A characterization of Markov processes enjoying the time-inversion property

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A characterization of Markov processes enjoying the

time-inversion property

Stephan Lawi

To cite this version:

Stephan Lawi. A characterization of Markov processes enjoying the time-inversion property. 2005.

�hal-00005074v2�

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time-inversion property

S.Lawi

∗

1st June 2005

Abstra t

We give a ne essary and su ient ondition for a homogeneous Markov pro ess taking valuesin

R

n

to enjoy the time-inversion propertyof degree

α

. The ondition sets theshape for the semigroup densities of the pro ess and allows to further extend the lass of known pro essessatisfying the time-inversion property. As an appli ation we re overthe result of Watanabein[24℄for ontinuousand onservativeMarkovpro esseson

R

+

. Asnewexamples wegeneralizeDunklpro essesand onstru tamatrix-valuedpro esswithjumpsrelatedtothe Wishartpro essbyaskew-produ trepresentation.

Keywords : homogeneousMarkovpro esses;time-inversionproperty;Besselpro esses;Dunkl pro- esses;Wishartpro esses;semi-stablepro esses.

Mathemati sSubje t Classi ation(2000): 60J25;60J60;60J65;60J99.

1 Introdu tion Let

{(X

t

, t ≥ 0); (P

x

)

x∈R

n

}

beahomogeneousMarkovpro esswithsemigroupdensities(assumed toexist):

P

t

(x, dy) = p

t

(x, y)dy.

(1.1) For all

x ∈ R

n

and some

α > 0

, the pro ess

{(t

α

_X

₁

t

, t > 0); P

x

}

is Markov and in general inhomogeneous.

Denition1.1. Thepro ess

{(X

t

, t ≥ 0); P

x

}

issaidtoenjoythetime-inversionpropertyofdegree

α

iftheMarkovpro ess

{(t

α

_X

1 t

, t > 0); P

x

}

ishomogeneous.

CelebratedexamplesofMarkovpro esses,knowntoenjoythispropertyfor

α = 1

,areBrownian motionswithdriftin

R

n

andBesselpro esseswithdrift(see[21,24℄). GallardoandYor[12℄re ently workedoutasu ient onditiononthesemigroupdensitiesforaMarkovpro esstoenjoythe time-inversionproperty. Their argumentextended the lass of pro essesto pro esseswith jumps su h astheDunklpro ess[23℄andmatrix-valuedpro essessu has theWishartpro ess[3℄. Theaimof thepaperisto showthat the former onditionis a tuallyne essaryandsu ientand to provide somenewexamples.

Se tion2 ontainsthemaintheoremofthepaper,whi hisprovedinse tion3using straightfor-wardanalyti alarguments. Se tion4 onsidersanappli ationofthetheoremtoMarkovpro esses on

R

+

. Theresultis showntobestrongenoughto entirely hara terizethe lassofdiusion pro- esseson

R

+

that enjoythe time-inversionpropertyand thusprovidea dierentproof than that

∗

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erty. WereviewhowthegeneralizedDunkl pro esststherequirementsofthetheorem andthen introdu ea matrix-valued pro ess with jumps. Therelation of the latterpro essto theWishart pro essmimi stherelationbetweentheone-dimensionalDunklandBesselpro esses.

2 Markov pro esses whi h enjoy time-inversion Fix

x ∈ R

n

. Re all that under

P

x

, the pro ess

(t

α

_X

₁

t

, t > 0)

is Markov, inhomogeneous, with transitionalprobabilitydensities

q

(x)

s,t

(z, y), (s < t; z, y ∈ R

n

)

,whi hsatisfythefollowingrelation:

E

x

h

f (t

α

X

1 t

)

¯

s

α

X

1 s

= z

i

=

Z

dy f (y) q

s,t

(x)

(z, y)

(2.1) where

q

s,t

(x)

(a, b) = t

−nα

p

1 t

¡x,

b

t

α

¢

p

1 s

−

1 t

¡

b

t

α

,

_s

a

α

¢

p

1 s

¡x,

a

s

α

¢

.

(2.2)

Throughoutthepaper,weassumethesemigroupdensities

p

t

(x, y)

tobepositiveoverthedomain andregularenoughtobeatleasttwi edierentiablein thespa eandtimevariables.

Thepro ess

(t

α

_X

1 t

, t > 0)

isnotuniquelydenedfromtheknowledgeofthesemigroupdensities

p

t

(x, y)

. A tually, there exists at least one transformation that leaves the semigroup densities

q

(x)

s,t

(a, b)

un hanged: Doob's

h

-transform.

Denition2.1. Doob's

h

-transformisthetransformation

T

h

: P

x

|

F

t

7→

h(X

t

)

h(x)

e

−νt

_P

x

|

F

t

.

(2.3)

forsomefun tion

h

andsome onstant

ν > 0

. Thisremark leadstoourrsttwoassertions.

Proposition2.2. Twopro esses relatedby

h

-transformsyieldthesamepro ess bytime-inversion. Proposition 2.3.

T

h

isanequivalen e relation overthe lass of homogeneous Markov pro esses.

It is hen e legitimate to resear h fora riterium to lassifyall pro esses that enjoy the time-inversionpropertyupto

h

-transforms. Thefollowingtheoremgivesa on isestatementofourmain result:

Theorem 2.4. The Markov pro ess

(t

α

_X

₁

t

, t > 0)

is homogeneous if and only if the semigroup densitiesof

(X

t

, t ≥ 0)

areof the form:

p

t

(x, y) = t

−

nα

2 Φ

³

x

t

α

2 ,

y

t

α

2 ´

θ

³

y

t

α

2 ´

exp

½

ρ

³

x

t

α

2 ´

+ ρ

³

y

t

α

2 ´

¾

(2.4) oriftheyarein

h

-transformrelationshipwithit. Thefun tions

Φ, θ, ρ

havethefollowing properties for

λ > 0

: 1.

Φ(λx, y) = Φ(x, λy)

; 2.

θ(λy) = λ

β

_θ(y)

for some

β ∈ R

; 3.

ρ(λx) = λ

2 α

ρ(x)

;

Moreover, ifthesymmetry ondition

Φ(x, y) = Φ(y, x)

issatised,thenthesemigroup densitiesare relatedasfollows:

q

(x)

t

(a, b) =

Φ (x, b)

Φ (x, a)

exp

n

tρ (x)

o

p

t

(a, b).

(2.5)

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Denition 2.5. A Markovpro ess

(X

t

, t ≥ 0)

is alled semi-stablewith index

γ

in the sense of Lamperti[18℄if:

n

(X

ct

, t ≥ 0); P

x

o

(d)

=

n

(c

γ

X

t

, t ≥ 0); P

x/c

γ

o

.

(2.6)

Asa onsequen e,thesemigroupdensitiesofasemi-stablepro esswithindex

γ

havethefollowing property:

p

t

(x, y) = t

−nγ

p

1 ³

x

t

γ

,

y

t

γ

´

.

(2.7)

Remarking that the expression forthe semigroupdensities in Theorem 2.4 satises this property for

γ = α/2

yieldsthefollowing orollary:

Corollary 2.6. A Markov pro ess that enjoys the time-inversion property of degree

α

is a semi-stablepro ess with index

α/2

, orisin

h

-transformrelationship withit. The onverseisnot true.

