Exponential functionals of brownian motion and class-one Whittaker functions

www.imstat.org/aihp 2011, Vol. 47, No. 4, 1096–1120

DOI:10.1214/10-AIHP401

© Association des Publications de l’Institut Henri Poincaré, 2011

Exponential functionals of Brownian motion and class-one Whittaker functions

Fabrice Baudoin

and Neil O’Connell

aDepartment of Mathematics, Purdue University, West Lafayette, IN 47906, USA. E-mail:fbaudoin@math.purdue.edu bMathematics Institute, University of Warwick, Coventry CV4 7AL, UK. E-mail:n.m.o-connell@warwick.ac.uk

Received 24 April 2009; revised 1 November 2010; accepted 10 November 2010

Abstract. We consider exponential functionals of a Brownian motion with drift inRⁿ, defined via a collection of linear functionals.

We give a characterisation of the Laplace transform of their joint law as the unique bounded solution, up to a constant factor, to a Schrödinger-type partial differential equation. We derive a similar equation for the probability density. We then characterise all diffusions which can be interpreted as having the law of the Brownian motion with drift conditioned on the law of its exponential functionals. In the case where the family of linear functionals is a set of simple roots, the Laplace transform of the joint law of the corresponding exponential functionals can be expressed in terms of a (class-one) Whittaker function associated with the corresponding root system. In this setting, we establish some basic properties of the corresponding diffusion processes.

Résumé. Nous étudions certaines fonctionelles d’un mouvement Brownien avec dérive dansRⁿqui sont définies par une collection de fonctionnelles linéaires. Nous donnons une caractérisation de la transformée de Laplace de leur loi jointe comme l’unique solu- tion bornée, à une constante près d’une équation aux dérivées partielles de type Schrödinger. Nous déduisons une équation similaire pour la densité. Nous caractérisons ensuite toutes les diffusions qui peuvent être interprétées comme ayant la loi d’un mouvement Brownien avec dérive conditionné par la loi de ses fonctionelles exponentielles. Dans le cas où la famille des fonctionelles est un ensemble de racines simples, la transformée de Laplace de la densité jointe des fonctionnelles exponentielles correspondantes peut être exprimée en termes d’une fonction de Whittaker de classe 1 associée au système. Dans ce cadre, nous établissons quelques propriétés du processus de diffusion correspondant.

MSC:60J65; 60J55; 37K10; 22E27

Keywords:Conditioned Brownian motion; Quantum Toda lattice

1. Introduction

Let(Xt, t≥0)be a standard one-dimensional Brownian motion with driftμ. In the paper [19], Matsumoto and Yor consider the process

log

t 0

e^2Xs−Xtds, t >0

,

and prove that it is a diffusion process with infinitesimal generator given by 1

2 d² dx²+

d dx logKμ

e^−x d

dx, (1)

1Supported in part by Science Foundation Ireland Grant number SFI 04-RP1-I512.

whereK_μis the Macdonald function. As explained in [19], this theorem can be regarded, by Brownian scaling and Laplace’s method, as a generalization of Pitman’s ‘2M−X’ theorem [25,26] which states that, ifM_t=max0≤s≤tX_s, then 2M−Xis a diffusion process with infinitesimal generator given by

1 2

d²

dx²+μcoth(μx)d

dx. (2)

Note thatμcoth(μx)=(−μ)coth(−μx)andK_μ=K_−μ. Supposeμ >0. Then the diffusion with generator (2) can be interpreted as a Brownian motion with driftμconditioned to stay positive. Similarly, the diffusion with gen- erator (1) can be interpreted as the Brownian motionX conditioned on its exponential functionalA=_∞

0 e^−2Xsds having a certain distribution (a Generalised Inverse Gaussian law) in a sense which can be made precise [2,3]. The relevance of these interpretations in the present context is as follows.

