Bessel bridge representation for heat kernel in hyperbolic space

0.38 MB·

by  Xue Cheng

Bessel bridge representation
for heat kernel in hyperbolic space
Xue Cheng
Department of Mathematical Finance
School of Mathematical Sciences
7 Peking University, Beijing, China
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0 email: [email protected]
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n Tai-Ho Wang
a
J Department of Mathematics, Baruch College, CUNY

5
1 Bernard Baruch Way, New York, NY10010

]
R e-mail: [email protected]
P
.
h
t Current Version: January 6, 2017
a
m
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1 Abstract
v
4
9 This article shows a Bessel bridge representation for the transition density of Brow-
1
nian motion on the Poincar´e space. This transition density is also referred to as
1
0 the heat kernel on the hyperbolic space in differential geometry literature. The rep-
.
1
resentation recovers the well-known closed form expression for the heat kernel on
0
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hyperbolic space in dimension three. However, the newly derived bridge represen-
1
: tation is different from the McKean kernel in dimension two and from the Gruet’s
v
i
X formula in higher dimensions. The methodology is also applicable to the derivation
r of an analogous Bessel bridge representations for heat kernel on a Cartan-Hadamard
a
radially symmetric space and for the transition density of hyperbolic Bessel process.
Keywords: Heat kernel in hyperbolic space, Heat kernel in radially symmetric space,
Asymptotic expansion, Brownian motion in hyperbolic space, Bessel bridge represen-
tation, Hyperbolic Bessel process
BESSEL BRIDGE REPRESENTATION FOR HEAT KERNEL IN
HYPERBOLIC SPACE
XUE CHENG AND TAI-HO WANG
Abstract. This article shows a Bessel bridge representation for the transition
density of Brownian motion on the Poincar´e space. This transition density is also
referred to as the heat kernel on the hyperbolic space in differential geometry lit-
erature. The representation recovers the well-known closed form expression for the
heat kernel on hyperbolic space in dimension three. However, the newly derived
bridge representation is different from the McKean kernel in dimension two and
from the Gruet’s formula in higher dimensions. The methodology is also applica-
ble to the derivation of an analogous Bessel bridge representations for heat kernel
on a Cartan-Hadamard radially symmetric space and for the transition density of
hyperbolic Bessel process.
1. Introduction
Heat kernel, also known as the fundamental solution for heat operator, plays a
crucial role in various branches of mathematics including analysis, differential geom-
etry, and probability theory. On Euclidean spaces, heat kernels have closed form
expression given by the Gaussian kernels, which also serve as the transition density
of Euclidean Brownian motions. Deriving, to some extent, analytical expression of
heat kernel on general curved space is more involved if not completely impossible.
Symmetry of the underlying space plays an important role. Hyperbolic space is one
of the symmetry spaces with constant negative curvature that has an expression for
heat kernel in analytic form. As we shall see throughout the article, due to symmetry,
expressions for heat kernels on hyperbolic space depend solely on geodesic distance.
Derivations of heat kernel on hyperbolic space in closed or quasi closed forms have
been done by various authors. We list notable a few as follows. McKean in [11]
presented a quasi closed form expression (up to an integral), nowadays known as the
McKean kernel, for heat kernel on two dimensional hyperbolic space, see (2.1) below.
