Skew-product representations of multidimensional Dunkl Markov processes

www.imstat.org/aihp 2008, Vol. 44, No. 4, 593–611

DOI: 10.1214/07-AIHP108

© Association des Publications de l’Institut Henri Poincaré, 2008

Skew-product representations of multidimensional Dunkl Markov processes

Oleksandr Chybiryakov

Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 175 rue du Chevaleret, F-75013 Paris.

E-mail: oleksandr.chybiryakov@m4x.org

Received 14 September 2005; revised 6 April 2006; accepted 17 October 2006

Abstract. In this paper we obtain skew-product representations of the multidimensional Dunkl processes which generalize the skew-product decomposition in dimension 1 obtained in L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martin- gales.Séminaire de Probabilités XXXIX, 2006. We also study the radial part of the Dunkl process, i.e. the projection of the Dunkl process on a Weyl chamber.

Résumé. Dans cet article nous obtenons des produits semi-directs des processus de Dunkl multidimensionnels qui généralisent ceux obtenus en dimension 1 dans L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martingales. InSéminaire de Probabilités XXXIX, 2006. Nous étudions les processus de Dunkl radiaux qui sont les projections des processus de Dunkl sur une chambre de Weyl.

MSC:60J75; 60J25

Keywords:Dunkl processes; Feller processes; Skew-product; Weyl group

1. Introduction

The study of the multidimensional Dunkl processes was originated in [18] and [20]. They were studied further in [7–

10]. These processes share some important properties with Brownian motion: for example, they are martingales and enjoy the chaotic representation property as well as the time-inversion property. To a Dunkl process we can associate two processes: its Euclidean norm and its radial part. The Euclidean norm of the Dunkl process is in fact a Bessel process and its radial part is a continuous Markov process taking values in a Weyl chamberC. Note that a particular case of the radial part process – Brownian motion in a Weyl chamber – is studied in [2].

It is well known that Brownian motion (and more generally any rotational invariant diffusion) can be decomposed as the skew-product of a certain radial process and a time-changed spherical Brownian motion (see [6,17]). In general, the following question can be raised: suppose given a groupGacting (not necessarily transitively) onRⁿand a Markov process onRⁿ, “invariant” by the action ofG. Does this lead to a certain skew-product decomposition of the Markov process? In this paper we answer this question in the case of the Dunkl processes and the groupW – the group of symmetries ofRⁿ. We also studyX^W – the radial part of the Dunkl process noting some analogies betweenX^W and Bessel processes.

The paper is organized as follows. In Section 2 we introduce some notations about martingale problems and ex- tended infinitesimal generators for Markov processes. In Section 3 we give the definition of the Dunkl process and its radial part. We also present the invariance property of the Dunkl process under the action of the groupO(Rⁿ).

Section 4 contains the study of the radial part of the Dunkl process. We give the condition on the parameters ofX^W which guarantees that it does not hit the walls of a Weyl chamberC. This is done by applying a famous argument due

to McKean (see Problem 7, p. 47 in [16]). As a part of the proof we find harmonic functions forX^Wwhich are analogs of harmonic functions for Bessel processes, i.e. the adequate power or logarithm function. Under the condition that X^Wdoes not hit the walls ofCit can be characterized as the unique solution of a stochastic integral equation or as the unique solution of the corresponding martingale problem (ifX^W hits the walls ofC it can still be characterized as a unique solution of the stochastic integral equation up toT₀– the first time it hits the walls ofC).

In Section 5 we generalize the skew-product decomposition given in [10] to multidimensional Dunkl processes.

From [9] it is known that the Dunkl process can jump only in the directions given by the roots of an associated root system. In Theorem 19 we construct the Dunkl process from its radial part by adding the jumps in these directions one by one. In order to add the jumps we do the time-change then add jumps in the fixed direction and then do the inverse time-change. In order to add the jumps in a fixed direction we use the perturbation of generators technique from [5].

We also see that under a certain invariance condition for the extended infinitesimal generator this perturbation is given by a skew-product with a Poisson process. From Lemma 18 and Theorem 19 we obtain that the Dunkl process can be characterized as the unique solution of a martingale problem. Note a similar result from [20], which is recalled in Theorem 4. Our result holds for a more general class of processes – the two-parameter Dunkl processes (see [10] in dimension 1, [7] and [15]), but we need to impose the condition which guarantees that the radial part of the Dunkl process does not hit the wall ofC.

2. Notations

We will constantly use martingale problems. Therefore, we need to recall here some definitions from [5] which will be used later.