SetJ = −mint≥0X_t and let J˜be an independent copy of J which is also independent ofX. Then the process X˜ =2 max{M− ˜J ,0} −X has the same law asXand, moreover,J˜= −mint≥0X˜_t. This is well known and can be seen for example as a consequence of the classical output theorem for theM/M/1 queue [23]. From this we can see that the process 2M−Xhas the same law as that ofXconditioned (in an appropriate sense) on the event thatJ=0;

in other words, 2M−Xis a diffusion with infinitesimal generator given by (2), started from zero. This basic idea can be used to obtain a multi-dimensional version of Pitman’s 2M−Xtheorem [4,5,24], which gives a representation of a Brownian motion conditioned to stay in a Weyl chamber inRⁿas a certain functional (which generalises 2M−X) of a Brownian motion inRⁿ.

Similarly [19], ifA˜is an independent copy ofA, which is also independent ofX, then the process X˜_t=log

1+ ˜A⁻¹ _t

0

e^2(Xs−Xt)ds

+X_t, t≥0,

has the same law asXand, moreover,A˜=_∞

0 e^−2X˜sds. This time, we conclude that, for eachε >0, the process log

ε+

_t

0

e^2(Xs−Xt)ds

+Xt, t≥0,

the same law as that of logε+X conditioned on the event A=ε. Carefully lettingε→0 yields the theorem of Matsumoto and Yor [2]. The probabilistic proofs of the multi-dimensional versions of Pitman’s 2M−X theorem given in the papers [4,24] carry over, in the same way, to the exponential functionals setting, although the task of letting the analogue of ε go to zero is a highly non-trivial problem in the general setting. Nevertheless, it gives a heuristic derivation that a certain functional of a Brownian motion inRⁿshould have the same law as a Brownian motion conditioned on a certain collection of its exponential functionals. This leads us to the question considered in the present paper.

We consider a Brownian motionB^(μ)inRⁿwith driftμ, and a collection of linear functionalsα1, . . . , αdsuch that the exponential functionals

Aⁱ_∞= _∞

0

e^{−2αi(Bs(μ))}ds, i=1, . . . , d,

are almost surely finite. Our aim is to understand which diffusion processes can arise when we condition on the law ofA_∞=(A¹_∞, . . . , A^d_∞). The first step is to understand the law ofA_∞. We show that the Laplace transform ofA_∞ satisfies a certain Schrödinger-type partial differential equation and proceed to characterise all diffusion processes which can be interpreted as having the law ofB^(μ)conditioned on the law ofA_∞.

In the case whenα₁, . . . , α_dis a simple system (see Section4for a definition), these diffusion processes are closely related to the quantum Toda lattice. The Schrödinger operator is

H=1

2Δ+

i

θ_i²e^−2αi,

whereθ∈R^d, and the corresponding diffusion process has infinitesimal generator given by 1

2Δ+ ∇logk_μ· ∇, (3)

wherek_μ is a particular eigenfunction ofH known as a class-one Whittaker function. In the casen=d=1 and α1(x)=x, the class-one Whittaker function iskμ(x)=Kμ(e^−x)and the infinitesimal generator is given by (1).

More generally, for a simple systemα₁, . . . , α_d, the diffusion process with generator given by (3) plays an analogous role, in the exponential functionals setting, as that of a Brownian motion conditioned to stay in the Weyl chamber {x∈Rⁿ: αi(x) >0, i=1, . . . , d}. These processes have already found an application in the paper [22], where the corresponding multi-dimensional version of the above theorem of Matsumoto and Yor in the ‘typeA’ case has been proved and used to determine the law of the partition function associated with a directed polymer model which was introduced in the paper [23].

The outline of the paper is as follows. In Section2we work in a general setting and establish a Schrödinger type partial differential equation satisfied by the characteristic function of exponential functionals of a multi-dimensional Brownian motion. We also study a family of martingales related to the conditional laws of exponential functionals that will later appear. In Section3, we identify a family of diffusions which can be interpreted as having the law of the Brownian motion with drift conditioned on the law of its exponential functionals. In Section4, we restrict our attention to the case where the collection of vectors used to define the exponential functionals is a simple system, and give an overview of relevant facts about class-one Whittaker functions. In Section5, we study properties of the conditioned processes in this setting. In the final section, we present some explicit results for the ‘typeA₂’ case.