A detailed derivation of the McKean kernel using Fourier transform, isometries, and
eigenvalues and eigenfuctions of the Laplace-Beltrami operator can be found in [1]
(Section 2 in Chapter X). The closed form expression for heat kernel on three di-
mensional hyperbolic space, see (2.2) below, and the Millson’s recursion formula for
1
2 X. CHENG AND T.-H. WANG
higher dimensional hyperbolic heat kernel were reported in [2]. A different proof of
Millson’s recusion formula based on the relationship between the heat kernel and the
wave kernel and the explicit formula for wave kernel on symmetry space was given
in [3]. The following expression obtained in [4] for the n-dimensional hyperbolic heat
kernel is known as the Gruet’s formula
e−(n−1)2t/8 (cid:18)n+1(cid:19)(cid:90) ∞ e(π2−b2)/2tsinh(b)sin(πb/t)
p (t,z,w) = √ Γ db,
Hn
π(2π)n/2 t 2 [cosh(b)+cosh(r)](n+1)/2
0
where r = d(z,w) is the geodesic distance between z,w ∈ Hn. We refer the interested
reader to [10] for a derivation of the Gruet’s formula and its relationship to the pricing
ofAsianoptions. Aprobabilisticapproach,whichisdifferentfromtheoneemployedin
thecurrentarticle,ofderivingtheheatkernelontwodimensionalhyperbolicspacecan
also be found in [7]. As closed form expression is concerned, [9] obtained expressions
for heat kernels on symmetric spaces of rank 1. Finally, it is worth mentioning that
a nice application of the McKean kernel in quantitative finance can be found in [5].
In this article, we prove yet another representation for heat kernel on hyperbolic
space: the Bessel bridge representation in Theorem 1. By working under geodesic
polar coordinates, Brownian motion in hyperbolic space is decomposed into a one
dimensional process in the radial part and a process on the unit sphere of codimen-
sion one. The radial part, also known as the hyperbolic Bessel process, is indeed a
Brownian motion with drift. Due to symmetry of hyperbolic space, the drift in the
radial part depends only on the geodesic distance. Girsanov’s theorem allows us to
define an equivalent probability measure on the underlying probability space through
a Radon-Nikodym derivative so that, in the new probability measure, the radial pro-
cess becomes a Bessel process of order n. We then substitute the stochastic integral
that results from the Radon-Nikodym derivative with a Riemann integral by applying
Ito’s formula. The bridge representation of hyperbolic heat kernel is thus obtained
by conditioning on the terminal point of the radial process in the new probability
measure. The whole procedure is implemented in the proof of Theorem 1. Similar
representation for the transition density of hyperbolic Bessel process is shown in The-
orem 2. With minor modifications, the same procedure is also applicable to the case
of general Cartan-Hadamard radially symmetric spaces and the result is summarized
in Theorem 3. We remark that, in the one dimensional case, the idea of bridge rep-
resentation for transition density of a diffusion first appeared, to our knowledge, in
[13], see also [14] for further discussions.
BESSEL BRIDGE REPRESENTATION HYPERBOLIC HEAT KERNEL 3
2. Heat kernel on hyperbolic space
Throughout the text, stochastic processes and random variables are assumed de-
fined on the complete filtered probability space (Ω,F,P,{F } ) satisfying the
t t∈[0,∞)
usual conditions. We shall denote the n-dimensional hyperbolic space by Hn and the
associated heat kernel on Hn between z,w ∈ Hn at time t by p (t,z,w).
Hn
2.1. The hyperbolic space. Conventionally, hyperbolic spaces are parametrized
by two isometrically equivalently models: the half-space model and the ball model.
The underlying space in the half-space model is the half plane Rn = {(x ,··· ,x ) :
+ 1 n
x > 0}, while the underlying space in the ball model is the open ball B = {y =
n n
(y ,··· ,y ) : (cid:107)y(cid:107) < 1}. The transformation between the two models can be found
1 n
for instance in [1] (p.264). In particular, for n = 2, the transformation between the
two models is given by the M¨obius transform, T : C → B ,
+ 2
z −i
w = T(z) = .
z +i
In the half-space model, the metric ds2 and the Laplace-Beltrami are given respec-
tively by
dx2 +···+dx2
ds2 = 1 n,
x2
n
(cid:0) (cid:1)
∆ = x2 ∂2 +···+∂2 +(2−n)x ∂ ;
M n x1 xn n n
whereas in the ball model, the metric is given, in polar coordinates (ρ,θ), θ ∈ Sn−1,
by
4
(cid:0) (cid:1)
ds2 = dρ2 +ρ2dθ2 ,
(1−ρ2)2
(1−ρ2)2 (cid:18) 1 1 (cid:19)
∆ = ∂2 + ∂ + ∆ ,
M 4 ρ ρ ρ ρ2 Sn−1
where dθ2 is the Riemann metric and ∆ the Laplace-Beltrami operator on the
Sn−1
(cid:0) (cid:1)
standard unit sphere Sn−1. Moreover, if we make the transformation ρ = tanh r ,
2
then (r,θ) becomes the geodesic polar coordinates for Hn. We shall be working
primarily in the geodesic polar coordinates (r,θ) ∈ [0,∞) × Sn−1 under which the
Riemann metric and the Laplace-Beltrami operator of Hn are given respectively as
ds2 = dr2 +sinh2rdθ,
1
∆ = ∂2 +(n−1)cothr∂ + ∆ .