3 Proof of the Theorem 3.1 Su ien y

If

p

t

(x, y)

satises the ondition(2.4), then from formula (2.2) the semigroupdensities

q

(x)

s,t

(a, b)

anbewrittenas

q

(x)

s,t

(a, b) = (t − s)

−

nα

2

3 Y

i=1

R

(i)

s,t

where

R

_s,t

(1)

=

Φ

µ

x t

α

2 ,

b

t

α

t

α

2 ¶

Φ

Ã

b

t

α

µ

_st

t − s

¶

α

2 ,

a

s

α

µ

_st

t − s

¶

α

2 !

Φ

³

x s

α

2 ,

a

s

α

s

α

2 ´

−1

R

s,t

(2)

= θ

µ b

t

α

t

α

2 ¶

θ

Ã

a

s

α

µ

_st

t − s

¶

α

2 !

θ

³

a

s

α

s

α

2 ´

−1

R

s,t

(3)

=

exp

(

ρ

¡x t

α

2 ¢ + ρ

µ b

t

α

t

α

2 ¶

+ ρ

Ã

b

t

α

µ

_st

t − s

¶

α

2 !

+ ρ

Ã

a

s

α

µ

_st

t − s

¶

α

2 !

−ρ

¡x s

α

2 ¢ − ρ

³

a

s

α

s

α

2 ´

)

Usingthepropertiesof

Φ, θ, ρ

des ribedin Theorem2.4,weobtain:

R

s,t

(1)

=

Φ (x, b)

Φ (x, a)

Φ

µ

_b

(t − s)

α

2 ,

a

(t − s)

α

2 ¶

R

s,t

(2)

= θ

µ

_b

(t − s)

α

2 ¶

R

s,t

(3)

=

exp

½

(t − s)ρ(x) + ρ

µ

a

(t − s)

α

2 ¶

+ ρ

µ

b

(t − s)

α

2 ¶¾

.

Hen ethereisnoseparatedependen eon

s

and

t

,butonlyonthedieren e

t − s

,whi hallowsto on ludethat

q

s,t

(x)

(a, b) = q

(x)

t−s

(a, b)

andproveshomogeneityfor thepro ess

(t

α

_X

₁

t

, t > 0)

. Ifin addition

Φ(x, y) = Φ(y, x)

,weobtain (2.5).

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Forsimpli ityofnotation,weprovethene essityof ondition(2.4)inthe ase

α = 1

. Theextension to

α > 0

is immediate by hange of variables

X

t

7→ X

α

t

. Re all rst the following denition of homogeneousfun tions:

Denition3.1. Afun tion

f : R

n

_{→ R}

that satisesfor

x = (x

1 , . . . , x

n

)

,

∀λ ∈ R

+

,

f (λx) = λ

β

f (x)

is alled homogeneous of degree

β ∈ R

. If

f ∈ C

1 _(R

n

₎

, Euler's HomogeneousFun tion Theorem givesane essaryandsu ient onditionforthefun tion

f (x)

tobehomogeneous:

n

X

i=1

x

i

∂

∂x

i

f (x) = βf (x).

Considerthefun tionof

2n + 1

variables:

l(x, y, t) = ln(p

t

(x, y)), x, y ∈ R

n

. From(2.2),

l

must satisfyfor

s = t − h

:

∂

∂t

· n ln

1 t

+ l

µ

x,

b

t

,

1 t

¶

+ l

µ b

t

,

a

t − h

,

1 t − h

−

1 t

¶

− l

µ

x,

a

t − h

,

1 t − h

¶¸

= 0.

(3.1) 3.2.1 The kernel

Φ(x, y)

Taking derivativeswithrespe tto

b

i

and

a

j

forsome

0 ≤ i, j ≤ n

yields:

∂

∂b

i

∂

∂a

j

∂

∂t

l

µ b

t

,

a

t − h

,

1 t − h

−

1 t

¶

= 0.

Ifweset

φ (x, y, t) =

∂

∂x

i

∂

∂y

j

l (x, y, t)

, thelatterbe omes

−

_t

2 _{(t − h)}

1 µ

φ +

b

t

· ∇

1 φ

¶

−

1 t(t − h)

2 µ

φ +

a

t − h

· ∇

2 φ

¶

+

1 t(t − h)

µ 1

t

2 −

1 (t − h)

2 ¶

∂φ = 0,

withthenotation

∇

1 = (

∂

∂x

1 , . . . ,

∂

∂x

n

)

T

_{, ∇}

2 = (

_∂y

∂

₁

, . . . ,

_∂y

∂

_n

)

T

and

φ = φ

³

b

t

,

a

t−h

,

1 t−h

−

1 t

´

. For larity,we hangevariablesto

z

1 =

b

t

,

z

2 =

a

t − h

,

t

1 = −

1 t

,

t

2 =

1 t − h

.

Then

φ = φ(z

1 , z

2 , t

1 + t

2 )

and

t

1 µ

φ + z

1 · ∇

1 φ + t

1 ∂φ

¶

= t

2 µ

φ + z

2 · ∇

2 φ + t

2 ∂φ

¶

,

orequivalently

t

1 µ

φ + z

1 · ∇

1 φ + (t

1 + t

2 ) ∂φ

¶

= t

2 µ

φ + z

2 · ∇

2 φ + (t

1 + t

2 ) ∂φ

¶

.

(3.2)

We hangevariableson emore:

u =

t

1 t

2 , v = t

1 + t

2

to get:

φ + z

1 · ∇

1 φ + v ∂φ = u¡φ + z

2 · ∇

2 φ + v ∂φ¢.

(3.3)

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φ(λx, µy, λµt) =

1 λµ

φ(x, y, t).

(3.4)

Proof. AstheLHSof equation(3.3)is independent of

u

,one an readilytaketheequivalent on-dition:

φ + z

1 · ∇

1 φ + v ∂φ =

0,

(3.5)

φ + z

2 · ∇

2 φ + v ∂φ =

0.

(3.6)

Let

g(λ) = φ(λz

1 , z

2 , λv)

and

h(µ) = φ(z

1 , µz

2 , µv)

. Equation(3.5)implies

g(λ)+λg

′

_{(λ) = 0,}

whi h solvesto

g(λ) = λ

−1

_g(1)

andhen e

φ(λz

1 , z

2 , λv) =

1 λ

φ(z

1 , z

2 , v).

Equation(3.6)implies

h(µ) + µh

′

_{(µ) = 0,}

whi hsolvesto

h(µ) = µ

−1

_h(1)

andhen e

φ(z

1 , µz

2 , µv) =

1 µ

φ(z

1 , z

2 , v).

Combiningthelattertwoequationsyields(3.4).

Conversely,if

g

ishomogeneousofdegree

−1

and

h

−1

,weget:

g(λ) + λg

′

_{(λ) = h(µ) + µh}

′

_{(µ) = 0,}

whi h isequivalentto(3.5)and(3.6)and on ludes theproof.