Let(S, d)be a metric space andDS[0,∞)(CS[0,∞)) the space of right continuous (continuous) functions from [0,∞)into (S, d)having left limits.P(S)denotes the set of Borel probability measures on S. For any x∈S, δ_x denotes the element ofP(S)with unit mass atx. LetLbe the space of all measurable functions onS.Ais a linear mapping whose domainD(A)is a subspace ofLand whose rangeR(A)lies inL. TypicallyD(A)will beC^∞_K(S)– the space of infinitely differentiable functions onSwith compact support.

LetX be a measurable stochastic process with values inS defined on some probability space(Ω,F, P ). For ν∈P(S)we say thatXis asolution of theDS[0,∞)martingale problem(A, ν)ifXis a process with sample paths inDS[0,∞),P(X₀∈ ·)=ν(·), and for anyu∈D(A)

u(X_t)−u(X₀)− _t

0

Au(X_s)ds

is an(Ft^X)-martingale, whereFt^X:=F{Xs, s≤t}, where in this paperF{Xs, s≤t}indicates the sigma-field gener- ated by variablesX_s,s≤t. LetUbe an open set ofSandXbe a process with sample paths inDS[0,∞). Define the (Ft^X)-stopping time

τ:=inf{t≥0|X_t∈/UorX_t−∈/U}.

ThenXis asolution of the stoppedDS[0,∞)martingale problem(A, ν, U )ifP (X0∈ ·)=ν(·),X_·=X_·∧τ a.s., and for anyu∈D(A)

u(X_t)−u(X₀)− _t∧τ

0

Au(X_s)ds

is an (Ft^X)-martingale. If there exists a unique solution of a (stopped) martingale problem we will say that the (stopped) martingale problem is well-posed.

In what follows it will be convenient to work with the extended infinitesimal generator (see [14,21], VII). We recall here the definition from ([21], VII).

IfXis a Markov process with respect to(Ft), a Borel functionfis said to belong to the domainDAof theextended infinitesimal generator(orextended generator) if there exists a Borel functiongsuch that, a.s.,_t

0|g(X_s)|ds <+∞

for everyt, and

f (Xt)−f (X0)− _t

0

g(Xs)ds

is a(Ft, P_x)-right-continuous martingale for everyx.

3. Preliminaries

Letx·ydenote the usual scalar product forxandyonRⁿ. For anyα∈Rⁿ\{0},σ_αdenotes the reflection with respect to the hyperplaneH_α⊂Rⁿorthogonal toα. For anyx∈Rⁿ, it is given by

σ_α(x)=x−2α·x α·αα.

For our purposes we need the following definition (see, for example, [19]):

Definition 1. LetR⊂Rⁿ\{0}be a finite set.ThenRis called a(restricted)root system,if 1. R∩Rα= {±α}for allα∈R;

2. σ_α(R)=Rfor allα∈R.

The subgroupW⊂O(Rⁿ)which is generated by the reflections{σ_α |α∈R}is called the Weyl reflection group associated withR.

One can prove that for any root systemRinRⁿ, the reflection groupWis finite and the set of reflections contained inWis exactly{σ_α, α∈R}(see [19]).

Example 2 (Root system of typeA_n−1). Lete₁, . . . , e_nbe the standard basis vectors ofRⁿ,then R=

±(e_i−e_j),1≤i < j≤n is a root system inRⁿ.

Example 3 (Root system of typeB_n). R= {±ei,1≤i≤n,±(ei±ej),1≤i < j≤n}is a root system onRⁿ. Without loss of generality we will suppose that ifR is a root system inRⁿthen for anyα∈R,α·α=2, so that for anyx∈Rⁿ

σ_α(x)=x−(α·x)α. (1)

For any given root systemRtakeβ∈Rⁿ\

α∈RHα, thenR₊= {α∈R|α·β >0}is its positive subsystem. For any α∈Reitherα∈R₊or−α∈R₊. Of course the choice ofR₊is not unique.

From [8,20] theDunkl processesX^(k)are a family of càdlàg Markov processes with extended generatorsLkwhere, for anyu∈C²_K(Rⁿ),Lkuis given by

Lku(x)=1

2 u(x)+

α∈R₊

k(α)

∇u(x)·α

x·α −u(x)−u(σ_αx)

(x·α)² , (2)

wherekis a nonnegative multiplicity function invariant by the finite reflection groupWassociated withR, i.e.k:R→ [0,+∞)andk◦w≡k, for anyw∈W. It is simple to see thatLk does not depend on the choice ofR₊. In what follows suppose thatX₀∈Rⁿ\

α∈RH_α a.s.