Hn r r sinh2r Sn−1
4 X. CHENG AND T.-H. WANG
We remark that the geodesic polar coordinates on Hn is a global diffeomorphism,
henceforth defines a global coordinates, since hyperbolic space is Cartan-Hadamard.
2.2. The heat kernel. Generally speaking, heat kernel on a differetiable manifold
M is a fundamental solution to the (probabilist’s) heat operator ∂ − 1∆ , where
t 2 M
∆ is the Laplace-Beltrami operator on M. The minimal heat kernel also serves
M
as the transition density of Brownian motion on M. We refer the reader to [6] for
expositions of Brownian motions on manifolds and their relationships to heat kernel.
For reader’s reference, we reproduce the heat kernel on the two and three dimensional
hyperbolic spaces in the following.
√
2e−t/8 (cid:90) ∞ ξe−ξ2
2t
p (z,w,t) = dξ (2.1)
H2 (cid:112)
(2πt)3/2 coshξ −coshd(z,w)
d(z,w)
and
e−r2 r
p (z,w,t) = 2t e−t , (2.2)
H3 2
(2πt)3/2 sinh(r)
where r = d(z,w) is the geodesic distance between z and w. Note that the heat
kernels given in (2.1) and (2.2) are densities with respect to the volume form.
In the following, we apply Girsanov’s theorem to derive an expression for the heat
kerneloverHn, forn ≥ 2, inwhichtheclosedformexpression(2.2)forH3 isrecovered.
NotethatsincetheLaplace-BeltramioperatoronHn isrotationallyinvariant,theheat
kernel, or equivalently the transition density for the Brownian motion on hyperbolic
space, is also rotationally invariant, hence a radial function.
Consider the processes (R ,Θ ) governed by the SDEs
t t
n−1
dR = dW + coth(R )dt, (2.3)
t t t
2
1
dΘ = dZ , (2.4)
t t
sinh(R )
t
whereW isastandardonedimensionalBrownianmotionandZ isaBrownianmotion
t t
on the standard sphere Sn−1, independent of W . The infinitesimal generator of the
t
process (R ,Θ ) is 1∆ . Thus, it represents a Brownian motion on Hn in geodesic
t t 2 Hn
polar coordinates. We set the initial condition Θ to be a random variable uniformly
0
distributed on Sn−1 so that the distribution of Θ remains uniformly distributed on
t
Sn−1 for all t. The main result of the article is given in the following theorem.
Theorem 1. (Bessel bridge representation)
Let z,w ∈ Hn. The heat kernel p (T,z,w) on the hyperbolic space Hn has the
Hn
BESSEL BRIDGE REPRESENTATION HYPERBOLIC HEAT KERNEL 5
following representation
pHn(T,z,w) = e−(n−81)2T (cid:16)sinrhr(cid:17)n−21 (2eπ−T2rT2)nE˜r(cid:34)e−(n−1)8(n−3)(cid:82)0T(cid:20)sinh21(Rt)−R1t2(cid:21)dt(cid:35), (2.5)
2
where r = r(z,w) is the geodesic distance between z and w. E˜ [·] denotes the con-
r
ditional expectation E˜[·|R = r], where R is a Bessel process of order n in the
T t
P˜-measure.