Note that theprevious resultremains valid for all se ond derivativesof

l(x, y, t)

with respe t to

x

i

and

y

j

,

∀i, j ∈ {1, . . . , n}

. By integration over

x

i

and

y

j

, one an thus already make an assumptiononthegeneralshapeofthefun tion

l

:

l(x, y, t) = k(x, y, t) + ϕ

1 (x, t) + ϕ

2 (y, t) + h

1 (x) + h

2 (y) + τ (t),

(3.7)

where

k : R

2n+1

_{→ R, ϕ}

1 , ϕ

2 : R

n+1

→ R

,

τ : R → R

and

h

1 , h

2 : R

n

_{→ R}

. Moreover, thekernel

k(x, y, t)

mustsatisfythefollowingproperty:

k(λx, µy, λµt) = k(x, y, t).

Thisisimmediatefrom(3.4)and

φ(λx, µy, λµt) =

∂

∂λx

i

∂

∂µy

j

l(λx, µy, λµt) =

∂

∂λx

i

∂

∂µy

j

k(λx, µy, λµt) =

1 λµ

∂

∂x

i

∂

∂y

j

k(λx, µy, λµt),

1 λµ

φ(x, y, t)

=

1 λµ

∂

∂x

i

∂

∂y

j

l(x, y, t) =

1 λµ

∂

∂x

i

∂

∂y

j

k(x, y, t).

Thispropertyisa tuallyequivalentto

k

µ

λ

√

x

t

,

y

√

t

, 1

¶

= k

µ x

√

t

, λ

y

√

t

, 1

¶

Setting

Φ

³

x

√

t

,

y

√

t

´

= exp k(x, y, t)

ompletestheproofoftherst onditionofthetheorem,i.e.

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3.2.2 The fun tion

ρ(x)

Wenowfo usonthefun tions

ϕ

1 (x, t), ϕ

2 (y, t)

. From(3.1),theymustsatisfy

∂

∂t

· ϕ

1 µ

x,

1 t

¶

+ ϕ

2 µ b

t

,

1 t

¶

+ ϕ

1 µ b

t

,

1 t − h

−

1 t

¶

+ ϕ

2 µ

_a

t − h

,

1 t − h

−

1 t

¶

− ϕ

1 µ

x,

1 t − h

¶

− ϕ

2 µ

a

t − h

,

1 t − h

¶¸

= 0.

(3.8)

Re allthat thevariables

a, b, x

areindependentfrom ea hother andthat

ϕ

1 , ϕ

2

are denedto be expli itlydependentontheirvariables. Sotakingthederivativeof(3.8)withrespe tto

x

i

gives

1 t

2 ∂

t

ψ

1 µ

x,

1 t

¶

=

1 (t − h)

2 ∂

t

ψ

1 µ

x,

1 t − h

¶

where

ψ

1 (x, t) =

∂

∂x

i

ϕ

1 (x, t)

. Hen e

ψ

1 (x, t)

mustbeoftheform

ψ

1 (x, t) =

1 t

ψ

11 (x)+ψ

12 (x)

,whi h implies

ϕ

1 (x, t) =

1 t

ϕ

11 (x) + ϕ

12 (x).

Weomit these ond termas it an always be re astaspart of

h(x)

in (3.7). There are so farno further onditionstoaddonthefun tion

ϕ

11 (x)

.

Takingthederivativeof(3.8)withrespe tto

b

i

leadsto

∂

∂b

i

∂

∂t

· ϕ

2 µ b

t

,

1 t

¶

+

t(t − h)

h

ϕ

11 µ b

t

¶¸

= 0,

whi h developsto

−

1 t

2 ∂

i

ϕ

2 −

b

t

3 · ∇∂

i

ϕ

2 −

1 t

3 ∂

t

∂

i

ϕ

2 +

2t − h

ht

∂

i

ϕ

11 −

t(t − h)

h

1 t

2 ∂

i

ϕ

11 −

t(t − h)

h

b

t

3 · ∇∂

i

ϕ

11 = 0.

Re allthat

ψ

11 (x) =

∂

∂x

i

ϕ

11 (x)

and set

ψ

2 (x, t) =

∂

∂x

i

ϕ

2 (x, t)

. Inthisnotation,weget

−

1 _t

ψ

2 −

b

t

2 · ∇ψ

2 −

1 t

2 ∂

t

ψ

2 +

2t − h

h

ψ

11 −

t(t − h)

h

1 t

ψ

11 −

t(t − h)

h

b

t

2 · ∇ψ

11 = 0.

Let

z =

b

t

, t

1 =

1 t

, t

2 =

1 t−h

,thenwehave

ψ

2 (z, t

1 ) + z · ∇ψ

2 (z, t

1 ) + t

1 ∂

t

ψ

2 (z, t

1 ) =

t

2 t

1 (t

2 − t

1 )

ψ

11 (z) −

1 t

2 − t

1 z · ∇ψ

11 (z).

AstheLHSisindependentof

t

2

,so mustbetheRHS,whi himplies

z · ∇ψ

11 (z) = ψ

11 (z).

Hen e

ψ

11

1

. Inthat ase,

ψ

2 (z, t

1 ) + z · ∇ψ

2 (z, t

1 ) + t

1 ∂

t

ψ

2 (z, t

1 ) =

1 t

1 ψ

11 (z).

Wesolvetheequationfor

g(λ) = ψ

2 (λz, λt

1 )

toget

g(λ) =

1 t

1 ψ

11 (z) +

1 λ

c(z, t

1 )

,where

c(z, t

1 )

turns out to be a homogeneous fun tion of degree

−1

. Integration over

z

i

yields some onditions for

ϕ

2 (z, t)

. On eagain,sin etheindex

i ∈ {1, . . . , n}

was hosenarbitrarily,weobtainthefollowing hara teristi s:

ϕ

1 (λz, λt) =

ϕ

11 (λz)

λt

with

ϕ

11 (λz) = λ

2 _ϕ

11 (z)

and

ϕ

2 (λz, λt) =

ϕ

21 (λz)

λt

+ ˆ

c(λz, λt),

where

ϕ

21 (λz) = λ

2 _ϕ

21 (z)

andeven

ϕ

21 (z) = ϕ

11 (z)

withoutlossofgenerality. Let

ρ(z) = ϕ

11 (z) =

ϕ

12 (z)

,thenwere overthethird onditionofthetheorem, thatis

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3.2.3 The fun tion

θ(y)

Inordertofurtherinvestigatethepropertiesof

c(z, t)

ˆ

,wederive(3.8)by

a

i

:

∂

∂a

i

∂

∂t

· 1

h

ϕ(a) + ˆ

c

µ

a

t − h

,

1 t − h

−

1 t

¶

− ˆc

µ

a

t − h

,

1 t − h

¶¸

= 0.

Thisleadsto

c(z, t

2 − t

1 ) + z · ∇c(z, t

2 − t

1 ) +

t

2

1 − t

2

2 t

2 ∂

t

c(z, t

2 − t

1 ) = c(z, t

2 ) + z · ∇c(z, t

2 ) + t

2 ∂

t

c(z, t

2 ).

