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arXiv:0811.0504v1 [math.PR] 4 Nov 2008

Symmetry, Integrability and Geometry: Methods and ApplicationsSIGMA 4 (2008), 074, 14 pages

First Hitting Time of the Boundary

of the Weyl Chamber by Radial Dunkl Processes⋆

Nizar DEMNI

SFB 701, Fakult¨ at f¨ ur Mathematik, Universit¨ at Bielefeld, Deutschland

E-mail: demni@math.uni-bielefeld.de

Received July 01, 2008, in final form October 24, 2008; Published online November 04, 2008

Original article is available at http://www.emis.de/journals/SIGMA/2008/074/

Abstract. We provide two equivalent approaches for computing the tail distribution of the

first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first

approach is based on a spectral problem with initial value. The second one expresses the tail

distribution by means of the W-invariant Dunkl–Hermite polynomials. Illustrative examples

are given by the irreducible root systems of types A, B, D. The paper ends with an interest

in the case of Brownian motions for which our formulae take determinantal forms.

Key words: radial Dunkl processes; Weyl chambers; hitting time; multivariate special func-

tions; generalized Hermite polynomials

2000 Mathematics Subject Classification: 33C20; 33C52; 60J60; 60J65

1 Motivation

The first exit time from cones by a multidimensional Brownian motion has been of great interest

for mathematicians [2, 8] and for theoretical physicists as well [5]. An old and famous example of

cones is provided by root systems in a finite dimensional Euclidean space, say (V,?·?) [15]. More

precisely, a root system R in V is a collection of non zero vectors from V such that σα(R) = R

for all α ∈ R, where σαis the reflection with respect to the hyperplane orthogonal to α:

σα(x) = x − 2?α,x?

?α,α?α,x ∈ V.

A simple system S is a basis of span(R) which induces a total ordering in R. A root α is positive

if it is a positive linear combination of elements of S. The set of positive roots is called a positive

subsystem and is denoted by R+. The cone C associated with R, known as the positive Weyl

chamber, is defined by

C := {x ∈ V, ?α,x? > 0 ∀α ∈ R+} = {x ∈ V, ?α,x? > 0 ∀α ∈ S}.

For such cones, explicit formulae for the first exit time were given in [8] and involve Pfaffians of

skew-symmetric matrices while [2] covers more general cones. During the last decade, a diffusion

process valued in C, the topological closure of C, was introduced and studied in a series of

papers [4, 11, 12, 13, 14] and generalizes the reflected Brownian motion, that is, the absolute

value of a real Brownian motion. This diffusion, known as the radial Dunkl process, is associated

with a root system and depends on a set of positive parameters called a multiplicity function

often denoted k. The latter is defined on the orbits of the action of the group generated by all

the reflections σα, α ∈ R, the reflection group, and is constant on each orbit. Let X denote

a radial Dunkl process starting at x ∈ C and define the first hitting time of ∂C by

T0:= inf{t,Xt∈ ∂C}.

⋆This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collection

is available at http://www.emis.de/journals/SIGMA/Dunkl operators.html

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2N. Demni

Let l : α ∈ R ?→ k(α) − 1/2 be the so called index function [4]. Then it was shown in [4] that

T0< ∞ almost surely (hereafter a.s.) if −1/2 ≤ l(α) < 0 for at least one α ∈ R, and T0= ∞

a.s. if l(α) ≥ 0 for all α ∈ R. Moreover, the tail distribution of T0may be computed from the

absolute-continuity relations derived in [4]. Two cases are distinguished

• 0 ≤ l(α) ≤ 1/2 for all α ∈ R with at least one α such that 0 < l(α) ≤ 1/2: the radial

Dunkl process with index −l hits ∂C a.s.;

• −1/2 ≤ l(α) < 0 for at least one α and l(β) ≥ 0 for at least one β: X itself hits ∂C a.s.

In the first case, we shall address the problem in its whole generality and specialize our results

to the types A, B, D, while in the second case we shall restrict ourselves to the type B. One of

the reasons is that the second case needs two values of the multiplicity function or equivalently

two orbits. Another reason is that, after a suitable change of the index function, we are led to

the first case, that is, to the case when both indices are positive.

After this panorama, we present another approach which is equivalent to the one used before

and has the merit to express the tail distribution through the W-invariant parts of the so-called

Dunkl–Hermite polynomials1. The last part is concerned with determinantal formulae obtained

for the first hitting time of ∂C by a multi-dimensional Brownian motion. These formulae have

to be compared to the ones obtained in [8].