Proof. Let (R ,Θ ) be the process satisfying (2.3):(2.4) with initial conditions R = 0
t t 0
and Θ being uniformly distributed on Sn−1. We start with calculating the expecta-
0
tion of an arbitrary bounded measurable radial function f as
(cid:90) (cid:90) ∞
E[f(R )] = f(r)p(T,r)sinhn−1rdrdω,
T
Sn−1 0
where dω is the volum form on Sn−1 and p(T,r) = p (T,z,w), r = r(z,w) denotes
Hn
the geodesic distance between z and w. For the radial process R , define the new
t
measure P˜ by the Radon-Nikodym derivative
dP˜
dP = e(cid:82)0Th(Rt)dWt−12(cid:82)0Th2(Rt)dt, (2.6)
(cid:2) (cid:3)
where h(r) = n−1 1 −coth(r) . Note that h is a bounded function, in fact, |h(r)| ≤
2 r
n−1 for all r ≥ 0. Thus (2.6) is a well-defined change of probability measure. There-
2
fore, Girsanov’s theorem implies that, under the measure P˜, W is a Browian motion
t
with drift h. Moreover, in the P˜-measure, the SDE for the radial process R becomes
t
n−1 dt
˜
dR = dW + ,
t t
2 R
t
which is a Bessel process of order n. Therefore, we have
(cid:20) dP(cid:21) (cid:104) (cid:105)
E[f(RT)] = E˜ f(RT)dP˜ = E˜ f(RT)e−(cid:82)0Th(Rt)dWt+12(cid:82)0Th2(Rt)dt
(cid:104) (cid:105)
= E˜ f(RT)e−(cid:82)0Th(Rt)dW˜t−12(cid:82)0Th2(Rt)dt . (2.7)
We substitute the stochastic integral in (2.7) with a Riemann integral by applying
Ito’s formula as follows. Let H be an antiderivative of h, i.e., H(cid:48) = h. Apparently,
(cid:0) (cid:1)
H(r) = n−1 ln r . Then by applying Ito’s formula we have
2 sinhr
(cid:90) T (cid:90) T (cid:20)h(cid:48)(R ) n−1h(R )(cid:21)
˜ t t
h(R )dW = H(R )−H(0)− + dt.
t t T
2 2 R
0 0 t
It follows that the exponent of the exponential term in (2.7) becomes
(cid:90) T 1 (cid:90) T
− h(R )dW˜ − h2(R )dt
t t t
2
0 0
6 X. CHENG AND T.-H. WANG
(cid:90) T (cid:20)h(cid:48)(R ) n−1 h(R ) h2(R )(cid:21)
t t t
= −H(R )+H(0)+ + − dt
T
2 2 R 2
0 t
(cid:20)sinh(R )(cid:21)n−21 (n−1)2 (n−1)(n−3) (cid:90) T (cid:20) 1 1 (cid:21)
T
= ln − T − − dt.
R 8 8 sinh2(R ) R2
T 0 t t
Hence, we have
(cid:34) (cid:26) (cid:27)n−1 (cid:20) (cid:21) (cid:35)
E[f(RT)] = e−(n−81)2TE˜ f(RT) sinh(RT) 2 e−(n−1)8(n−3)(cid:82)0T sinh21(Rt)−R1t2 dt .
R
T
In particular, when n = 3, the last expression has a much simpler form:
(cid:20) (cid:21)
sinh(R )
E[f(R )] = e−TE˜ f(R ) T .