The semi-group densities of the Dunkl process with the generator(2)are given by p_t^(k)(x, y)= 1

c_kt^γ+n/2exp

−|x|²+ |y|² 2t

D_k

x

√t, y

√t

ω_k(y), x, y∈Rⁿ, (3)

whereD_k(u, v) >0 is the Dunkl kernel (for the properties of the Dunkl kernel see [19]),γ:=

α∈R₊k(α),ω_k(y)=

α∈R₊|α·y|^2k(α)andc_k=

Rⁿe^−|x|2/2ω_k(x)dx(see [18]).

We will need the following result from [20].

Theorem 4. Let(P_t)_t≥0be the semigroup of the Dunkl process given by its density(3)and letLk be given by(2).Let (X_t)_t≥0be a càdlàg process onRⁿsuch that its Euclidean norm process(X_t)_t≥0on[0,+∞[is continuous.Then the following statements are equivalent.

1. (Xt)t≥0is the Dunkl process associated with the semigroup(Pt)t≥0. 2. For anyf ∈C²(Rⁿ)

M_t^f

t≥0:=

f (X_t)−f (X0)− _t

0 Lkf (X_s)ds

t≥0

is a local martingale.

3. For anyf ∈C²_K(Rⁿ),(M_t^f)_t≥0is a martingale.

Consider now a fixed Weyl chamberC of the root-systemR which is a connected component ofRⁿ\

α∈RH_α. The spaceRⁿ/WofW-orbits inRⁿcan be identified toC, i.e. there exists a homeomorphism φ:Rⁿ/W→ C. Denote X_t^W:=π(X_t^(k))– aradial part of the Dunkl process(or aradial Dunkl process), where

π:=φ◦π₁ (4)

andπ1:Rⁿ→Rⁿ/W denotes the canonical projection. From [8],X^W is a Markov process with extended generator L^Wk u(x)=1

2 u(x)+

α∈R+

k(α)∇u(x)·α

x·α , (5)

for anyu∈C²₀(C), such that ∇u(x)·α=0 forx∈Hα,α∈R₊. The semigroup densities ofX^W are of the form p^W_t (x, y)= 1

c_kt^γ+n/2exp

−|x|²+ |y|² 2t

D_k^W

x

√t, y

√t

ω_k(y), x, y∈ C, (6)

where

D^W_k (u, v)=

w∈W

D_k(u, wv). (7)

From [20] the Dunkl process(X_t)_t≥0is a Feller process and its Euclidean norm(X_t)_t≥0is a Bessel process of indexγ+n/2−1. It is easy to see that(X_t^W)t≥0defined above is also a Feller process.

In order to state the next result, we denote the trajectory of the Dunkl process starting at x by (X^(k,R,x)_t )t≥0. The following proposition is an analog for the Dunkl–Markov processes of the rotational invariance property of the Brownian motion inRⁿ.

Proposition 5. Let(X_t^(k,R,x))_t≥0,x∈Rⁿ be the Dunkl process,O(Rⁿ)– the orthogonal group ofRⁿ.Then for any θ∈O(Rⁿ)

θ X_t^(k,R,x)

t≥0,x∈Rⁿ (d)=

X^(k_t ^{θ,θ R,θ x)}

t≥0,x∈Rⁿ,

wherek_θ:θ Rα→k(^tθ α),^tθis the transpose ofθ,θ R= {θ α, α∈R}.

Proof. It will be convenient to denoteL^R_k the extended generator of(X^(k,R,x)_t )_t≥0,x∈Rⁿ given by (2). By Theorem 4 it is enough to prove that for anyu∈C²(Rⁿ)andx∈Rⁿ

L^{θ R}_kθ u(θ x)=L^R_k(u◦θ )(x).

From (1) one has (u◦θ )

σα(x)

=u

θ x−(α·x)θ α

=u

θ x−(θ α·θ x)θ α

=u

σθ α(θ x)

and

∇(u◦θ )(x)·α=t

θ∇u(θ x)

·α= ∇u(θ x)·θ α.

Then

L^R_k(u◦θ )(x)= 1

2 u(θ x)+

θ α∈θ R₊

k(α)

∇u(θ x)·θ α

θ x·θ α −u(θ x)−u(σ_{θ α}(θ x)) (θ x·θ α)²

=L^{θ R}_kθ u(θ x).