Sin e

c(z, t)

is homogeneous of degree

−1

, the RHS is zero. This only proves that

∂

∂t

c(z, t) =

∂

∂t

∂

∂z

i

ˆ

c(z, t) = 0, ∀i

. Hen e,

c(z, t) = ˆ

ˆ

c

1 (z) + ˆ

c

2 (t)

. Goingba ktoequation(3.8)derivedby

b

i

gives thistimefor

ˆ

c(z, t)

,

∂

∂b

i

∂

∂t

· ˆ

c

µ b

t

,

1 t

¶¸

=

∂

∂b

i

∂

∂t

· ˆ

c

1 µ b

t

¶¸

= 0,

whi h learlyimplies

ˆ

c

1 (

b

t

) = ˆ

c

11 (b) + ˆ

c

12 (

1 t

)

. Usingequation(3.8)on emoregives

∂

∂t

· ˆ

c

12 µ 1

t

¶

+ ˆ

c

2 µ 1

t

¶

+ ˆ

c

2 µ

₁

t − h

−

1 t

¶

− ˆc

2 µ

₁

t − h

¶¸

= 0,

(3.9) whi h isequivalentto

t

2

1 c

ˆ

′

12 (t

1 ) + t

1

2 c

ˆ

′

2 (t

1 ) + (t

1 + t

2 )(t

2 − t

1 ) ˆ

c

′

2 (t

2 − t

1 ) − t

2

2 c

ˆ

′

2 (t

2 ) = 0.

Weset

g(t) = tˆ

c

′

2 (t)

sothat

t

2

1 c

ˆ

′

12 (t

1 ) + t

1 g(t

1 ) = t

2 ¡g(t

2 ) − g(t

2 − t

1 )

¢ − t

1 g(t

2 − t

1 ).

Assuming

lim

t

1 →0

LHS

t

1 =

β

2 ∈ R

with

|β| < ∞

, we divide theequation by

t

1

and takethelimit as

t

1 → 0

. This yields

1

2 β = t

2 g

′

(t

2 ) − g(t

2 ),

whi h solvesfor

g(t) = −

1

2 β + γt, γ ∈ R

. Hen e,

ˆ

c

2 (t) = −

1

2 β ln t + γt

and

ˆ

c

12 (t) = β ln t,

whereanyadditional onstanttermhasbeenset to 0. It willbe onvenientto set

γ = 0

. Aterm

γ 6= 0

orrespondstotheindependent aseofananefun tionof

t

inthesum(3.7)andisin luded inthedis ussionfor

τ (t)

. Tosummarize,thefun tion

ˆ

c(z, t)

mustbeexpressedas

ˆ

c(z, t) = ˆ

c

1 (z) −

β

2 ln t,

where

c

1 (z)

has the followingproperty:

ˆ

c

1 (λz) = ˆ

c

1 (z) + β ln λ.

Let

θ(z) = exp ˆ

c

1 (z)

. We then re over ondition2ofthetheorem,i.e.

θ(λz) = λ

β

θ(z).

3.2.4 The fun tions

h

1 (x), h

2 (y)

The obvious solution to (3.1) in this ase is

h(z) = h

1 (z) = −h

2 (z)

.

exp h

hen e denes an

(9)

3.2.5 The fun tion

τ (t)

Itremainstoexpli itlyformulatetheformofthefun tion

τ : R

+

→ R

thatisdened tosatisfy

∂

∂t

· −n ln t + τ

µ 1

t

¶

+ τ

µ

₁

t − h

−

1 t

¶

− τ

µ

₁

t − h

¶¸

= 0.

Butthisequationissimilarto(3.9),soweimmediatelyhavethesolution(

γ ∈ R

):

τ (t) = −

n

₂

ln t + γt.

On e again

e

γt

an bein luded in an

h

-transform,so

γ

isset to 0and

e

τ (t)

_{= t}

−

n

2

givestherst fa torinthesemigroupdensities(2.4).

3.2.6 The general ase

α > 0

For

α > 0

,weusethe hangeofvariables

X

t

7→ X

α

t

. Thesemigroupdensitiesfor

X

α

t

be ome

p

α

t

(x, y) = p

t

³

x

α

1 , y

1 α

´

J(y)

where

J(y)

isthe Ja obianof theinverse transformation, that is

J(y) = α

−n

_y

1 α

−1

1 · · · y

1 α

−1

n

, and weusetheslightabuseofnotation

x

1 α

=

³

x

α

1

1 , . . . , x

1 α

n

´

. Thesemigroupdensities anbere astinto

p

α

t

(x, y) = t

−

nα

2 Φ

α

³

x

t

α

2 ,

y

t

α

2 ´

θ

α

³

y

t

α

2 ´

exp

½

ρ

α

³

x

t

α

2 ´

+ ρ

α

³

y

t

α

2 ´

¾

where

Φ

α

_{(x, y) = Φ}

³

_x

1 α

, y

1 α

´

satises ondition 1,

θ

α

_{(y) = θ}

³

_y

1 α

´

J(y)

satises ondition 2for

¯

β =

β+n

_α

− n

and

ρ

α

_{(x) = ρ}

³

_y

1 α

´

satises ondition3ofequation(2.4).

4 Appli ation to diusions on

R

+

The aseofthediusionson

R

+

wasentirely hara terizedbyWatanabein[24℄. Itwasshownthat onlyBesselpro essesinthewidesense(whi hwere allthedenitionbelow)enjoythetime-inversion propertyofdegree1.

Denition4.1. Forsome

ν > −1

and

c ≥ 0

,thediusionpro essgeneratedby

L =

1 ₂

∂

2 ∂x

2 +

µ 2ν + 1

2x

+

h

′

c

(x)

h

c

(x)

¶

_∂

∂x

(4.1)

is alledBessel pro essin the widesense . Thefun tion

h

c

(x)

isgivenby

h

c

(x) = 2

ν

Γ(ν + 1) (

√

2c x)

−ν

_I

ν

(

√

2c x),

(4.2)

where

I

ν

isthemodiedBesselfun tion.

Remark 4.2 . TheBesselpro essin thewidesense isin

h

-transformrelationship,for

h ≡ h

c

, with theBesselpro ess.

Weshowthat theresultin [24℄is a onsequen eof Theorem2.4, whi h hasthefollowing one-dimensionalformulation:

(10)

Theorem 4.3. TheMarkov pro ess

(t

α

_X

1 t

, t > 0)

on

R

+

ishomogeneousif andonly if the semi-group densitiesof

(X

t

, t ≥ 0)

areof the form:

p

t

(x, y) = t

−

α

2 (1+β)

φ

³

xy

t

α

´

y

β

exp

(

−

k

2

2 Ã

x

α

2 t

+

y

α

2 t

!)

(4.3)

for

k > 0

, or if it is in

h

-transform relationship with it. Moreover, the semigroup densities are relatedasfollows:

q

t

(x)

(a, b) =

φ (xb)

φ (xa)

exp

µ

−t

k

2

2 x

2 α

¶

p

t

(a, b).

(4.4)

Proof. Theorem2.4formulatedon

R

+

implies

Φ(x, y) = φ(xy)

,

θ(y) = y

β

and

ρ(x) = −

k

2

2 x

2 α

, k >

0

,forthe ondition1-3tobesatisedinonedimension.

Weidentifyfurtherthe lassofdiusionpro essesandprovideadierentprooffortheresultin [24℄.

