2λ λ

with 0 < s < t and λ = (e2bt − 1)/(2b) (see [3]). For s and r0 tending to zero, the above expression becomes zd (r, t) = rd−1 wd (r, t)

(3.4)

with wd (r, t) = Lith. Math. J., 53(3):241–263, 2013.

1 2d/2−1 Γ(d/2)λd/2

e−r

2

/(2λ)

.

(3.5)

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The reflecting Ornstein–Uhlenbeck process is a particular case of the process (X+ d (t))t0 with bi (x) = bxi and σi (x) = 1, i = 1, 2, . . . , d. By Theorem 4 and (2.5) we obtain that the distance from the origin of the reflecting Ornstein–Uhlenbeck process, which we denote by (Od+ (t))t0 , satisfies the following stochastic integral equation: t  Od+ (t)

1(0,R)

= 0



   R2 Bd (s) − 2 1[R,∞) Bd (s) dW (s) Bd (s)

t  1(0,R)

+





0

t + 0



  R2 Bd (s) − 2 1[R,∞) Bd (s) Bd (s)



 d−1 + bBd (s) ds 2Bd (s)

  R2 1[R,∞) Bd (s) ds − Lt (R). 3 Bd (s)

(3.6)

By exploiting Theorem 1 we have that the reflecting radial Ornstein–Uhlenbeck process (Od+ (t))t0 admits the probability density function given by  2  R2d R zd+ (r, t) = rd−1 wd (r, t) + d+1 wd ,t , r r while the kernel function is defined as  wd+ (r, t)

= wd (r, t) + wd

 R2 ,t . r

It is possible to obtain a partial differential equation satisfied by zd+ (r, t) and wd+ (r, t). The next theorem provides these results. Theorem 5. The kernel function wd+ (r, t) of the process (Od+ (t))t0 is a solution to the following Cauchy problem: ⎧ ∂ + ⎪ w (r, t) = Lwd+ (r, t), 0 < r  R, ⎪ ⎨ ∂t d (3.7) wd+ (r, 0) = δ(r), ⎪ ⎪ ⎩ ∂ w+ (r, t) = 0, ∂r

d

r=R

where 

   1 ∂2 d−1 ∂ + wd (r, t), + br 2 ∂r2 2r ∂r  2       2  1 r4 ∂ 2 R 3−d R R4 ∂ Lwd + wd ,t = − 2b ,t . r 2 R4 ∂r2 r r3 ∂r r Lwd (r, t) =

Furthermore, the probability density function zd+ (r, t) is a solution to the following Cauchy problem: ⎧ ∂ + ⎪ z (r, t) = L∗ zd+ (r, t), 0 < r  R, ⎪ ⎨ ∂t d zd+ (r, 0) = δ(r), ⎪ ⎪ ⎩ ∂ z + (r, t) = 0, ∂t d

t=0

(3.8)

Reflecting diffusion processes

253

where     1 ∂2 d−1 ∂ d−1 − − b zd (r, t), + br + 2 ∂r2 2r ∂r 2r2  2        2  1 r4 ∂ 2 R4 R d+1 d−1 R R4 ∂ L ∗ zd + − 2 b zd ,t = + 2b + ,t . r 2 R4 ∂r2 r r3 ∂r r2 r4 r L∗ zd (r, t) =



Proof. The kernel wd (r, t) satisfies the following partial differential equation:   ∂ d−1 ∂wd (r, t) 1 ∂ 2 wd (r, t) + wd (r, t) = Lwd (r, t) = + br 2 ∂t 2 ∂r 2r ∂r while the density function zd (r, t) is a solution to the partial differential equation    1 ∂2 ∂ ∂ d−1 ∗ zd (r, t) − zd (r, t) = L zd (r, t) = + br zd (r, t) ∂t 2 ∂r2 ∂r 2r     1 ∂2 d−1 ∂ d−1 = zd (r, t) − − b zd (r, t) + br zd (r, t) + 2 ∂r2 2r ∂r 2r2 with r > 0 (see [3]). Therefore, if we consider r = R2 /r > R (with 0 < r  R), we have that     1 ∂2 ∂ d−1 ∂     wd (r , t) + + 2br wd (r , t) . wd (r , t) = ∂t 2 ∂r2 r ∂r