4. Study of the Markov processX^W

Consider the Weyl chamberC= {x∈Rⁿ|x·α >0, α∈R₊}. We now study the radial part of the Dunkl process, i.e.

the Markov process(X_t^W)_t≥0with extended generator given by (5).(X^W_t )_t≥0is a continuous Feller process with its values inC.

Denote

¯

ω_k(x)=

α∈R₊

(α·x)^k(α), (8)

then for anyu∈C²₀(C), such that∇u(x)·α=0 forx∈H_α,α∈R₊, andx∈C L^W_k u(x)=1

2 u(x)+

α∈R₊

k(α)∇u(x)·α x·α

=1

2 u(x)+ ∇u(x)· ∇logω¯_k(x).

The following proposition gives a condition on the functionk:R→R₊in the definition of the extended genera- tor (5), which ensures that the processX^W, such thatX₀^W∈Ca.s., never touches∂C.

Proposition 6. LetX^W be the radial Dunkl process,X₀^W∈Ca.s.,with extended generator given by(5).Suppose that for anyα∈R,k(α)≥¹₂.Define

T₀:=inf

t >0|X^W_t ∈∂C

, (9)

then

T0= +∞ a.s.

Remark 7. Suppose that for anyα∈R, k(α)≥ ¹₂. From Proposition6 P_x(T₀= +∞)=1, whereP_x denotes the law ofX^W started atx∈C.Lety∈∂C.For anyε >0,using(6),one obtains that P_y(X_ε^W∈H_α∩ C)=0.Since

∂C⊂(

α∈RH_α)∩ C P_y

X_ε^W∈C

=1.

Hence,by the same argument as in([13],p. 162), P_y

X_t^W∈C,∀ε < t <+∞

=E_y P_XW

ε

X^W_t ∈C,∀0< t <+∞

=1, where expectationE_yis computed under the lawP_y.Lettingε→0one obtains

P_y(T0= +∞)=1.

Hence Proposition6is true ifX₀^W∈∂C.

In order to prove this proposition we need the following two lemmas.

Lemma 8. Forx∈Cdefine δ(x):=

α∈R₊

(α·x)^1−2k(α), (10)

thenδis harmonic inC forL^W_k given by(5),i.e.L^W_k δ(x)=0for anyx∈C.

Remark 9. It is important in the following proof that the functionkin the definition ofL^W_k is constant on the orbits ofRunder the action of the associated Weyl group.

Proof of Lemma 8. Fori=1, . . . , mlet Rⁱ be the orbits ofR under the action of the associated Weyl groupW. DenoteR₊ⁱ =Rⁱ∩R₊. Then for any{i₁, . . . , i_l} ⊂ {1, . . . , m}

Rˆ₊=R₊ⁱ¹∪ · · · ∪Rⁱ₊^l

is also a positive subsystem of the root systemRˆ=Rⁱ¹∪ · · · ∪R^il inRⁿ (the two conditions in Definition 1 forRˆ are easily verified). Without loss of generality it is enough to consider the irreducible case and in that case there are at most two orbits. Hence, it is enough to prove the lemma form=2. Denote

π_i(x)=

α∈Rⁱ₊

(α·x).

Sincek:R→R₊is constant onR₊ⁱ define k_i:=k(α), α∈R₊ⁱ ,

then

¯

ω_k(x)=π₁^k1(x)π₂^k2(x) and

δ(x)=π₁^1−2k1(x)π₂^1−2k2(x).

We want to prove thatL^W_k δ(x)=0 for anyx∈C. From Theorem 4.2.6 in ([4], p. 140), for any root systemR, if π(x)=

α∈R₊(α·x), then π(x)=0.

Note that

π^1−2k+2

∇π^1−2k· ∇logπ^k

=0. (11)

Indeed

π^1−2k= ∇ · ∇π^1−2k= ∇ ·

(1−2k)π^−2k∇π

=(1−2k)

−2kπ^−2k−1(∇π· ∇π )+π^−2k π

= −2k(1−2k)π^−2k−1(∇π· ∇π ).

On the other hand 2

∇π^1−2k· ∇logπ^k

=2k(1−2k)π^−2k−1(∇π· ∇π )

and the result(11)follows.

Denote

¯

π:=π₁^1−2k1 and

ˆ π:=π₁^k1. We need to check that

A=

π₁^1−2k1π₂^1−2k2 +2

∇

π₁^1−2k1π₂^1−2k2

· ∇log

π₁^k1π₂^k2

=0 or

π π¯ ₂^1−2k2 +2

∇

¯ π π₂^1−2k2

· ∇log ˆ π π₂^k2

=0.