Proposition 4.4. If

(X

t

, t ≥ 0)

is a diusion pro ess and

(tX

1 t

, t > 0)

is homogeneous and onservative, thenboth arene essarily (possibly time-s aled) Besselpro esses inthe widesense. Proof. If

(X

t

, t ≥ 0)

isadiusionpro ess,thenitsinnitesimalgeneratorhasthefollowinggeneral stru ture:

L = s(y)

∂

2 ∂y

2 + µ(y)

∂

∂y

where it remains to identify the fun tions

s(y)

and

µ(y)

. For a xed

x > 0

, let

L

(x)

be the innitesimal generator of

(tX

1 t

, t > 0)

. From equation (2.5),

L

(x)

has the following relationship with

L

:

L

(x)

_{: f (b) 7→}

1 φ(xb)

L¡φ(xb)f(b)¢ −

k

2

2 x

2 _{f (b),}

whi h developsto

L

(x)

f (b) = s(b)f

′′

(b) +

½

µ(b) + 2x s(b)

φ

′

(xb)

φ(xb)

¾

f

′

(b) + U (x, b)f (b).

For thepro ess tobe onservative,werequire

U (x, b) = 0

,whi himplies nokillingin theinterior ofthedomain,thatis

s(b) x

2 φ

′′

_(xb)

φ(xb)

+ µ(b) x

φ

′

_(xb)

φ(xb)

−

k

2

2 x

2 _{= 0.}

We hangevariablesto

z = xb

toobtain

s(z/x) φ

′′

(z) +

µ(z/x)

x

φ

′

_{(z) −}

k

2

2 φ(z) = 0.

Sin ethelattermustbevalidforall

x > 0

,weareledtoset:

s(b) =

σ

2

for

σ > 0

and

µ(b) =

σ

2

2 2ν+1

b

for

ν > −1

. Thisyieldsthefollowingequation

1

2 φ

′′

_{(z) +}

2ν + 1

2z

φ

′

_{(z) −}

k

2 2σ

2 φ(z) = 0.

The general solution (non-singular at 0 and up to a onstant fa tor) is expressed through the modiedBesselfun tionoftherstkindas follows:

φ(z) =

µ kz

σ

¶

−ν

I

ν

µ kz

σ

¶

.

(11)

p

t

(x, y) = N t

−

1+β

2 ³

xy

t

´

−ν

I

ν

µ k

σ

xy

t

¶

y

β

exp

½

−

k

2

2 µ x

2 t

+

y

2 t

¶¾

,

where

N

is a normalizationfa tor. The additional ondition that

lim

t→0

p

t

(x, y) = δ(x − y)

implies

β = 2ν + 1

and

k =

1 σ

, whi h leadsto the semigroupdensities ofa time-s aled Besselpro ess of dimension

ν

:

p

t

(x, y) =

y

σ

2 _t

³

y

x

´

ν

I

ν

³

xy

σ

2 _t

´

exp

½

−

x

2 _{+ y}

2 2σ

2 _t

¾

.

Theinnitesimalgeneratorfor

(tX

1 t

, t > 0)

isgivenby

L

(x)

=

σ

2

2 ∂

2 ∂b

2 + σ

2 ½ 2ν + 1

2b

+ x

φ

′

_(xb)

φ(xb)

¾

_∂

∂b

,

whereonere ognizesexpression(4.1)for

h

c

(b) = φ(xb)

andatime-s ale

t 7→ σ

2 _t

.

Proposition4.4smoothlyextendstoanypower

α > 0

oftheBesselpro essthroughthemapping

X

t

7→ X

t

α

. Inparti ular, itis worthremarkingthat the ase

α = 2

givesrisetosquares ofBessel pro esses,whi hleadstothefollowingresult:

Proposition 4.5. If

(X

t

, t ≥ 0)

is a diusion pro ess and

(t

2 _X

1 t

, t > 0)

is homogeneous and onservative, thenboth arene essarily (possiblytime-s aled)squaresofBessel pro essesinthewide sense.

5 Examples

5.1 Generalized Dunkl pro esses and Ja obi-Dunkl pro esses 5.1.1 MultidimensionalDunkl pro esses

Webrieyreviewthe onstru tionoftheDunklpro essin

R

n

(see[22,23℄). Denition5.1. TheDunklpro essin

R

n

istheMarkov àdlàgpro esswithinnitesimalgenerator

1

2 L

(k)

₌

1

2 n

X

i=1

T

i

2

(5.1)

where

T

i

, 1 ≤ i ≤ n,

isaone-dimensionaldierential-dieren eoperatordenedfor

u ∈ C

1 _(R

n

₎

by

T

i

u(x) =

∂u(x)

∂x

i

+

X

α∈R

+

k(α)α

i

u(x) − u(σ

α

x)

hα, xi

.

(5.2)

h·, ·i

is theusual s alarprodu t.

R

is arootsystem in

R

n

and

R

+

apositivesubsystem.

k(α)

is anon-negative multipli ity fun tion dened on

R

and invariantby the nite ree tion group

W

asso iatedwith

R

.

σ

α

is theree tionoperator withrespe tto thehyperplane

H

α

orthogonal to

α

su hthat

σ

α

x = x − hα, xiα

andfor onvenien e

hα, αi = 2

(see[8,9℄).

AresultobtainedbyM.Rösler[22℄yieldsthesemigroupdensitiesas follows:

p

(k)

t

(x, y) =

1 c

k

t

γ+n/2

exp

µ

−

|x|

2 + |y|

2 2t

¶

D

k

µ x

√

t

,

y

√

t

¶

ω

k

(y)

(5.3)

(12)

where

D

k

(x, y) > 0

is the Dunkl kernel,

ω

k

(y) =

Y

α∈R

+

|hα, yi|

2k(α)

the weight fun tion whi h is

homogeneousofdegree

2γ = 2

X

α∈R

+

k(α)

and

c

k

=

Z

R

n

e

−

|x|2

2 ω

k

(x)dx

.

Followinga thorough study of theproperties of theone-dimensional Dunkl pro ess in [13℄, it was remarkedin[12℄ thattheDunklpro essin

R

n

enjoysthetime-inversionpropertyofdegree1. Consideringthat theDunklkernelsatises

D

k

(x, y) = D

k

(y, x)

and D

k

(µx, y) = D

k

(x, µy),

(5.4)

theproofisstraightforwardwith

Φ(x, y) ≡ D

k

(x, y),

θ(y) ≡ ω

k

(y),

ρ(x) ≡ −

|x|

2

2 .

(5.5)

By equation (2.5), the semigroup densities of the time-inverted pro ess is even in

h

-transform relationshipwiththesemigroupdensitiesoftheoriginalDunklpro ess:

q

t

(x)

(a, b) =

D

k

(x, b)

D

k

(x, a)

exp

µ

−

|x|

2

2 t

¶

p

(k)

t

(a, b).