T 2 T
R
T
Finally, since R in P˜ measure is a Bessel process of order n, we end up with
t
(cid:90) (cid:90) ∞
f(r)p(T,r)sinhn−1rdrdω = E[f(R )]
T
Sn−1 0
(cid:34) (cid:26) (cid:27)n−1 (cid:20) (cid:21) (cid:35)
= e−(n−81)2TE˜ f(RT) sinh(RT) 2 e−(n−1)8(n−3)(cid:82)0T sinh21(Rt)−R1t2 dt
R
T
= e−(n−1)2T Γ(cid:0)n2(cid:1) (cid:90) (cid:90) ∞f(r)(cid:18)sinhr(cid:19)n−21 ×
8
2πn r
2 Sn−1 0
E˜r(cid:34)e−(n−1)8(n−3)(cid:82)0T(cid:20)sinh21(Rt)−R1t2(cid:21)dt(cid:35) (22rTn)−n1Γe−(cid:0)2rnT2(cid:1)drdω
2
2
= e−(n−1)2T (cid:90) (cid:90) ∞f(r)(cid:16) r (cid:17)n−21 e−2rT2 ×
8
sinhr (2πT)n
Sn−1 0 2
(cid:34) (cid:20) (cid:21) (cid:35)
E˜r e−(n−1)8(n−3)(cid:82)0T sinh21(Rt)−R1t2 dt sinhn−1rdrdω,
where in passing to the penultimate equality we used the probability density p of
B
the Bessel process R given by
t
2rn−1e−r2
2t
p (t,r) =
B (2t)nΓ(cid:0)n(cid:1)
2
2
which satisfies the Fokker-Planck equation
1 n−1 (cid:16)u(cid:17)
∂ u = ∂2u− ∂
t 2 r 2 r r
with initial condition u(r,0) = δ(r), the Dirac delta function centered at 0. Also
n
note that the normalizing constant 2π2 comes from the volume of Sn−1. Thus,
Γ(n)
2
we obtain the bridge representation (2.5) for the transition density of hyperbolic
Brownian motion. (cid:3)
BESSEL BRIDGE REPRESENTATION HYPERBOLIC HEAT KERNEL 7
Remark 1.
Note that when n = 3 the representation (2.5) reduces to
r e−r2
p (T,z,w) = e−T 2T
H3 2 sinhr(2πT)3
2
which coincides with the closed form expression (2.2). However, for n = 2, (2.5) reads
pH2(T,z,w) = e−T8(cid:114) r e−2rT2 E˜r(cid:34)e18(cid:82)0T(cid:26)sinh21(Rt)−R1t2(cid:27)dt(cid:35).
sinhr 2πT
Notice that
(1) This expression is different from the McKean kernel (2.1) or the Gruet’s for-
mula in the sense that a) the power of 2πT is in the correct dimension (n = 1)
2
and b) the “Gaussian” term e−r2 is factored outfront naturally.
2T
(2) The integrand in the exponential term, i.e., the function φ(x) := 1 − 1 is
sinh2x x2
increasing in [0,∞) with lim φ(x) = −1 and lim φ(x) = 0. Therefore,
x→0+ 3 x→∞
φ is bounded above by 0 and below by −1.
3
As applications of the bridge representation (2.5), a series expansion and an asymp-
toticexpansioninsmalltimeforthehyperbolicheatkernelarealmoststraightforward.
For notational simplicity, hereafter in this subsection we shall denote by
(cid:20) (cid:21)
(n−1)(n−3) 1 1
g(r) = − − .
8 sinh2(r) r2
Note that g is strickly decreasing and |g(r)| ≤ (n−1)(n−3) for all r > 0.
24
Corollary 1. The hyperbolic heat kernel p has the following series expansion
Hn
p (T,z,w) (2.8)
Hn
= e−(n−81)2T (cid:16)sinrhr(cid:17)n−21 (2eπ−T2rT2)ne(cid:82)0Tg(rt)dt (cid:88)∞ Tk!kE˜r(cid:34)(cid:18)(cid:90) 1g(RTs)−g(rTs)ds(cid:19)k(cid:35),
2 0
k=0
where r , for t ∈ [0,T], is defined by
t
(cid:16) (cid:17)
r = g−1 E˜ [g(R )] . (2.9)
t r t
In other words, g(r ) is an unbiased estimator for g(R ) in the Bessel bridge measure.
t t
Proof. It sufficies to deal with the conditional expectation term in (2.5).
(cid:34) (cid:26) (cid:27) (cid:35)
E˜r e−(n−1)8(n−3)(cid:82)0T sinh21(Rt)−R1t2 dt
8 X. CHENG AND T.-H. WANG
(cid:104) (cid:105)
= e(cid:82)0Tg(rt)dtE˜r e(cid:82)0T{g(Rt)−g(rt)}dt
(cid:34) (cid:35)
= e(cid:82)0Tg(rt)dtE˜r (cid:88)∞ 1 (cid:18)(cid:90) T {g(Rt)−g(rt)}dt(cid:19)k
k!