(3.9)

(3.10)

It is easy to see that ∂ R2 ∂ = − 2 , ∂r r ∂r

∂2 2R2 ∂ R4 ∂ 2 = + ∂r2 r3 ∂r r4 ∂r2

r2 ∂ ∂ = − , ∂r R2 ∂r

r4 ∂2 = ∂r2 R4

or, equivalently,



 2 ∂ ∂2 + . ∂r2 r ∂r

(3.11)

Plugging the above expressions into (3.10), we obtain the following equality:  2   2    2     2  ∂ 1 r4 ∂ 2 R R R 3−d R R4 ∂ ˜ wd wd , t = Lwd ,t = ,t + − 2b 3 wd ,t , 4 2 ∂t r r 2 R ∂r r r r ∂r r which leads to Eq. (3.7). It is not hard to check that the initial and the boundary conditions appearing in problem (3.7) hold. Now, we focus our attention on the density zd . By (3.9) the same substitutions adopted in the previous calculations lead to the following equality:  2    2     2  1 r4 ∂ 2 ∂ R R d+1 R R4 ∂ zd zd ,t = ,t + + 2b 3 zd ,t 4 2 ∂t r 2 R ∂r r r r ∂r r  2   2  r d−1 R + − b zd ,t , 4 R 2 r and therefore the result (3.8) follows.   Lith. Math. J., 53(3):241–263, 2013.

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Remark 2. For d = 2, we are able to obtain the distribution function of (O2+ (t))t0 as follows:

P O2+ (t) < R

 

R

R rw2 (r, t) dr + R4

= 0

 2  1 R w , t dr 2 r3 r

0

R

∞ rw2 (r, t) dr +

=

rw2 (r, t) dr = 1 − e−R

2

/(2λ)

+ e−R

4

/(2R2 λ)

,

R2 /R

0

which, for small values of R , becomes P

O2+ (t)








R2 ∼1− 1− 2λ

 =

R2 . 2λ

Furthermore, we observe that

 2 2 4 2 4 2 P R1 < O2+ (t) < R2 = e−R1 /(2λ) − e−R2 /(2λ) + e−R /(2R2 λ) − e−R /(2R1 λ) . Remark 3. We observe that the kernel for the reflecting Ornstein–Uhlenbeck process in spherical coordinates admits the following representation: wd+ (r, θd−1 , t) =

 2 Γ(d/2) + 1 4 2  wd (r, t) = d/2 d/2 d/2 e−r /(2λ) + e−R /(2λr ) , d/2 2π 2 π λ

where θd−1 = (θ1 , . . . , θd−2 , φ) with θi ∈ [0, π], i = 1, . . . , d − 2, φ ∈ [0, 2π], and 0 < r  R, which in Cartesian coordinates becomes wd+ (xd , t) =



1 2d/2 π d/2 λd/2 xd d−1

e−xd 

2

/(2λ)

+ e−R

4

/(2λxd 2 )



with xd ∈ Rd . Analogously, for the density function, we have that zd+ (xd , t)

=



1

e

2d/2 π d/2 λd/2

−xd 2 /(2λ)

 R2d −R4 /(2λxd 2 ) + e . xd 2d

Remark 4. Let m  1, we have that the mth-order moment of Od+ (t) reads E



m Od+ (t)

R =

R r

d−1+m

wd (r, t) dr +

0

r

w d+1−m d

 R2 , t dr r

0

R =



R2d



∞ r

d−1+m

wd (r, t) dr + R

0

1 = d/2−1 2 Γ(d/2)