One has

π π¯ ₂^1−2k2

=π₂^1−2k2 π¯ + ¯π π₂^1−2k2

+2

∇ ¯π· ∇π₂^1−2k2 and

∇

¯

π π₂^1−2k2

· ∇log ˆ π π₂^k2

=

¯

π∇π₂^1−2k2+π₂^1−2k2∇ ¯π

·

∇logπ₂^k2+ ∇logπˆ

=

¯

π∇π₂^1−2k2· ∇logπ₂^k2 +

¯

π∇π₂^1−2k2· ∇logπˆ +

π₂^1−2k2∇ ¯π

· ∇logπ₂^k2 +

π₂^1−2k2∇ ¯π

· ∇logπˆ , then

A=π₂^1−2k2 π¯+2π₂^1−2k2(∇ ¯π· ∇logπ )ˆ + ¯π π₂^1−2k2

+2π¯

∇π₂^1−2k2· ∇logπ₂^k2 +2

∇ ¯π· ∇π₂^1−2k2 +2

¯

π∇π₂^1−2k2

· ∇logπˆ +2

π₂^1−2k2∇ ¯π

· ∇logπ₂^k2 . But

¯

π+2(∇ ¯π· ∇logπ )ˆ =0 and

π₂^1−2k2 +2

∇π₂^1−2k2· ∇logπ₂^k2

=0, hence

A=2

∇ ¯π· ∇π₂^1−2k2 +2

¯

π∇π₂^1−2k2

· ∇logπˆ +2

π₂^1−2k2∇ ¯π

· ∇logπ₂^k2

=2π π¯ ₂^1−2k2

∇logπ¯ · ∇logπ₂^1−2k2 +

∇logπ₂^1−2k2· ∇logπˆ +

∇logπ¯ · ∇logπ₂^k2 . But one has

∇logπ¯· ∇logπ₂^1−2k2

=

∇logπ₁^1−2k1· ∇logπ₂^1−2k2

=(1−2k1)(1−2k2)(∇logπ₁· ∇logπ₂), ∇logπ₂^1−2k2· ∇logπˆ

=k₁(1−2k2)(∇logπ₁· ∇logπ₂), ∇logπ¯· ∇logπ₂^k2

=k2(1−2k1)(∇logπ1· ∇logπ2)

and

(∇logπ1· ∇logπ2)= 1 π1π2

(∇π1· ∇π2)

= 1 2π1π2

(π1π2)−π1 π2−π2 π1

=0.

HenceA=0 and the lemma is proven.

In the same way one can prove the following lemma.

Lemma 10. Forx∈Cdefine δ(x)¯ :=

α∈R₊,k(α)=1/2

(α·x)^1−2k(α)log

α∈R₊,k(α)=1/2

(α·x), (12)

thenδ¯is harmonic inC forL^W_k given by(5).

Proof. Suppose thatk(α)=¹₂ for anyα∈Rthen for anyu∈C²(C),x∈C L^W1/2u(x)=1

2

u(x)+

∇u(x)· ∇logπ(x) , whereπ(x)=

α∈R₊(α·x). Then logπ(x)= − 1

π²(x)

∇π(x)· ∇π(x) + 1

π(x) π(x)= − 1 π²(x)

∇π(x)· ∇π(x)

and

logπ(x)+

∇logπ(x)· ∇logπ(x)

=0.

Denote

¯

π₁:=

α∈R₊,k(α)=1/2

(α·x)^1−2k(α)=

1≤i≤m,k_i=1/2

π_i^1−2ki, ˆ

π₁:=

α∈R₊,k(α)=1/2

(α·x)^k(α)=

1≤i≤m,k_i=1/2

π_i^ki,

¯

π2:=log

α∈R₊,k(α)=1/2

(α·x), ˆ

π2:=

α∈R₊,k(α)=1/2

(α·x)^1/2,

thenω¯k= ˆπ1πˆ2andδ¯= ¯π1π¯2. In order to prove Lemma 10 one needs to check that (π¯1π¯2)+2

∇(π¯1π¯2)· ∇log(πˆ1πˆ2)

=0, but it is equal to

¯ π2

π¯1+2

∇ ¯π1· ∇log(πˆ1) + ¯π1

π¯2+2

∇ ¯π2· ∇log(πˆ2) +2(∇ ¯π1· ∇ ¯π2)+2π¯2(∇ ¯π1· ∇logπˆ2)+2π¯1(∇ ¯π2· ∇logπˆ1)

and in the same way as in the proof of Lemma 8 one can see that it is equal to zero.