2. Exponential functionals and associated partial differential equations

In this section, we work in a general setting and establish a Schrödinger type partial differential equation satisfied by the characteristic function of exponential functionals of a multi-dimensional Brownian motion. We also study a family of martingales related to the conditional laws of exponential functionals that will later appear.

Letα1, . . . , αdbe a collection of distinct, non-zero vectors inRⁿsuch that Ω= x∈Rⁿ: α_i(x) >0∀i

(4) is non-empty. LetB^(μ)be a standard Brownian motion inRⁿwith driftμ∈Ω. For 0≤t≤ ∞, set

Aⁱ_t= _t

0

e^{−2αi(Bs(μ))}ds, i=1, . . . , d.

Here,α_i(β)=(α_i, β)where(·,·)denotes the usual inner product onRⁿ. 2.1. Partial differential equation for the characteristic function

The process(B_t^(μ), A_t)_t≥0is a diffusion with generator 1

2Δ_x+(μ,∇x)+ d

i=1

e^−2αi(x) ∂

∂ai

.

We first check that this operator is hypoelliptic.

Proposition 2.1. The operator 1

2Δ_x+(μ,∇x)+ d

i=1

e^−2αi(x) ∂

∂a_i

is hypoelliptic onR^n+dand therefore,fort >0the random variable(B_t^(μ), A_t)admits a smooth density with respect to the Lebesgue measure.

Proof. We use Hörmander’s theorem. Since theα_i’s are pairwise different and non-zero, there existsv∈Rⁿsuch that i =j ⇒ αi(v) =αj(v).

Consider now the vector field V =

n

i=1

vi

∂

∂x_i,

and let us denote T =

d

i=1

e^−2αi(x) ∂

∂ai

.

The Lie bracket betweenV andT is given by LVT = [V , T] = −2

d

i=1

α_i(v)e^−2αi(x) ∂

∂a_i.

Similarly, by iterating this bracketktimes, we get L^kVT =(−1)^k2^k

d

i=1

αi(v)^ke^−2αi(x) ∂

∂a_i.

Since the α_i’s are pairwise different and non-zero, we deduce from the Van der Monde determinant that at every x∈Rⁿthe family

L^k_VT ,1≤k≤d

is a basis ofR^d. It implies that the Lie bracket generating condition of Hörmander is satisfied so that the operator

1

2Δx+(μ,∇x)+d

i=1e^{−2αi(x) ∂}_∂a

i is hypoelliptic.

Let nowθ∈R^dand, forx∈Rⁿ, define g_μ^θ(t, x)=E

e^{−di=1θi2e−2αi (x)Ait}

, t≥0, and

j_μ^θ(x)=E

e^{−di=1θi2e−2αi (x)Ai∞} .

Proposition 2.2.

(1) The semigroup generated by the Schrödinger operator 1

2Δ+(μ,∇)− d

i=1

θ_i²e^−2αi(x)

admits a heat kernelq_μ^θ(t, x, y)and we have

g_μ^θ(t, x)=

Rⁿq_μ^θ(t, x, y)dy.

(2) The functionj_μ^θ is the unique bounded function that satisfies the partial differential equation 1

2Δj_μ^θ(x)+

μ,∇j_μ^θ(x)

= _d

i=1

θ_i²e^−2αi(x)

j_μ^θ(x)

and the limit condition

x→∞lim,x∈Ωj_μ^θ(x)=1.

Proof.

(1) It is a straightforward consequence of the Feynman–Kac formula thatq_μ^θ(t, x, y)exists and is given by q_μ^θ(t, x, y)=E

e^{−di=1θi2e−2αi (x)Ait}B_t^(μ)=y−x 1

(2πt )^n/2e^{−y−x−μt2/(2t )}. Integrating this with respect toy, we obtain

g_μ^θ(t, x)=

Rⁿq_μ^θ(t, x, y)dy.