2m

wd (y, t) dy

R2 y= r



R

 R rd−1+m 0

y

d−1−m

1 λd/2

e−r

2

∞ /(2λ)

dr + R2m

rd−1−m R

1 λd/2

 e−r

2

/(2λ)

dr

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Reflecting diffusion processes

 R 2 /2λ 1 = d/2−1 2(d+m)/2−1 λm/2 e−w w(d+m)/2−1 dw 2 Γ(d/2) 0

R2m 2(d−m)/2−1 + λm/2

∞ e

−w

w

(d−m)/2−1





dw

r2 w= 2λ



R2 /2λ

=

     1 R2m d + m R2 d − m R2 (2λ)m/2 γ + Γ , , , Γ(d/2) 2 2λ 2 2λ (2λ)m/2

where γ(·,·) is the incomplete gamma function, and Γ(·,·) represents its complement. For d = 3, the mean value of (Od+ (t))t0 becomes   R 2 /2λ ∞ √ 2 R2 + −w −w E O3 (t) = √ 2λ e w dw + √ e dw π 2λ 



R2 /2λ

0

√ √ 2  2 2λ  2R −R2 /(2λ) −R2 /(2λ) e = √ 1−e + √ . π πλ

3.3

Brownian motion case

From the reflecting Ornstein–Uhlenbeck process we derive as particular case, the reflecting Brownian motion. It is well known that, for b = 0, the Ornstein–Uhlenbeck process reduces to the standard Brownian motion, and, consequently, Bd (t) becomes the classical d-dimensional Bessel process. Let fd (r, t) and gd (r, t) be the kernel function and the density function, respectively, of (Bd (t))t0 , that is, gd (r, t) = rd−1 fd (r, t),

where fd (r, t) =

1 2d/2−1 Γ(d/2)td/2

e−r

2

/(2t)

.

Let us denote by (Φ+ d (t))t0 the reflecting Bessel process. Theorem 2 permits us to write explicitly the kernel function fd+ (r, t) and the density law gd+ (r, t) of the reflecting Bessel process, while (3.6), for b = 0, represents the stochastic differential equation governing (Φ+ d (t))t0 . By setting b = 0 in the statements of Theorem 5 we obtain that ∂ + f (r, t) = Mfd+ (r, t), ∂t d

where  Mfd (r, t) =

 Mfd Lith. Math. J., 53(3):241–263, 2013.

1 ∂2 + 2 ∂r2



  d−1 ∂ fd (r, t), 2r ∂r

      2  1 r4 ∂ 2 R2 3−d ∂ R + fd ,t = ,t . 4 2 r 2 R ∂r r ∂r r

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O. Aryasova, A. De Gregorio, and E. Orsingher

Analogously, the density function gd+ (r, t) is a solution to the following partial differential equation: ∂ + g (r, t) = M∗ gd+ (r, t), ∂t d

where 

1 ∂2 M gd (r, t) = − 2 ∂r2 ∗

∗



M gd



   d−1 ∂ d−1 gd (r, t), + 2r ∂r 2r2

       2  1 r4 ∂ 2 R2 d+1 ∂ d−1 R + gd ,t = + ,t . 4 2 2 r 2 R ∂r r ∂r r r

Remark 5. We observe that the Laplace transform of the density function of the process (Φ+ d (t))t0 becomes L

gd+ (r, t)



∞ (s) =

e−st gd+ (r, t) dt

0

rd−1 = d/2−1 2 Γ(d/2)