(5.6)

5.1.2 Generalized Dunkl pro esses

InanattempttogeneralizetheDunklpro ess,weextendthedenitionoftheinnitesimalgenerator to

L

(k,λ)

f (x) =

1

2 ∆f (x) +

X

α∈R

+

k(α)

h∇f(x), αi

hx, αi

+

X

α∈R

+

λ(α)

f (σ

α

x) − f(x)

hx, αi

2

(5.7)

where

∆

istheusualLapla ian,

f ∈ C

2 _(R

n

₎

and

λ(α)

isanon-negativemultipli ityfun tiondened on

R

andinvariantbytheniteree tiongroup

W

,similarlyas

k(α)

. WeretrievetheDunklpro ess for

λ(α) = k(α)

. Notethatthese pro essesarenolongermartingalesfor

λ(α) 6= k(α)

.

Theone-dimensional asewasintrodu edin[13℄, wherethesemigroupdensitieswereexpli itly derived,

p

(k,λ)

t

(x, y) =

1 t

k−

1

2 y

k−

1

2 exp

µ

−

x

2 _{+ y}

2 2t

¶

D

k,λ

³

xy

t

´

(5.8) withthegeneralizedDunklkernel,

(ν = k −

1

2 , µ =

√

ν

2 _{+ 4λ)}

,

D

k,λ

(z) = 1

{y∈R

−

}

1 2z

ν

(I

ν

− I

µ

) (−z) + 1

{y∈R

+

}

1 2z

ν

(I

ν

+ I

µ

) (z) ,

(5.9) for

z =

xy

t

. From(5.4),thegeneralizedDunklkernelsatises

D

k,λ

(x, y) = D

k,λ

(y, x)

and D

k,λ

(µx, y) = D

k,λ

(x, µy),

(5.10)

whi hreadilyimpliesthatthegeneralizedDunklpro essalsoenjoysthetime-inversionpropertyof degree1. Thesemigroupdensitieswerederivedasanappli ationoftheskew-produ trepresentation of thegeneralizedDunkl pro ess

(X

t

, t ≥ 0)

in terms ofits absolute value(a Besselpro ess)and anindependentPoissonpro ess

N

(λ)

t

:

X

t

(d)

= |X

t

|(−1)

N

(λ)

At

(5.11) where

A

t

=

R

t

0 ds

X

2 s

.

Inthe

n

-dimensional ase,theappli ation oftheskew-produ t representation derivedby Chy-biryakov[6℄showsthat thegeneralizedDunklpro ess enjoystime-inversionforsomespe i root

(13)

Proposition5.2. Let

(X

t

, t ≥ 0)

bethegeneralizedDunklpro essgeneratedby(5.7),with

(X

W

t

, t ≥

0)

itsradialpart,i.e. thepro ess onnedtoaWeyl hamber. Let

R

+

≡ {α

1 , . . . , α

l

}

forsome

l ∈ N

be the orresponding positive root system andlet

(N

i

t

, t ≥ 0), i = 1, . . . , l

be independent Poisson pro esses of respe tive intensities

λ(α

i

)

. Then

X

t

may be represented as

Y

l

t

, whi h is dened by indu tionasfollows:

Y

t

0 = X

t

W

and Y

t

i

= σ

N

i

Ai

_t

α

i

Y

t

i−1

, i = 1, . . . , l,

where

A

i

t

=

Z

t

0 ds

hY

s

i−1

, α

i

2

.

Theprooffollowstheargumentin [6℄, whilerepla ing

k(α

i

)

by

λ(α

i

)

appropriately.

Fromnowon,let

R

+

≡ {α

1 , . . . , α

l

}

forsome

l ≤ n

beanorthogonalpositiverootsystem,that is

hα

i

, α

j

i = 2δ

ij

. Forthisparti ularrootsystem,one anshowthatthegeneralizedDunklpro ess enjoystime-inversionofdegree1. Werstprovethefollowingabsolute ontinuityrelation: Lemma 5.3. Let

(X

W

t

, t ≥ 0)

be theradial Dunklpro ess withinnitesimal generator

L

W

k

f (x) =

1

2 ∆f (x) +

l

X

i=1

k(α

i

)

hα

i

, ∇f(x)i

hα

i

, xi

.

(5.12) Fixed

ν ∈ {1, . . . , l}

. Let

k

′

_(α)

be another oe ient fun tion on the root system

R

+

su h that

k

′

_(α

ν

) > k(α

ν

)

and

k

′

_(α

i

) = k(α

i

)

for

i 6= ν

. Then, denoting

P

(k)

x

the law of the radial Dunkl pro ess

X

W

t

startingfrom

x

, wehave

P

(k

x

′

)

¯

F

t

=

µ hα

ν

, X

t

W

i

hα

ν

, xi

¶

k

′

(α

ν

)−k(α

ν

)

exp

"

−

¡k

′

_(α

_ν

_{) −}

1

2 ¢

2 −

¡k(α

ν

) −

1 ₂

¢

2

2 Z

t

0 ds

hα

ν

, X

s

W

i

2 #

·P

(k)

x

¯

F

t

.

(5.13) Proof. Let

k

0 (α)

bea oe ientsu h that

k

0 (α

ν

) =

1

2

for somexed

i ∈ {1, . . . , l}

.

X

W

t

hasthe followingmartingalede omposition(see[11℄):

X

t

W

= x + B

(k

0 )

t

+

l

X

i=1

k

0 (α

i

)

Z

t

0 ds

hα

i

, X

s

W

i

α

i

where

B

(k

0 )

t

isa

(P

(k

0 )

x

, F

t

)

-Brownianmotion. Considerthelo almartingale

L

(k

t

′

)

= exp

Ã

_µ

k

′

(α

ν

) −

1

2 ¶ Z

t

0 hα

ν

, dB

s

(k

0 )

i

hα

ν

, X

s

W

i

−

¡k

′

_(α

_ν

_{) −}

1

2 ¢

2

2 Z

t

0 ds

hα

ν

, X

s

W

i

2 !

,

for some oe ient fun tion

k

′

_(α)

su h that

k

′

_(α

ν

) >

1 ₂

and

k

′

_(α

i

) = k

0 (α

i

)

for

i 6= ν

. TheIt formulafor

ln

¡hα

ν

, X

W

t

i

¢

ombinedwiththeorthogonalityoftherootsyields

L

(k

t

′

)

=

µ hα

ν

, X

t

W

i

hα

ν

, xi

¶

k

′

(α

ν

)−

1

2 exp

Ã

−

¡k

′

_(α

ν

) −

1 ₂

¢

2

2 Z

t

0 ds

hα

ν

, X

s

W

i

2 !

.

Denethenewlaw

P

(k

′

₎

x

|

F

t

= L

(k

′

₎

t

· P

(k

0 )

x

|

F

t

. BytheGirsanovtheorem,

B

t

(k

′

)

= B

(k

0 )

t

−

µ

k

′

(α

ν

) −

1

2 ¶ Z

t

0 ds

hα

ν

, X

s

W

i

isa

(P

(k

′

₎

x

, F

t

)

-Brownianmotionandhen e,

X

t

W

= x + B

(k

′

₎

t

+

l

X

i=1

k

′

(α

i

)

Z

t

0 ds

hα

i

, X

s

W

i

α

i

(14)

isaradialDunklpro essunder

(P

(k

′

₎

x

, F

t

)

.

Dene

k

asanother oe ientontherootsystemthatsatisesthe onditionsenun iatedinthe lemma. Theabsolute ontinuityrelationisthena onsequen eofthesu essiveappli ationofthe latterresulttotheindi es

k

′

and

k

.