(2) It is again a straightforward consequence of the Feynman–Kac formula that j_μ^θ solves the partial differential equation, and the limit condition is easily checked. Let us now prove uniqueness. We have to show that ifφis a bounded solution of the equation that satisfies

x→∞lim,x∈Ωφ(x)=0,

thenφ=0. For that, let us observe that under the above conditions, forx∈Rⁿ, the process φ

B_t^(μ)+x exp

− d

i=1

θ_i²e^−2αi(x)Aⁱ_t

is a bounded martingale that goes to 0 whent → +∞. It follows that this martingale is identically zero almost surely, which impliesφ=0.

For later reference, we rephrase the second part of the previous proposition as follows:

Corollary 2.3. The functionh^θ_μ(x)=e^μ(x)j_μ^θ(x)is the unique solution to 1

2Δh^θ_μ(x)− d

i=1

θ_i²e^−2αi(x)h^θ_μ(x)=1

2μ²h^θ_μ(x), (5)

such thate^−μ(x)h^θ_μ(x)is bounded and

x→∞lim,x∈Ωe^−μ(x)h^θ_μ(x)=1.

Example 2.1. The following example has been widely studied(see,for example, [8,19]and references therein).Sup- posen=d=1,θ₁²=1/2andα₁(x)=x.Then

A_∞= _∞

0

e^−2(Bt+μt)dt, μ >0,

where(B_t, t≥0)is a standard one-dimensional Brownian motion,and j_μ^θ(x)=E

exp

−1

2e^−2xA_∞

, x∈R. In this case,h^θ_μ(x)=e^μxj_μ^θ(x)solves the equation

d²

dx²−e^−2x

h^θ_μ=μ²h^θ_μ.

This equation is easily solved by means of Bessel functions.By taking into account the boundary condition when x→ +∞,we recover the formula[19],Theorem6.2:

j_μ^θ(x)=2^1−μ

(μ)e^−μxKμ

e^−x

, (6)

whereK_μis the Macdonald function[18]:

K_μ(x)=1 2

x 2

μ _+∞

0

e^{−t−x2/(4t )}

t^1+μ dt. (7)

The formula(6)can also be derived using the fact[8]that A_∞ has the same law as1/2γμ,whereγ_μ is a gamma distributed random variable with parameterμ.

Example 2.2. The following example has also been studied in the literature[12,14].Supposen=1,d=2,θ₁²=θ₂²= 1/2,α1(x)=x andα2(x)=^x₂.Then

A¹_∞= _∞

0

e^−2(Bt+μt)dt, A²_∞= _∞

0

e^−(Bt+μt)dt, μ >0, where(B_t, t≥0)is a standard one-dimensional Brownian motion,and

j_μ^θ(x)=E

exp

−1

2e^−2xA¹_∞−1 2e^−xA²_∞

, x∈R. In this case,h^θ_μ(x)=e^μxj_μ^θ(x)solves the equation

d²

dx²−e^−x−e^−2x

h^θ_μ=μ²h^θ_μ.

This is Schrödinger’s equation with the so-called Morse potential.It is solved by means of Whittaker functions and by taking into account the boundary condition whenx→ +∞,we get

j_μ^θ(x)=2^μ−1/2(1+μ)

(2μ) e^(−μ+1/2)xW_−1/2,μ 2e^−x

,

whereW_k,μis the Whittaker function(see[18],p. 279):

W_k,μ(x)= x^ke^−x/2 (1/2+μ−k)

_+∞

0

e^−tt^μ−k−1/2

1+ t x

μ+k−1/2

dt.

2.2. Conditional densities

We prove now that the random variableA_∞has a smooth density with respect to the Lebesgue measure ofR^d and moreover give an expression of the conditional densities only in terms of this density.

Proposition 2.4. The random variableA_∞has a smooth densitypwith respect to the Lebesgue measure ofR^d and fort≥0

P(A_∞∈dy|Ft)=e^{2di=1αi(Bt(μ))}p

e^2α1(Bt(μ))

y₁−A¹_t

, . . . ,e^2αd(Bt(μ))

y_d−A^d_t

1(0,y1)×···×(0,yn)(A_t)dy, whereFis the natural filtration ofB^(μ).