∞ e

−st e

−r2 /2t

td/2

R2d dt + d/2−1 2 Γ(d/2)rd+1

0

∞ e

−st e

−R4 /2tr 2

td/2

dt

0

2−d/4+3/2 s(d−2)/4 r = Γ(d/2)

 d/2

√

Rd−2 Kd/2−1 ( 2sr) + 2 Kd/2−1 r

√

2sR2 r

 ,

s > 0,

where   ∞ 1 z ν 2 Kν (z) = e−t−z /(4t) t−ν−1 dt 2 2 0

with ν ∈ R is the Bessel function of imaginary argument. Since  K1/2 (x) =

π 2x

1/2

e−x

for d = 3, the following equality holds: √ √

   2 L g3+ (r, t) (s) = 2 re− 2sr + e− 2sR /r .

4 4.1

Reflecting hyperbolic Brownian motions

A brief introduction to the hyperbolic framework

Let us start this section by summing up basic definitions and well-known results on the hyperbolic Brownian motion (see [6, 7] and [12]). The d-dimensional hyperbolic Brownian motion with d  2 is a diffusion process defined on the Poincaré upper half-space Hd with generator given by  d 1 2  ∂2 ∂ x − (d − 2)xd . 2 d ∂xd ∂x2i i=1

257

Reflecting diffusion processes

The hyperbolic distance η from the origin Od = (0d−1 , 1) of a point xd ∈ Hd is expressed by means of the following equality:

d−1 2 x + x2d + 1 cosh η = i=1 i . 2xd The main object of interest in the analysis of the hyperbolic Brownian motion is the process (ηd (t))t0 , which represents the hyperbolic distance process of the Brownian motion in Hd . The kernel ud (η, t) of (ηd (t))t0 is a solution to the following partial differential equation:   ∂ 1 ∂ ∂ d−1 sinh η ud (η, t) ud (η, t) = ∂t ∂η 2 sinhd−1 η ∂η with initial condition ud (η, 0) = δ(η). For d = 2, we have (see [4, 12]) e−t/4 u2 (η, t) = √ √ π( 2t)3

∞

φe−φ /4t dφ, cosh φ − cosh η 2

√ η

η > 0,

and, for d = 3, we have (see [11]) e−t ηe−η /4t u3 (η, t) = √ 3/2 , sinh η 2 πt 2

η > 0.

In general, the following equality holds: ud+2 (η, t) = −

e−dt ∂ ud (η, t) 2π sinh η ∂η

(4.1)

with d = 1, 2, . . . . Formula (4.1) is known as Millson’s formula (see [5]). The probability density function of (ηd (t))t0 , the process describing the hyperbolic distance from Od , is equal to pd (η, t) = ud (η, t) sinhd−1 η,

η > 0,

which is a solution to the following partial differential equation:  2  1 d−1 ∂ ∂ ∂ d−1 − pd (η, t). pd (η, t) = + ∂t 2 sinhd−1 η ∂η 2 tanh η ∂η sinh2 η Furthermore, by considering the infinitesimal generator P , it is possible to show that (ηd (t))t0 satisfies the following stochastic differential equation: dηd (t) =

d−1 dt + dW (t), 2 tanh ηd (t)

where W (t) is a standard Wiener process. Lith. Math. J., 53(3):241–263, 2013.

ηd (0) = 0,

(4.2)

258

4.2

O. Aryasova, A. De Gregorio, and E. Orsingher

Reflecting in spheres of the Poincaré half-space

The radial component of the reflecting hyperbolic Bessel process inside the hyperbolic d-dimensional sphere with radius S and center Od is defined by  ηd (t), ηd (t) ∈ (0, S), ηd+ (t) = S2 ηd (t) ∈ [S, ∞). ηd (t) , In the previous definition, we have used the hyperbolic counterpart of the circle inversion, that is, ηη  = S 2 ,

where η and η  are the hyperbolic distances from the origin of two points belonging to the same geodesic curve, and S is the radius of a hyperbolic disc. Theorem 6. The kernel related to (ηd+ (t))t0 is equal to  2  S + ud (η, t) = ud (η, t) + ud ,t , η

0 < η  S,

while the density function is given by d−1 p+ ud (η, t) + d (η, t) = (sinh η)

 2   2 d−1  2  S S S sinh ud ,t , η η η

0 < η  S.