2First approach

2.1First case

Let us denote by Plxthe law of (Xt)t≥0starting from x ∈ C and of index function l. Let Elx

be the corresponding expectation. Recall that [4, p. 38, Proposition 2.15c], if l(α) ≥ 0 for all

α ∈ R+, then

Recall that the semi group density of X is given by [4, 19]

P−l

x(T0> t) = El

x

?

α∈R+

?α,Xt?

?α,x?

−2l(α)

.

pk

t(x,y) =

1

cktγ+m/2e−(|x|2+|y|2)/2tDW

k

?x

√t,

y

√t

?

ω2

k(y), x,y ∈ C,

where γ =

?

α∈R+

k(α) and m = dimV is the rank of R. The weight function ωkis given by

ωk(y) =

?

α∈R+

?α,y?k(α)

and DW

k

is the generalized Bessel function. Thus,

P−l

x(T0> t) =

?

?

α∈R+

?α,x?2l(α)e−|x|2/2t

cktγ+m/2

?

C

e−|y|2/2tDW

k

?x

?x

√t,

y

√t

??

??

α∈R+

?α,y?dy

=

α∈R+

?α,x

√t?2l(α)e−|x|2/2t

ck

?

C

e−|y|2/2DW

k

√t,y

α∈R+

?α,y?dy

1They were called generalized Hermite polynomials in [20] but we prefer calling them as above to avoid the

confusion with the generalized Hermite polynomials introduced by Lassalle [17].

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First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes3

=

?

α∈R+

?α,x

√t?2l(α)e−|x|2/2t

ck

g

?x

√t

?

,

where

g(x) :=

?

C

e−|y|2/2DW

k(x,y)

?

α∈R+

?α,y?dy.(1)

Our key result is stated as follows:

Theorem 1. Let Tibe the i-th Dunkl derivative and ∆k=

m ?

i=1

T2

ithe Dunkl Laplacian. Define

Jx

k:= −∆x

k+

m

?

i=1

xi∂x

i:= −∆x

k+ Ex

1,

where Ex

1:=

m ?

i=1

xi∂x

iis the Euler operator and the superscript indicates the derivative action.

Then

Jx

k

?e−|y|2/2DW

k(x,y)?= Ey

1

?e−|y|2/2DW

k(x,y)?.

if = ∂x

Proof. Recall that if f is W-invariant then Tx

(see [20]). On the one hand [20]

if and that Tx

iDk(x,y) = yiDk(x,y)

∆x

kDW

k(x,y) =

m

?

i=1

y2

i

?

w∈W

Dk(x,wy) = |y|2DW

k(x,y).

On the other hand,

Ex

1DW

k(x,y) =

?

?

1DW

w∈W

m

?

?x,wy?Dk(x,wy) =

i=1

xiTx

iDk(x,wy) =

?

w∈W

?

m

?

i=1

(xi)(wy)iDk(x,wy)

=

w∈W

w∈W

?w−1x,y?Dk(x,wy)

= Ey

k(x,y),

where the last equality follows from Dk(x,wy) = Dk(w−1x,y) since Dk(wx,wy) = Dk(x,y) for

all w ∈ W. The result follows from an easy computation.

Remark 1. The appearance of the operator Jkis not a mere coincidence and will be explained

when presenting the second approach.

?

Corollary 1. g is an eigenfunction of −Jkcorresponding to the eigenvalue m + |R+|.

Proof. Theorem 1 and an integration by parts give

−Jx

kg(x) = −

?

C

Ey

1

?

e−|y|2/2DW

k(x,y)

? ?

e−|y|2/2DW

α∈R+

?α,y?dy

= −

m

?

m

?

i=1

?

C

yi

?

α∈R+

?α,y?∂y

i

?

k(x,y)

?

dy

=

i=1

?

C

e−|y|2/2DW

k(x,y)∂i

yi

?

α∈R+

?α,y?

dy

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4N. Demni

=

?

C

e−|y|2/2DW

k(x,y)

?

α∈R+

?α,y?

m

?

i=1

1 +

?

α∈R+

αiyi

?α,y?

dy.

The proof ends after summing over i.

?