Nowas anappli ation ofProposition5.2,weprovethefollowing:

Proposition 5.4. Let

R

+

≡ {α

1 , . . . , α

l

}

be an orthogonal positive root systemfor

l ≤ n

. Then the generalized Dunklpro ess

(X

t

, t ≥ 0)

generatedby (5.7) enjoysthe time-inversion property of degree1.

Proof. Usingorthogonalityoftheroots,remarkthat

hα

i

, σ

j

xi

2 = hα

i

, x − hα

j

, xiα

j

i

2 = hα

i

, xi

2 ,

whi h impliesin parti ular

hα

i

, Y

t

i

2 = hα

i

, X

t

W

i

2 ,

sothat theindu tiverepresentationof

X

t

in Proposition5.2be omes

X

t

=

l

Y

i=1

σ

N

i

Ai

_t

α

i

X

W

t

for A

i

t

=

Z

t

0 ds

hX

W

s

, α

i

2 .

TheradialpartofaDunklpro essenjoysthetime-inversionpropertyofdegree1. Weneedtoshow thatthesemigroupdensitiesofthegeneralizedDunklpro essarerelatedtothesemigroupdensities ofitsradialparts. For

f ∈ C

2 _(R

n

₎

,

E

x

£f(Y

t

i

)¤ = E

x

£f(Y

t

i−1

)1

{N

i

Ai

_t

is even}

¤ + E

x

£f(σ

α

i

Y

i−1

t

)1

{N

i

Ai

_t

is odd}

¤.

Sin e

P(N

i

u

is even) =

1

2 (1 + exp(−2λ(α

i

)u))

,weobtain

E

x

£f(Y

t

i

)¤ = E

x

· f (Y

t

i−1

)

1

2 ¡1 + exp(−2λ(α

i

)A

i

t

)

¢

¸

+ E

x

· f (σ

α

i

Y

t

i−1

)

1

2 ¡1 − exp(−2λ(α

i

)A

i

t

)

¢

¸

.

Theexpe tation

E

x

£f(X

t

)

¤

anthusbeevaluatedbyindu tionon

i ∈ {1, . . . , l}

. Itfollowsthatthe semigroupdensitiesof

X

t

an beexpressedas theprodu tof thesemigroupdensitiesofits radial partstimesafun tioninvolvingexpe tationsoftheform

E

(k)

x

· exp (−2λ(α

ν

)A

ν

t

)

¯

X

t

W

= y

¸

,

for

ν ∈ {1, . . . , l}

. FromLemma5.3,

E

(k)

x

· exp (−2λ(α

ν

)A

ν

t

)

¯

X

t

W

= y

¸

=

p

(k

′

)

t

(x, y)

p

(k)

t

(x, y)

µ hα

ν

, yi

hα

ν

, xi

¶

k(α

ν

)−k

′

(α

ν

)

,

where

k

′

_(α

_i

_{) = k(α}

_i

₎

for

i 6= ν

and

k

′

_(α

_ν

_{) =}

1

2 +

q

¡k(α

ν

) −

1 ₂

¢

2 + 4λ(α

ν

)

. The form of the semigroupdensitiesin (5.3)impliesthattheexpe tation isaratioofDunklkernels,

E

(k)

x

· exp (−2λ(α

ν

)A

ν

t

)

¯

X

W

t

= y

¸

=

c

k

c

k

′

D

k

′

³

x

√

t

,

y

√

t

´

D

k

³

x

√

t

,

y

√

t

´

w

k

′

³

y

√

t

´

w

k

³

y

√

t

´

Ã hα

ν

,

√

y

_t

i

hα

ν

,

√

x

_t

i

!

k(α

ν

)−k

′

(α

ν

)

,

whi h redu esto

E

(k)

x

· exp (−2λ(α

ν

)A

ν

t

)

¯

X

_t

W

= y

¸

=

c

k

c

D

k

′

³

x

√

t

,

y

√

t

´

³

x

y

´

µ

hα

ν

,

y

√

t

ihα

ν

,

x

√

t

i

¶

k

′

(α

ν

)−k(α

ν

)

,

(15)

bydenitionof

k

′

_(α)

. The onditional expe tationthussatises ondition1ofTheorem2.4. As a onsequen e,the onditionsofTheorem2.4aresatisedfor

Φ(x, y) ≡ D

k,λ

(x, y),

θ(y) ≡ ω

k

(y),

ρ(x) ≡ −

|x|

2

where

D

k,λ

(x, y)

isageneralizedDunklkernelgivenexpli itlyintermsofradialDunklkernelsand satisfyingequivalent onditions(see(5.4)).

5.1.3 Ja obi-Dunkl pro esses

Gallardoetal. [5℄derivedtheJa obi-Dunklpro essasthehyperboli analogoftheone-dimensional Dunklpro ess. Itisdenedas thepro essgeneratedby

L

(α,β)

_{f (x) =}

∂

2 f (x)

∂x

2 +

A

′

_(x)

A(x)

∂f (x)

∂x

+

∂

∂x

µ A

′

_(x)

A(x)

¶ µ f (x) − f(−x)

2 ¶

,

(5.14)

where

A(x) =

¡sinh

2 (x)

¢

α+

1

2 _¡cosh

2 (x)

¢

β+

1

2

. From the expression of the semigroupdensities de-veloped in [5℄, this pro ess does not enjoy the time-inversion property. Its radial part however orrespondstotheJa obipro essofindex

(α, β)

on

R

+

(see [15,16℄). Theinnitesimalgenerator oftheJa obipro ess,expressedby

L

(α,β)

f (x) =

1

2 ∂

2 _{f (x)}

∂x

2 +

A

′

_(x)

A(x)

∂f (x)

∂x

,

(5.15)

isin

h

-transformrelationshipwiththeLapla ianoperator for

h(x) =

pA(x)

. Sin e theBrownian motionenjoysthetime-inversionpropertyofdegree1,so doestheJa obipro essbyTheorem2.4.

5.2 Matrix-valued pro esses 5.2.1 Eigenvaluepro esses

Dysonin [10℄ des ribed theeigenvaluesofaHermitian Brownianmotionas thejoint evolutionof independentBrownianmotions onditionedneverto ollide(seealso [14℄ and[4℄). Itwas further remarked that the pro ess version of the Gaussian orthogonal ensemble does not admit su h a representationforits eigenvalues. Thisworkwas extendedby KönigandO'Connellin [17℄ tothe pro essversionoftheLaguerreensemble,denominatedtheLaguerrepro essanddenedasfollows: Denition5.5. Let

B

t

bean

n × m

matrixwithindependentstandard omplexBrownianentries. TheLaguerrepro essisthematrix-valuedpro essdened by

{X

t

= B

′

t

B

t

, t ≥ 0}

,where

B

′

t

isthe transposeof

B

t

.

From [17℄, theeigenvaluesof theLaguerrepro essevolvelike

m

independent squaredBessel pro- esses onditionedneverto ollide. Nosu h representation howeverexists forthe asewhere the entries of

B

t

arerealBrownianmotions,i.e. theWishartpro ess onsideredbyBruin [3℄.