Using Lemmas 8 and 10 one obtains the following result, which will lead to a presentation ofX^W (see Corollaries 13 and 14).

Lemma 11. Letω¯_kbe defined by(8)andk(α)≥¹₂for anyα∈R.Suppose thatX₀∈Ca.s.,then there exists a unique solutionXof the stochastic integral equation

X_t=X₀+β_t+ _t

0

∇logω¯_k(X_s)ds, (13)

where(βt)t≥0is a Brownian motion onRⁿ.FurthermoreP(∀t >0, Xt∈C)=1.

Proof. We will follow the proof of the similar argument given in ([1] Lemma 3.2). Since the function∇logω¯_k∈ C^∞(C)Eq. (13) has a unique (strong) maximal solution inC, defined up to timeζ, whereζ is an explosion time or the exit time fromC.

Case 1(k(α)=¹₂ for anyα∈R). From Lemma 8 the functionδdefined by (10) is harmonic and positive onC.

By Ito’s formula one deduces that{δ(Xt), t < ζ}is a positive local martingale, thus it converges a.s. whent→ζ. But δ= +∞on∂C, thereforeXt → +∞whent→ζ. On the other hand by Ito’s formula fort < ζ

X_t²= X0²+2 t

0

(X_s·dβs)+2 t

0

γds+t n

= X₀²+2 _t

0

X_sdβ˜_s+t (n+2γ ), whereγ=

α∈R₊k(α)andβ˜_t =_t

0X_s⁻¹(X_s·dβs),t < ζ is a real-valued Brownian motion up to timeζ. This shows thatX_t²is the square of a(n+2γ )-dimensional Bessel process up to timeζ (started atX₀²>0 a.s.).

SinceX_t²→ +∞, by standard results (see [21], XI.1)ζ = +∞a.s. This implies thatT₀= +∞a.s., whereT₀is defined by (9).

Case 2(There existsα∈Rsuch thatk(α)=¹₂). From Lemma 10 and Ito’s formula,{¯δ(X_t), t < ζ}is a continuous local martingale. Let At := ¯δ(X),δ(X)¯ t for t < ζ andτt :=inf{s≥0|As =t}, then by Theorem 1.7 in ([21], Chapter V)B_t= ¯δ(X_τt)− ¯δ(X₀)will be a real-valued Brownian motion up to timeA_ζ. Ifζ <+∞we have already seen thatX_tcannot tend to+∞. Hence, ift→ζ, eitherδ(X¯ _t)tends to+∞or to−∞. Therefore, eitherB_t tends to+∞or to−∞, whent→A_ζ, but that is impossible for Brownian motion, either becauseA_ζ <+∞, or because

A_ζ= +∞andB_t≥0(B_t≤0)infinitely often whent→ +∞.

Proof of Proposition 6. Let C_m:=

x∈Cx·α > 1

m, α∈R,x< m

. (14)

Takeg_m∈C^∞(Rⁿ)such thatg_m≡logω¯_k onC_m+1andg_m≡0 onRⁿ\C_m+2. For anyu∈C^∞_K(Rⁿ), defineLmby Lmu(x)=1

2 u(x)+

∇u(x)· ∇g_m(x) .

Let(X_t^W)t≥0be the radial part of the Dunkl process andτˆm:=inf{s >0|X^W_s ∈/Cm}. For anyf ∈C^∞_K(Rⁿ)there existsf˜∈C^∞_K(C)such thatf˜≡f onCm+1. Then from(5)

f˜ X_t^W_{∧ ˆτ}

m

− ˜f X₀^W

− _{t∧ ˆτm}

0

L^W_k f˜ X_s^W

ds=f X^W_{t∧ ˆτ}

m

−f X^W₀

− _{t∧ ˆτm}

0

Lmf (X^W_s )ds (15) is a martingale.

LetXbe the solution of(13)such that P(X₀∈ ·)=P

X^W₀ ∈ ·

= ¯ν(·) (16)

forν¯∈P(C). Letτ¯_m:=inf{s >0|X_s∈/C_m}. Then from Lemma 11 one deduces that

¯

τ_m→ +∞ a.s. (17)

Furthermore from(13)and Ito’s formula one deduces that f (Xt∧ ¯τ_m)−f (X0)−

_{t∧ ¯τm}

0 Lmf (Xs)ds (18)

is a martingale forf∈C^∞_K(Rⁿ).