Remark 2. The crucial advantage in our approach is that we do not need the explicit expression

of DW

k

can be expressed through multivariate hypergeometric series [1,

6, 7], it cannot in general help to compute the function g.

k. Moreover, though DW

2.1.1The A-type

This root system is characterized by

R = {±(ei− ej), 1 ≤ i < j ≤ m},

S = {ei− ei+1, 1 ≤ i ≤ m − 1},

R+= {ei− ej, 1 ≤ i < j ≤ m},

C = {y ∈ Rm, y1> y2> ··· > ym}.

The reflection group W is the permutations group Smand there is one orbit so that k = k1> 0

thereby γ = k1m(m − 1)/2. Moreover, DW

1

|W|DW

k

is given by (see [1, p. 212–214])2

k(x,y) =0F(1/k1)

0

(x,y).

Hence, letting y ?→ V (y) be the Vandermonde function, one writes:

?2l1|W|e−|x|2/2t

P−l

x(T0> t) = V

?x

√t

ck

?

C

e−|y|2/20F(1/k1)

0

?x

√t,y

?

V (y)dy,

where l1= k1− 1/2. Besides, Jkacts on W-invariant functions as

−Jx

k= Dx

0− Ex

1:=

m

?

i=1

∂2,x

i

+ 2k1

?

i?=j

1

xi− xj∂x

i−

m

?

i=1

xi∂x

i.

Finally, since g is W-invariant, then

(Dx

0− Ex

1)g(x) = mm + 1

2

g(x),

g(0) = m!

?

C

e−|y|2/2V (y)dy =

?

Rme−|y|2/2|V (y)|dy.

In order to write down g, let us recall that the multivariate Gauss hypergeometric function

2F(1/k1)

1

(e,b,c,·) (see [3] for the definition) is the unique symmetric eigenfunction that equals 1

at 0 of (see [3, p. 585])

m

?

i=1

zi(1 − zi)∂2,z

i

+ 2k1

?

i?=j

zi(1 − zi)

zi− zj

∂z

i

+

m

?

i=1

[c − k1(m − 1) − (e + b + 1 − k1(m − 1))zi]∂z

i

(2)

2Authors used another normalization so that there is factor

meter denoted there by α is the inverse of k1.

√2 in both arguments. Moreover the Jack para-

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First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes5

associated with the eigenvalue meb. Letting z = (1/2)(1 −x/√b), that is zi= (1/2)(1 −xi/√b)

for all 1 ≤ i ≤ m and e = (m + 1)/2,

c = k1(m − 1) +1

2[e + b + 1 − k1(m − 1)] =b

2+k1

2(m − 1) +m + 3

4

,

then, the resulting function is an eigenfunction of

m

?

i=1

?

1 −x2

i

b

?

∂2,x

i

+ 2k1

?

i?=j

(1 − x2

xi− xj

i/b)

∂x

i−

m

?

i=1

?

b +m + 3

2

− k1(m − 1)

?xi

b∂x

i

and Dx

0− Ex

1is the limiting operator as b tends to infinity. Hence,

Proposition 1. For 1/2 < k1≤ 1,

g(x) = g(0)C(m,k1) lim

b→∞2F(1/k1)

1

?m + 1

2

,b,b

2+k1

2(m − 1) +m + 3

4

,1

2

?

1 −

x

√b

??

,

where

C(m,k1)−1= lim

b→∞2F(1/k1)

1

?m + 1

2

,b,b

2+k1

2(m − 1) +m + 3

4

,1

2

?

.

Remark 3. One cannot exchange the infinite sum and the limit operation. Indeed, expand the

generalized Pochhammer symbol as (see [1, p. 191])

(a)τ=

m

?

i=1

(a − k1(i − 1))τi=

m

?

i=1

Γ(a − k1(i − 1) + τi)

Γ(a − k1(i − 1))

and use Stirling formula to see that each term in the above product is equivalent to

(a + k1(m − 1) + τi)τi

for large enough positive a. Moreover, since J(1/k1)

τ

is homogeneous, one has

J(1/k1)

τ

?1

2

?

1 −

x

√b

??

= 2−pJ(1/k1)

τ

?

1 −

x

√b

?

,|τ| = p.

It follows that

(b)τ

(b/2 + (m − 1)k1/2 + (m + 3)/4)τJ(1/k1)

τ

?1

2

?

1 −

x

√b

??

≈ J(1/k1)

τ

(1)

for large positive b. Thus, the above Gauss hypergeometric function reduces to

1F(1/k1)

0

?m + 1

2

,1

?

.

Unfortunately, the above series diverges since [1]

1F(1/k1)

0

(a,x) =

m

?

i=1

(1 − xi)−a.