Themainresultof[17℄isthatbothoftheabovementionedeigenvaluespro esses(inthe omplex Brownian ase) an beobtainedas the

h

-transformofpro esseswith

m

independent omponents. Thejointeigenvaluespro essisthusin

h

-transformrelationshipwithapro essthatenjoysthetime inversionproperty ofdegree1in the aseofthe

m

-dimensionalBrownianmotionand degree2in the aseofthe

m

-dimensionalsquaredBesselpro ess,asmadeexpli itinthefollowingproposition: Proposition5.6. Let

p

t

(x

i

, y

i

) (i = 1, . . . , m)

bethesemigroupdensitiesofsquaredBesselpro esses (respe tively Brownian motions), andlet

h(x) =

m

Y

i<j

(16)

for

x = (x

1 , . . . , x

m

)

. Then,thesemigroupdensitiesofthejointeigenvaluespro essoftheLaguerre pro ess(respe tively theHermitianBrownian motion) are given by

˜

p

t

(x, y) =

h(y)

h(x)

m

Y

i=1

p

t

(x

i

, y

i

)

(5.17)

withrespe ttothe Lebesguemeasure

dy =

m

Y

i=1

dy

i

.

It follows immediately by Theorem 2.4 that the eigenvalues pro esses enjoy the time-inversion property. Moreoverby Proposition 2.2, they yield the same pro ess under time-inversion as the

m

-dimensionalBrownianmotionor the

m

-dimensional squaredBesselpro essrespe tively.

5.2.2 Wishart pro esses The Wishart pro ess WIS(

δ, tI

m

,

1 t

x

), introdu ed by Bru in [2℄, is a ontinuous Markov pro ess takingvaluesin thespa eofrealsymmetri positivedenite

m × m

matri es

S

+

m

. Itissolutionto thefollowingsto hasti dierentialequation:

dX

t

=

pX

t

dB

t

+ dB

t

′

pX

t

+ δI

m

dt,

X

0 = x,

(5.18)

where

B

t

isan

m × m

matrixwith Brownian entries and

I

m

the identitymatrix. Furtherresults havebeenobtainedin [1℄ and [3℄. In[7℄, among othermajorndings abouttheWishart pro ess, thetransitionprobabilitydensitiesexpressedwithrespe ttotheLebesguemeasure

dy =

Y

i≤j

(dy

ij

)

werederivedintermsofgeneralizedBesselfun tions(werefertotheappendixforthedenition):

p

t

(x, y) =

1 (2t)

m(m+1)

2 exp

µ

−

_2t

1 T r(x + y)

¶

_{µ det(y)}

det(x)

¶

δ−m−1

4 ˜

_I

_δ−m−1

2 ³

xy

4t

2 ´

,

(5.19) for

x, y ∈ S

+

m

and

δ > m − 1

. From the shape of its densities, the Wishart pro ess was stated in [12℄ as an exampleof Markovpro essesenjoying thetime-inversionproperty ofdegree 2. The hypothesisof Theorem2.4isindeedsatisedfor

n =

1

2 m(m + 1)

and

Φ(x, y) ≡ (det(x) det(y))

−

δ−m−1

4 ˜

_I

_δ−m−1

2 ³

xy

4 ´

,

θ(y) ≡

₂

1 n

(det(y))

δ−m−1

2 ,

ρ(x) ≡ −

1

2 T r(x).

(5.20) Nextweuseaskew-produ trepresentation,asfortheDunklpro ess,toelaborateontheWishart pro ess and derivea matrix-valuedpro esswith jumps. The skew-produ t representationallows theexpressionofthesemigroupdensitiesintermsoftheWisharttransitionprobabilitydensities. Denition 5.7. Let

(N

(λ)

t

, t ≥ 0)

be a Poisson pro ess with intensity

λ

. Let

(X

t

, t ≥ 0)

be a Wishart pro ess WIS(

δ, tI

m

,

1 t

x

) independent of the Poisson pro ess. The skew-Wishart pro ess

(X

t

(λ)

, t ≥ 0)

isdenedthroughtheskew-produ t

X

t

(λ)

= X

t

(−1)

N

(λ)

At

(5.21) where

A

t

=

Z

t

0 T r(X

s

−1

)ds

.

Proposition 5.8. The transition probability densities of the skew-Wishart pro ess are related to the semigroup densities

p

t

(x, y)

ofthe Wishart pro ess

X

t

asfollows

p

(λ)

t

(x, y) = p

t

(x, |y|)

(

1 _{y∈S

+

m

}

1

2 Ã

1 +

Ã ˜I

_˜

ν

′

I

ν

!

³

xy

4t

2 ´

!

+ 1

_{y∈S

−

m

}

1

2 Ã

1 −

Ã ˜I

_˜

ν

′

I

ν

!

µ −xy

4t

2 ¶

!)

.

(5.22)

√

(17)

Proof. Let

(P

t

)

t>0

bethesemigroupoftheDunkl-Wishartpro ess. For

x > 0

and

f ∈ C

c

(M

m

(R))

,

P

t

f (x) = E

x

h

f (X

t

(λ)

)

i

= E

x

· f (X

t

) 1

_{N

(λ)

At

is even}

¸

+ E

x

· f (−X

t

) 1

_{N

(λ)

At

is odd}

¸

.

With

P(N

(λ)

u

is even) =

1 ₂

(1 + exp(−2λu))

,wehave

P

t

f (x) = E

x

· f (X

t

)

1

2 (1 + exp(−2λA

t

))

¸

+ E

x

· f (−X

t

)

1

2 (1 − exp(−2λA

t

))

¸

.

(5.23) Let

Q

(ν

′

₎

x

with

ν

′

₌

δ

′

−m−1

2

denote the probability lawof aWishart pro essWIS

(δ

′

_{, tI,}

1 t

x)

, and

Q

(ν)

x

with

ν =

δ−m−1

2

theprobabilitylawof

X

t

. A ordingto Theorem 1.2 (Remark 2.3) in [7℄, theprobabilitylawsarerelatedasfollows:

Q

(ν

_x

′

)

¯

F

t

=

µ det X

t

det x

¶

ν′−ν

2 exp

µ

−

ν

′2

− ν

2

2 Z

t

0 T r(X

s

−1

)ds

¶

· Q

(ν)

x

¯

F

t

,

fromwhi hwededu e

p

(ν

′

)

t

(x, y)

p

(ν)

_t

(x, y)

=

µ det y

det x

¶

ν′−ν

2 Q

(ν)

_x

· exp

µ

−

ν

′2

− ν

2

2 Z

t

0 T r(X

s

−1

)ds

¶ ¯

¯

X

t

= y

¸

.

Thus,fromtheexpressionofthesemigroupdensitiesin(5.19), wehave

E

(ν)

_x

· exp

µ

−2λ

Z

t

0 T r(X

s

−1

)ds

¶ ¯

¯

X

t

= y

¸

=

Ã ˜

_I

√

ν

2 _+4λ

˜

_I

ν

!

³

xy

4t

2 ´

.

Combiningthelatterwith(5.23)yieldsthesemigroupdensitiesfortheskew-Wishart pro ess. The skew-Wishart is an example of matrix-valued pro ess with jumps that enjoys the time-inversionpropertyofdegree2. Indeed,bysetting