0
k=0
(cid:34) (cid:35)
= e(cid:82)0Tg(rt)dt (cid:88)∞ 1 E˜r (cid:18)(cid:90) T {g(Rt)−g(rt)}dt(cid:19)k
k!
0
k=0
(cid:82)T
by dominating convergence theorem since the random variable {g(R )−g(r )}dt
0 t t
is bounded. In fact,
(cid:12)(cid:12)(cid:90) T (cid:12)(cid:12) (n−1)(n−3)
(cid:12) {g(R )−g(r )}dt(cid:12) ≤ T almost surely.
t t
(cid:12) (cid:12) 12
0
Finally, by making the change of variable t = Ts we obtain the series expansion
(2.8). (cid:3)
We remark that in fact we have the freedom of selecting the deterministic path r
t
in the series expansion (2.8). We choose the path as such since it serves as a first
order “unbiased estimator” in the small time asymptotic expansion in the corollary
that follows.
Corollary 2. As T → 0+, the hyperbolic heat kernel p has the following small time
Hn
asymptotic expansion up to second order
p (T,z,w) (2.10)
Hn
= e−(n−81)2T (cid:16) r (cid:17)n−21 e−2rT2 e(cid:82)0Tg(rt)dt (cid:8)1+O(T2)(cid:9),
sinhr (2πT)n
2
where r is given in (2.9).
t
Proof. Consider the infinite series on the right hand side of (2.8),
(cid:34) (cid:35)
(cid:88)∞ Tk (cid:18)(cid:90) 1 (cid:19)k
E˜ g(R )−g(r )ds
r Ts Ts
k!
0
k=0
(cid:20)(cid:90) 1 (cid:21)
= 1+T E˜ {g(R )−g(r )}du +O(T2)
r Tu Tu
0
= 1+O(T2)
by the definition of the path r . (cid:3)
t
Note that if we choose a different path r than (2.9), then the asymptotic expansion
t
in (2.10) is of order T only.
BESSEL BRIDGE REPRESENTATION HYPERBOLIC HEAT KERNEL 9
Lastly, by na¨ıvely choosing r as the straight line connecting 0 and r as well as
t
the unbiased estimator (2.9), in Figure 1 we illustrate numerically the accuracy of
the asymptotic expansion (2.10), compared with the Gruet’s formula. As shown in
the plots, the unbiased estimator does a pretty decent job; whereas the straight line
approximation is off for high dimensions.
dimension = 2 , t = 1 dimension = 5 , t = 1
5
1
0.10 0.00
y y
sit sit
n n
e 6 e
d 0 d
0. 5
0
0
0
2 0.
0
0.
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
geodesic distance geodesic distance
dimension = 10 , t = 1 dimension = 15 , t = 1
4
1
y 4e−08 y 2.0e−
densit 2e−08 densit 1.0e−14
0
0 0
0 +
+ e
e 0
0 0.0 0.5 1.0 1.5 2.0 0. 0.0 0.5 1.0 1.5 2.0
geodesic distance geodesic distance
Figure 1. Plots of hyperbolic heat kernel at time 1 in various dimen-
sions. Approximation of r in (2.10) by straight line in green, by the
t
unbiasd estimator (2.9) in blue. Gruet’s formula in red.
2.3. Transition density of hyperbolic Bessel process. The radial part R of
t
hyerbolic Brownian motion satisfying (2.3) is also referred to as the hyperbolic Bessel
process. HyperbolicBesselprocessesandthecalculationsoftheirrelatedmomentsare
extensively explored in recent papers [8] and [12]. By the same token as in Theorem 1,
we may as well derive a Bessel bridge representation for the hyperbolic Bessel process.
The advantage of the bridge represetation is that the expression is consistent across
dimensions. However, formulas given in [12] (see Theorem 3.3), obtained by applying
the Millson’s recursion formula, become more and more intractable as dimension goes
higher.