By Theorem 3.3 in ([5], p. 379) for anyν∈P(Rⁿ)the D_Rⁿ[0,∞)martingale problem(Lm, ν)is well posed.

Then by Theorem 6.1 in ([5], p. 216) for eachν∈P(Rⁿ)the stoppedD_Rⁿ[0,∞)martingale problem(Lm, ν, Cm)is well-posed. But from (15), (18) and (16)X_{·∧ ˆ}^W_τ

m andX_{·∧ ¯}τ_mare solutions of(Lm,ν, C¯ m). Hence X^W_{t∧ ˆτ}

m

t≥0 (d)=

X_{t∧ ¯τm}

t≥0 (19)

andP(τˆ_m< t )=P(τ¯_m< t ), for anyt >0. Hence from (17) ˆ

τ_m→ +∞ a.s.

and passing to the limit asm→ ∞in(19)one obtains that X^W_t

t≥0

(d)=(Xt)t≥0

and

T0= +∞ a.s.,

whereT0is defined by (9).

SinceX_t^W=π(Xt), whereXt is the Dunkl process one obtains Corollary 12. Let(Xt)t≥0be the Dunkl process,such thatX0∈Rⁿ\

α∈R₊Hα a.s.,with extended generator given by(2).Suppose that,for anyα∈R,k(α)≥¹₂.Define

T₀:=inf

t >0X_t∈

α∈R₊

H_α

,

then

T0= +∞ a.s.

The proof of Proposition 6 leads to

Corollary 13. Let(X^W_t )t≥0be the radial Dunkl process,such thatX0∈Ca.s.,with extended generator given by(5).

Suppose that for anyα∈R k(α)≥ ¹₂.ThenX^W is the unique solution to the stochastic integral equation Xt=X0+βt+

_t

0 ∇logω¯k(Xs)ds,

where(β_t)_t≥0is a Brownian motion inRⁿandω¯_kis given by(8).This solution is strong in the sense of([12],IV).

Denoteν(·):=P(X₀^W∈ ·),thenX^Wis a unique solution to theCC[0,∞)martingale problem(Lˆ^W_k , ν),whereLˆ^W_k is the restriction ofL^W_k onC^∞_K(C).

Suppose now that there existsα∈R such thatk(α) < ¹₂ and consider the radial Dunkl process. The preceding results extend to this case if one works tillT₀– the first time the process hits the walls of the Weyl chamber. Loosely speaking, here,T₀is considered as if it were an “explosion time.” One gets the following corollary.

Corollary 14. Let(X^W_t )t≥0be the radial Dunkl process,such thatX0∈Ca.s.,with extended generator given by(5).

Suppose that there existsα∈Rsuch thatk(α) <¹₂.LetT0be defined by(9).Then(X^W_t , t < T0)is the unique solution to the stochastic integral equation

X_t=X0+β_t+ _t

0 ∇logω¯_k(X_s)ds, t < T0,

where(β_t)_t≥0is a Brownian motion inRⁿandω¯_kis given by(8).This solution is strong in the sense of([12],IV).

Proof. Is analogous to the proof of Lemma 11 and Proposition 6.

Remark 15. One can show that the result of Corollary14is true fort≥0andX^W“reflects instantly” when it reaches

∂C(see[3]).

5. Skew-product decomposition of jumps

LetXbe a Dunkl process,π:Rⁿ→ Cbe given by(4). SinceXis càdlàg andπ(X)is continuous it will be useful to introduce the spaceD^π_Rn[0,∞)defined by

D^π_Rn[0,∞):=

ω∈D_Rⁿ[0,∞)|π◦ω∈C_C[0,∞)

. (20)

LetE:=Rⁿ\

α∈RH_α. Denote D^π_E[0,∞):=

ω∈DE[0,∞)|π◦ω∈CC[0,∞) .

We will say thatXis asolution of theD^π_E[0,∞)(D^π_Rn[0,∞))martingale problem(A, ν)ifXis a process with sample paths inD^π_E[0,∞)(D^π_Rn[0,∞)) andXis a solution of theDE[0,∞)(D_Rⁿ[0,∞)) martingale problem(A, ν).