Proof. If we denote byφthe characteristic function ofA_∞: φ(λ)=E

e^−(λ,A∞)

, λ₁, . . . , λ_d>0, then,

E

e^{−di=1λiAi∞}|Ft

=e^{−di=1λiAit}E

e^{−di=1λi(Ai∞−Ait)}|Ft

=e^−(λ,At)φ

e^{−2α1(Bt(μ))}λ₁, . . . ,e^{−2αd(Bt(μ))}λ_d .

Therefore, the process e^−(λ,At)φ(e^{−2α1(Bt(μ))}λ1, . . . ,e^{−2αd(Bt(μ))}λd) is a martingale. This implies that the function e^−(λ,a)φ(e^−2α1(x)λ1, . . . ,e^−2αd(x)λd)is harmonic for the operator ¹₂Δx+(μ,∇x)+d

i=1e^{−2αi(x) ∂}_∂a

i. This opera- tor being hypoelliptic, this implies thatA_∞has a smooth density with respect to the Lebesgue measure ofR^d. The result about the conditional densities stems from the injectivity of the Laplace transform.

In particular, we deduce from the previous proposition that if fory∈R^d₊, we denote q(x, a, y)=e^2di=1αi(x)p

e^2α1(x)(y1−a1), . . . ,e^2αd(x)(y_d−a_d)

for 0< a_i < y_i, x∈R^d, then the processq(B_t^(μ), A_t, y)1(0,y₁)×···×(0,y_n)(A_t)is a martingale. It implies that for any y∈R^d₊,q(x, a, y)satisfies the following partial differential equation:

1

2Δ_xq+(μ,∇xq)+ d

i=1

e^−2αi(x)∂q

∂a_i =0.

It also implies thatpis a solution of the partial differential equation:

d

i,j=1

(α_i, α_j)y_iy_j ∂²p

∂yi∂yj + d

i=1

α_i(μ)+ α_i²+2 d

j=1

(α_i, α_j)

y_i−1 2

∂p

∂yi

= − _d

i,j=1

(α_i, α_j)+ d

i=1

α_i(μ)

p.

Example 2.3. Supposen=d=1,θ₁²=1/2andα1(x)=x.ThenA_∞is distributed as1/2γμ,whereγμis a gamma law with parameterμ,that is

p(y)= 1 2^μ(μ)

e^−1/(2y)

y^1+μ 1_R>0(y),

and we have

q(x, a, y)= 1 2^μ(μ)

e^{−2μx−(1/2)(e−2x/(y−a))}

(y−a)^1+μ 1_R>0(y−a).

Example 2.4. Supposen=1,d=2,α₁(x)=xandα₂(x)=^x₂.Then,as seen before, A¹_∞=

_∞

0

e^−2(Bt+μt)dt, A²_∞= _∞

0

e^−(Bt+μt)dt, μ >0, and forλ₁, λ₂>0,

E

e^{−(1/2)λ21A∞1 −(1/2)λ22A∞2}

=2^μ−1/2λ^μ₁^−1/2(μ+1/2+λ²₂/(2λ1)) (2μ) W_−λ2

2/(2λ1),μ(2λ1)

= e^−λ1 (2μ)

_+∞

0

e^−tt^{μ+λ22/(2λ1)−1/2}(2λ1+t )^{μ−λ22/(2λ1)−1/2}dt.

By using in the previous integral the change of variablet=_eλ^2λ1s−¹1,we deduce the following nice formula E

e^{−(1/2)λ21A1∞}A²_∞=s P

A²_∞∈ds

= λ^2μ₁ ⁺¹ 2(2μ)

e^{−λ1cotanh(λ1s/2)}

(sinh(λ1s/2))^2μ+1ds, s >0.

This conditional Laplace transform can be inverted(see,for instance, [9])but,unlike the one-dimensional case,it does not seem to lead to a nice formula forp:

p(y₁, y₂)= 2^2μ (2μ)√

2π

+∞

j,k=0

(−1)^j2^j j!