Proof. In order to prove the statement of this theorem, we can suitably adapt the proof of Theorem 2. It is + not hard to verify that the reflecting condition for the kernel u+ d (η, t) of ηd (t) is satisfied, that is,   ∂ + = 0, ud (η, t) ∂η η=S and that, for the probability density p+ d (η, t), we have S p+ d (η, t) dη = 1.

 

0

Remark 6. Since Eq. (4.2) holds, we can apply the Meyer–Itô formula as in the proof of Theorem 4. Therefore, we conclude that ηd+ (t) is a solution of the following stochastic differential equation: t  ηd+ (t)

1(0,S)

= 0



 t     S2 S2 ηd (s) − 2 1[S,∞) ηd (s) dηd (s) + η (s) ds − Lt (S) 1 d [S,∞) ηd (s) ηd3 (s) 

0

t  1(0,S)

=



0

t  1(0,S)

+ 0

     S2 d−1 S2 ηd (s) − 2 1[S,∞) ηd (s) ηd (s) ds 1 + 2 tanh ηd (s) ηd3 (s) [S,∞) ηd (s)







   R2 ηd (s) − 2 1[S,∞) ηd (s) dW (s) − Lt (S), ηd (s) 

Reflecting diffusion processes

259

where 1 Lt (S) = lim ε→0 ε

t

  1[S,S+ε) ηd (s) ds.

0

Theorem 7. For the reflecting hyperbolic Brownian motion, we have that ⎧ + ∂ + ⎪ ⎪ ∂t ud (η, t) = Pud (η, t), ⎪ ⎨ u+ d (η, 0) = δ(η), ⎪ ⎪  ⎪∂ + ⎩  ∂η ud (η, t) η=S = 0,

(4.3)

where   ∂ 1 ∂ d−1 Pud (η, t) = sinh η ud (η, t), ∂η 2 sinhd−1 η ∂η  Pud

     2  1 η4 S2 2 ∂ S S 2 /η 2 ∂2 ud ,t = − (d − 1) + ,t , η 2 S4 η tanh(S 2 /η) ∂η ∂η 2 η

and

⎧∂ + p (η, t) = P ∗ p+ ⎪ d (η, t), ⎪ ⎨ ∂t d p+ d (η, 0) = δ(η), ⎪ ⎪  ⎩∂ +  ∂t pd (η, t) t=0 = 0,

(4.4)

(4.5)

(4.6)

where  2  1 d−1 ∂ ∂ d−1 P pd (η, t) = − + pd (η, t), 2 sinhd−1 η ∂η 2 tanh η ∂η sinh2 η ∗

∗

P pd



(4.7)

  4    1 S2 η 2 ∂ S 2 /η 2 ∂2 ,t = + (d − 1) + η η tanh(S 2 /η) ∂η ∂η 2 2 sinhd−1 (S 2 /η) S 4   2  d−1 S + p ,t . d 2 η sinh (S 2 /η)

(4.8)

Proof. The proof of (4.3) and (4.6) follows the same steps of the proof of Theorem 5, where Eqs. (4.5) and (4.8) are obtained from Eqs. (4.4) and (4.7), respectively, by applying (3.11).   Remark 7. Since, for d = 2, we have that



∞

P η2 (t) > η = 2 η Lith. Math. J., 53(3):241–263, 2013.

e−t/4 φe−φ /4t  cosh φ − cosh η dφ √ √ 3 π( 2t) 2

260

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(see [12]), the following result holds: P

η2+ (t)

S



e−t/4 √ √ 3 sinh η π( 2t)

>η = η

S

e−t/4 √ √ 3 π( 2t)

+ η S 2 /η

=

∞ η

= S 2 /η

2

 2  2  ∞ 2 S S φe−φ /4t  sinh dφ dη η η cosh φ − cosh(S 2 /η) S 2 /η

e−t/4 √ √ 3 sinh η π( 2t)