Letλ >0 andα∈R₊. LetD(A)⊂C^∞_K(E)andR(A)⊂C^∞_K(E). Suppose that for anyν∈P(Rⁿ)there exists X – a solution of theD^π_Rn[0,∞)martingale problem for(A, ν). We will need the following construction from ([5], p. 256). LetΩ=_∞

k=1(D_Rⁿ[0,∞)× [0,∞)) and(X_k, Δ_k)denote the coordinate random variables. DefineGk= F(X_l, Δ_l: l≤k)andG^k=F(X_l, Δ_l: l≥k). Then there is a probability distribution onΩ such that for eachk X_k is a solution of theD^π_Rn[0,∞)martingale problem forA,Δ_k is independent ofF(X₁, . . . , X_k, Δ₁, . . . , Δ_k−1)and exponentially distributed with parameterλ, and forA₁∈GkandA₂∈G^k+1,

P(A₁∩A₂)=E IA1P

A₂|X_k+1(0)=σ_α

X_k(Δ_k)

(21) andP(X1(0)∈ ·)=ν(·). Defineτ0=0,τ_k=_k

i=1Δ_i, andN_t=kforτ_k≤t < τ_k+1. Note thatNis a Poisson process with parameterλ. Define

Y (t )=X_k+1(t−τ_k), τ_k≤t < τ_k+1, (22)

andFt :=F_t^Y∨F_t^N.

Notation 16. We will denoteY in(22)byX∗αN.

Lemma 17. LetD(A)⊂C^∞_K(E)andR(A)⊂C^∞_K(E).Suppose that for anyν∈P(E)there exists a solutionXof theD^π_E[0,∞)martingale problem(A, ν).Then for anyν∈P(E)there exists a Poisson processN with parameter λsuch thatY =X∗αN,where∗α is defined by(22),is a solution of theD^π_E[0,∞)martingale problem(Aλ,α, ν), where for anyu∈D(A)andx∈E

Aλ,αu(x)=Au(x)+λ

u(σαx)−u(x) . Furthermore if for anyu∈D(A)andx∈E

A(u◦σ_α)(x)=Au σ_α(x)

, (23)

then there exists a solution of(Aλ,α, ν)given by (Y_t)_t≥0:=

σ_α^NtX_t

t≥0,

whereNis a Poisson process with parameterλindependent ofX.

Proof. We follow the proof of Proposition 10.2 in ([5], p. 256). Since E is not complete one shall consider the D^π_Rn[0,∞)martingale problem (A,ν). For any˜ ν˜∈P(Rⁿ)letX˜0 be such thatP(X˜0∈ ·)= ˜ν(·). If ν(E)˜ =0, then X˜_t:= ˜X₀, for anyt≥0, is a solution of theD^π_Rn[0,∞)martingale problem(A,ν). If˜ ν(E) >˜ 0 and(X_t)_t≥0is a solu- tion of theD^π_E[0,∞)martingale problem(A,ν)¯ withν(¯ ·)= ˜ν(·)/ν(E),˜ ν¯∈P(E), thenX˜_t:=X_tI_{{ ˜X0∈E}}+ ˜X0I_{{ ˜X0∈/E}} is a solution of theD^π_Rn[0,∞) martingale problem (A,ν). Taking˜ μ(x,·)=δ_σα(x)(·) and using Proposition 10.2 in ([5], p. 256) one obtains that Y = ˜X∗αN is a solution of the D_Rⁿ[0,∞) martingale problem(Aλ,α,ν). Let˜

˜

ν=ν∈P(E), thenX˜ ≡Xis a solution of theD^π_E[0,∞)martingale problem(A, ν). Furthermore, from (21) P

k≥1

Xk+1(0)=σα

Xk(Δk)

=1

and P

k≥1

Y (τ_k)=σ_α

Y (τ_k−)

=1.

Hence P

k≥1

π Y (τ_k)

=π

Y (τ_k−)

=1.

Since X_k has paths in D^π_E[0,∞) for any k≥1 π(Y )is a.s. continuous and Y is a process with sample paths in DE[0,∞). HenceY is a solution of theD^π_E[0,∞)martingale problem(Aλ,α, ν).

Suppose now that(23)is true. LetN be a Poisson process independent ofX andτ_k :=inf{s≥0|N_s=k}. Let Yt=σα^NtXt. Note that for anyu∈C^∞_K(E)

M_t^u:=u(X_t)−u(X₀)− _t

0

Au(X_s)ds is a(Ft^X)-martingale.

Since(τi)i≥0are independent fromM^u, for anyk≥0 u

X

(t∨τ2k)∧τ2k+1

−u X(τ2k)

−

_{(t∨τ2k)∧τ2k+1}

τ2k

Au(Xs)ds (24)