(j+2μ+1+k) k!(j+2μ+1)

1

y₁^j/2+μ+3/2e^{−(1+y2(k+j+μ+1/2))2/(4y1)}

×Dj+2μ+2

1+y₂(k+j+μ+1/2)

√y₁

,

whereD_ν is the parabolic cylinder function such that _+∞

0

e^{−θ t}

t^1+νe^{−a2/(4t )}D_2ν+1 a

√t

dt=√

π2^ν+1/2θ^νe^−a

√2θ,

that is

D_ν(x)=

√2

√πe^x2/4 _+∞

0

t^νe^−t2/2cos

xt−πν 2

dt, ν >−1.

3. Brownian motion conditioned on its exponential functionals

In this section, we study the Doob transforms of the process(B_t^(μ), At)associated with the conditioning ofA_∞. We first start with the bridges which are the extremal points.

Lemma 3.1 (Equation of the bridges). Lety∈R^d₊.The law of the process(B_t+μt )_t≥0conditioned by A_∞=y

solves the following stochastic differential equation:

dXt=

μ+(∇xlnq)

X_t, t

0

e^−2α(Xs)ds, y

dt+dβt,

where(β_t)_t≥0is a standard Brownian motion.

Proof. This follows directly from Proposition2.4and Girsanov’s theorem.

Example 3.1. The following example is considered[21]. Supposen=d =1, θ₁²=1/2 and α₁(x)=x. Then the equation becomes

dXt=

−μ+ e^−2Xt y−t

0e^−2Xsds

dt+dβt.

LetP^μbe the law ofB^(μ)andπ be the coordinate process on the space of continuous functionsR₊→Rⁿ. Ifνis a probability measure onR^d₊, in what follows (see [3]), we call the probability

R^d₊P^μ

· _+∞

0

e^−2α1(πs)ds=y₁, . . . , _+∞

0

e^−2αd(πs)ds=y_d

ν(dy),

the law of the process(B_t+μt )_t≥0conditioned by A_∞^law=ν.

Proposition 3.1. Let v be a bounded and positive function such that

R^dv(y)p(y)dy =1. The law of the process (B_t+μt )_t≥0conditioned by

A_∞^law=v(x)p(x)dx

solves the following stochastic differential equation:

dXt=

μ+F_v t

0

e^−2α1(Xs)ds, . . . , _t

0

e^−2αd(Xs)ds, Xt

dt+dβt, where,(βt)t≥0is a standard Brownian motion andFv:R^d×Rⁿ→Rⁿis given by

F_v(a, x)=(∇xlnφv)(a, x) with

φ_v(a, x)=

R^dp(z)v

a₁+e^−2α1(x)z₁, . . . , a_d+e^−2αd(x)z_d dz.

Proof. Following [3], we have to write the stochastic differential equation associated with the conditioning A_∞^law=p(x)v(x)dx.

But E

v(A_∞)|Ft

=e²

_d

i=1αi(B_t^(μ))

R^dp

e^2α1(Bt(μ))

y1−A¹_t

, . . . ,e^2αd(Bt(μ))

yd−A^d_t v(y)dy

=φ_v

A_t, B_t^(μ) ,

so that we get the expected conditioned stochastic differential equation by Girsanov theorem.

In the previous proposition, the driftF_v(a, x)depends only onxif, and only if, v(x)=e^{−di=1θi2xi}

j_μ^θ(0) for someθ∈R^d. Therefore:

Corollary 3.2. Forθ∈R^d,the law of the process(B_t+μt )_t≥0conditioned by A_∞^law= e^{−di=1θi2xi}

j_μ^θ(0) p(x)dx

is the law of a Markov process.Moreover,in that case,it solves in law the following stochastic differential equation

dXt= ∇lnh^θ_μ(X_t)dt+dβt. (8)

We now show that the pathwise uniqueness property holds for the stochastic differential equation (8). In what follows, we denote

L^θμ= ∇lnh^θ_μ∇ +1 2Δ.