η

∞

φe−φ /4t √ dφ dη cosh φ − cosh η

∞ η

2

√ η

+

e−t/4 φe−φ /4t √ √ 3 π( 2t) 2

η

=2 S 2 /η

sinh η dη dφ cosh φ − cosh η

φ √ η

∞

2

S 2 /η

e−t/4 φe−φ /4t √ √ 3 π( 2t)

S 2 /η

φe−φ /4t √ dφ dη cosh φ − cosh η

sinh η dη dφ cosh φ − cosh η

   2  2 e−t/4 φe−φ /4t  S cosh φ − cosh η − cosh φ − cosh dφ √ √ 3 η π( 2t)

S 2 /η

+2

e−t/4 φe−φ /4t  cosh φ − cosh η dφ √ √ 3 π( 2t) 2

η

∞ =2

e−t/4 φe−φ /4t  cosh φ − cosh η dφ √ √ 3 π( 2t) 2

η

∞ −2 S 2 /η

e−t/4 φe−φ /4t √ √ 3 π( 2t) 2





S2 cosh φ − cosh η

 dφ

 S2 . = P η2 (t) > η − P η2 (t) > η





Also, for d = 3, we can find the same relationship between the distribution of η3+ (t) and η3 (t), that is,  

+ 

 S2 P η3 (t) > η = P η3 (t) > η − P η3 (t) > . η

261

Reflecting diffusion processes

Remark 8. By means of the Millson formula (4.1) we get that

e−dt P ηd+2 (t) > η} = − 2π

∞ sinhd η

∂ ud (η, t) dη ∂η

η

e−dt e−dt = sinhd ηud (η, t) + d 2π 2π ! since lim ud (η, t) = 0

∞ cosh η sinhd−1 ηud (η, t) dη η

η→∞

=

  e−dt  sinhd ηud (η, t) + dE cosh ηd (t)1{ηd (t)>η} 2π

for d  2. Then we obtain



S 2 /η

+ P ηd+2 (t) > η =

sinhd+1 ηud+2 (η, t) dη η

  2  2  S e−dt d d S sinh ηud (η, t) − sinh ud = ,t 2π η η    + dE cosh ηd (t)1{η<ηd (t)
Orthogonal reflection in spheres of the Poincaré disc

The hyperbolic Brownian motion is also analyzed in the unit-radius Poincaré disc D = {w = reiθ : |r| < 1}, which is an alternative Euclidean model to the planar hyperbolic space. By using the conformal Cayley mapping (see [2]) z−i iz + 1 = −iz + 1 z+i it is possible to transform H2 = {z = x + iy, y > 0} into the unit-radius disc D. Since the Cartesian coordinates (x, y) and the polar coordinates (r, θ) are related by w=

x=

2r cos θ , r2 − 2r sin θ + 1

y=

1 − r2 , r2 − 2r sin θ + 1

the hyperbolic Laplacian in polar coordinates becomes  2      ∂2 (1 − r2 )2 1 ∂ ∂ 1 ∂2 ∂ y2 + = r + ∂x2 ∂y 2 22 r ∂r ∂r r2 ∂θ2 (see [2] or [12]). Moreover, between the hyperbolic distance in the Poincaré upper half-plane η (which coincides with the hyperbolic distance in the Poincaré disc) and r (the Euclidean distance), the following relation holds: 1+r η = log (4.9) 1−r Lith. Math. J., 53(3):241–263, 2013.