Let us observe that for the generator L^θμ, we have a useful intertwining with the Schrödinger operator ¹₂Δ− _d

i=1θ_i²e^−2αi(x)−¹₂μ²that will be used several times in the sequel.

Proposition 3.3.

h^θ_μL^θμ=

1 2Δ−

d

i=1

θ_i²e^−2αi(x)−1 2μ²

h^θ_μ.

Proof. Iff is a smooth function then we have

1 2Δ−

d

i=1

θ_i²e^−2αi(x)−1 2μ²

h^θ_μf

=1 2

Δh^θ_μ f+1

2(Δf )h^θ_μ+ ∇f∇h^θ_μ− _d

i=1

θ_i²e^−2αi(x)+1 2μ²

h^θ_μf

.

Since 1 2Δh^θ_μ=

_d

i=1

θ_i²e^−2αi(x)+1 2μ²

h^θ_μ,

the result readily follows.

We can now deduce:

Theorem 3.1. Letθ∈R^d.If(β_t)_t≥0is a Brownian motion,then forx0∈Rⁿ,there exists a unique process(X^x_t⁰)_t≥0

adapted to the filtration of(βt)t≥0such that:

X^x_t⁰=x₀+ _t

0 ∇lnh^θ_μ X_s^x0

ds+β_t, t≥0. (9)

Moreover,in law,the process(X^x_t⁰)_t≥0is equal to(B_t+μt+x₀)_t≥0conditioned by:

_+∞

0

e^{−2αi(Bt+μt+x0)}dt

1≤i≤d

law= e^{−di=1θi2yi}e^2di=1αi(x0)p(e^2α1(x0)y₁, . . . ,e^2αd(x0)y_d)

j_μ^θ(x0) dy.

Proof. Let x₀∈Rⁿ. Since the function∇lnh^θ_μ is locally Lipschitz, up to an explosion time ewe have a unique solution X_t^x0 for Eq. (9). Our goal is now to show that almost surely e= +∞. For that, we construct a suitable Lyapunov function for the generatorL^θ_μ.

Let

U (x)=cosh 2(μ, x) h^θ_μ(x) .

It is easily seen that whenx→ +∞,U (x)→ +∞. Moreover, from the intertwining, h^θ_μL^θμU=

1 2Δ−

d

i=1

θ_i²e^−2αi(x)−1 2μ²

cosh 2(μ, x)≤3

2μ²cosh 2(μ, x).

Therefore L^θ_μU≤3

2μ²U.

It implies that the process (e^{−(3/2)μ2t∧e}U (X_t^x_∧⁰_e))t≥0 is a positive supermartingale. Since U (x) → +∞

whenx→ +∞, we deduce that almost surelye= +∞.

Consequently, there is a unique solution(X^x_t⁰)_t≥0 for Eq. (9). The second part of the theorem is a direct conse-

quence of Corollary3.2and uniqueness in law for Eq. (9).

Example 3.2. Supposen=d=1andθ₁²=1/2andα₁(x)=x.Then L^θμ=

μ+e^−xK_μ−1(e^−x) K_μ(e^−x)

d dx +1

2 d²

dx². (10)

Let us denotep^μ,θ_t (x, y)the heat kernel ofL^θ_μ.From the intertwining,we have p^μ,θ_t (x, y)=K_μ(e^−y)

K_μ(e^−x)q_t^μ,θ(x, y), whereq_t^μ,θ(x, y)is the heat kernel of ¹₂(^d2

dx² −e^−2x−μ²).This kernel can be explicitly computed(see[1]or[20], Remark4.1):

q_t^μ,θ(x, y)=e^−(μ2/2)t _+∞

0

exp

−ξ

2−e^−2x+e^−2y 2ξ

Θ

e^−x−y ξ , t

dξ ξ , with

Θ(r, t)= r

√2π³te^{π2/(2t )} _+∞

0

e^{−ξ2/(2t )}e^−rcoshξsinhξsinπξ t dξ.

We deduce from that

p^μ,θ_t (−∞, y)=2e^{−μ2t /2}Θ e^−y, t

Kμ

e^−y ,