262

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or, equivalently, η r = tanh . 2

Let us denote by (D(t))t0 the process defined by the Euclidean distance (from the origin) of the hyperbolic Brownian motion in D. The kernel of (D(t))t0 satisfies the following equation: ∂ (1 − r2 )2 k(r, t) = ∂t 22



  ∂ 1 ∂ r k(r, t) r ∂r ∂r

with initial condition k(r, 0) = δ(r). The stochastic differential equation solved by the process (D(t))t0 reads dD(t) =

(1 − D2 (t))2 1 − D2 (t) √ dW (t). dt + 4D(t) 2

We consider the reflecting hyperbolic Brownian motion in the Poincaré disc D. The reflection is performed by means of the circular inversion with respect to a hyperbolic disc (inside D) with radius V and center (0, 0). We denote by (D+ (t))t0 the radial component of the reflecting process in D. By means of the same arguments as in the previous sections we can immediately obtain the kernel and the probability density function of (D+ (t))t0 . For instance, we have that the kernel function becomes 

 V2 k (η, t) = k(η, t) + k ,t η +

with η > 0. By means of (4.9) the kernel of (D+ (t))t0 is expressed in terms of the Euclidean radial component as follows:     1+r V2 + ,t + k k (r, t) = k log ,t 1−r log 1+r 1−r with 0 < r  tanh(V /2). The probability density function h+ (η, t) of D+ (t) has the same form of the reflecting Brownian motion in the Poincaré upper half-space, that is,  +

h (η, t) = sinh ηk(η, t) +

Since

V η

2



  2  V2 V sinh k ,t . η η

  1+r 2r sinh log = , 1−r 1 − r2

the density (4.10) in polar coordinates becomes   2r 1 + r2 2 h (r, t) = k log ,t 1 − r2 1 − r2 1 − r2     V2 V2 2 V2 + sinh ,t k ,t 1+r 1 − r2 log 1+r log 1+r log2 1−r 1−r 1−r +

with 0 < r  tanh(V /2).

(4.10)

Reflecting diffusion processes

263

References 1. V. Cammarota and E. Orsingher, Hitting spheres on hyperbolic spaces, Theory Probab. Appl., 57:560–587, 2012. 2. A. Comtet and C. Monthus, Diffusion in a one-dimensional random medium and hyperbolic Brownian motion, J. Phys. A, Math. Gen., 29:1331–1345, 1996. 3. B. Eie, Generalized Bessel process corresponding to an Ornstein–Uhlenbeck process, Scand. J. Stat., 10:247–250, 1983. 4. M.E. Gertsenshtein and V.B. Vasiliev, Waveguides with random inhomogeneities and Brownian motion in the Lobachevsky plane, Theory Probab. Appl., 3:391–398, 1959. 5. A. Grigor’yan and M. Noguchi, The heat kernel on hyperbolic space, Bull. London Math. Soc., 30:643–650, 1998. 6. J.C. Gruet, Semi-groupe du mouvement brownien hyperbolique, Stochastics Stochastics Rep., 56:53–61, 1996. 7. J.C. Gruet, A note on hyperbolic von Mises distributions, Bernoulli, 6:1007–1020, 2000. 8. N. Ikeda and H. Matsumoto, Brownian motion on the hyperbolic plane and Selberg trace formula, J. Funct. Anal., 163:63–110, 1999. 9. K. Ito and H.P. McKean, Diffusion Processes and Their Sample Paths, Springer, Berlin, 1996. 10. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1998. 11. F.I. Karpelevich, V.N. Tutubalin, and M.C. Shur, Limit theorems for the composition of distributions in the Lobachevsky plane, Theory Probab. Appl., 3:399–401, 1959. 12. L. Lao and E. Orsingher, Hyperbolic and fractional hyperbolic Brownian motion, Stochastics, 79:505–522, 2007. 13. N.I. Portenko, Generalized Diffusion Processes, Transl. Math. Monogr., Vol. 83, Amer. Math. Soc., Providence, RI, 1990. 14. P. Protter, Stochastic Integration and Differential Equations, Springer, Berlin, 2004. 15. A.K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Mat. Sb., Nov. Ser., 93(1):129–149, 1974 (in Russian). English transl.: Math. USSR, Sb., 22(1):129–149, 1974.

Lith. Math. J., 53(3):241–263, 2013.