Brownian Motion, Reflection Groups and Tanaka Formula

Brownian Motion, Reflection Groups and Tanaka Formula

NIZARDEMNI(*) - DOMINIQUELEÂPINGLE(**)

ABSTRACT- In the setting of finite reflection groups, we prove that the projection of a Brownian motion onto a closed Weyl chamber is another Brownian motion normally reflected on the walls of the chamber. Our proof is probabilistic and the decomposition we obtain may be seen as a multidimensional extension of Ta- naka's formula for linear Brownian motion. The paper is closed with a descrip- tion of the boundary process through the local times of the distances from the initial process to the facets.

1. Introduction

The study of stochastic processes in relation to root systems has known a considerable growth during the last decade (see [4], [5] and references therein). In this paper, a reflection groupWassociated with a root systemR is acting on a finite-dimensional Euclidean spaceV, and we project aV-va- lued Brownian motion onto the positive closed Weyl chamberC. In fact, each x2Vis conjugated to a uniquey2Cso that the projectionpmaps vectors in V into their orbits under the action ofW, namelyp:V !V=W. Using the terminology of Dunkl processes,p(X) is often called theW-invariant or radial part of the initial Brownian motionX([4], [5]), yet should not be confused with the Brownian motion in a Weyl chamber considered in [1] which behaves as a Brownian motion conditioned to stay in the open chamberC. The main result of this paper is the explicit semimartingale decomposition of the projected processp(X). Actually, the latter process is shown to behave as a V-valued Brownian motion in the interior ofCand to reflect normally at the

(*) Indirizzo dell'A.: Universite de Rennes I, IRMAR, F-35042 Rennes E-mail: nizar.demni@univ-rennes1.fr

(**) Indirizzo dell'A.: Universite d'OrleÂans, MAPMO-FDP, F-45067 OrleÂans E-mail: dominique.lepingle@univ-orleans.fr

boundary@C. The proof uses two ingredients of algebraic and probabilistic nature respectively : the expression ofpas the composition of a finite number of one dimensional projections and Tanaka formula for the absolute value of the linear Brownian motion. Another combination of tools from both re- flection groups theory and probability theory leads to a precise description of the boundary process by means of the local time of the distance fromXto the facets.

The paper is organized as follows. Section 2 is devoted to basic facts on the theory of reflection groups and to a thorough study of reflections acting on facets of a root system. An illustration of this study is given in Section 3 where we investigate the evolution of a simple random walk on a triangular lattice in the plane with the dihedral groupD3operating on the lattice. We prove the main result in Section 4. Finally, Section 5 is devoted to the description of the boundary process. Note that the problem has previously been tackled in [3] and taken up again in [5], however the proof of Theorem 7.1 displayed in [5] p. 216 and quoted from [3] is not correct. Moreover, our approach is different and more general since we do not assumeCto be a simplex, an assumption that ceases to hold for some root systems.

2. Root systems and reflected facets

We give a short introduction to the theory of reflection groups and refer to [6] for more details. LetVbe an Euclidean space with dimensionN. For a2Va unit vector we denote bys_athe orthogonal reflection with respect to the hyperplaneH_a:ˆ fx2V:a:xˆ0g:

sa(x) :ˆ x 2 (a:x)a: …1†

A finite subsetRof unit vectors inVis called areduced root systemif for all a2R

R\Ra ˆ fa; ag; s_a(R) ˆ R:

The groupWO(N) which is generated by the reflectionsfsa;a2Rgis called the reflection group associated with R. Each hyperplane

H_b:ˆ fx2V :b:xˆ0gwithb2Vn [_a2RH_aseparates the root systemR

intoR‡andR . Such a setR‡is called apositive subsystemand defines the positive Weyl chamber Cby

C:ˆ fx2V: a:x>0 8a2R‡g:

A subsetSofR‡is calledsimpleifSis a vector basis for span(R) and if each positive root is a linear combination of elements ofSwith coefficients being all nonnegative. The elements ofSare calledsimple roots. Such a subset exists and is unique onceR_‡is fixed. Moreover (Theorem 1.5 in [6]),Wis generated by the reflectionsfs_a;a2Sg. The positive Weyl chamber may also be written

Cˆ fx2V : a:x>0 8a2Sg: …2†

Its closure is called theclosed Weyl chamber

Cˆ fx2V : a:x0 8a2Sg:

With any partitionR‡ˆI‡[I₀[I we associate thefacet Fˆ fx2V: a:x>0 8a2I_‡;b:xˆ0 8b2I₀;g:x50 8g2I g: We note (I_‡(F);I₀(F);I (F)) the partition corresponding to the facetF. We callFthe set of nonempty facets withCardfI₀g ˆ1. For a facetF2 Fwith I₀(F)ˆ fbg,FH_bandH_bis called thesupportofF. The following result is an easy consequence of the isometry property ofsaand of the fact that sa(R‡n fag)ˆR‡n fag(Proposition 1.4 in [6]).

LEMMA1. For F2 F anda2S, s_a(F)2 Fand I0(sa(F)) ˆ fag if I0(F)ˆ fag

ˆ s_a(I₀(F)) if I₀(F)6ˆ fag: For F2 F and w2W, w(F)2 F and

I0(w(F)) ˆ w(I0(F)) if w(I0(F))2R‡

ˆ w(I0(F)) if w(I0(F))2R :

For any a2S we write F_a for the facet F associated with I₀ˆ fag;I ˆ ;:

F_aˆ fx2V: a:xˆ0;b:x>0 8b2Sn fagg:

It is aface([2], p. 61) of the chamberCwith support thewall Ha. For any simple roota2Swe introduce the mappingradefined onV by

ra(x) :ˆ x if a:x0

:ˆ sa(x) if a:x0:

A concise formula is

ra(x)ˆ x‡ 2 (a:x) a: …3†

Let w2W, and let wˆs_al s_a1 be a reduced decomposition of w wherea_i;iˆ1;. . .;lare simple roots andlˆl(w) is the length ofw. From Theorem 2.4 in [1] and a result of Matsumoto ([2], Ch.I V, No.1.5, Prop. 5) we know that the operatorral ra1 depends only onwand not on a par- ticular reduced decomposition. We denoterw this operator. Of particular interest is the operatorr_w0wherew₀denotes the (unique) longest element in W: l(w₀)ˆCard(R_‡). The following lemma follows immediately from Corollary 2.9 in [1] and plays a key role in the proof of our main result.

LEMMA2. If w0is the longest element in W, then rw0takes values in the closed Weyl chamber C and for any x2V,p(x):ˆr_w0(x)is the unique point of the orbit W:x which lies in C.

The following lemmas shed some light on the action ofraandp.

LEMMA3. Leta2S.

(1) x2 [b2R‡Hb()ra(x)2 [b2R‡Hb: (2) Let F2 F with support H_b.

(a) Ifa2I‡(F), then ra(F)ˆF.

Iffag ˆI0(F), then ra(F)ˆFandaˆb.

Ifa2I (F), then r_a(F)2 F and r_a(F)H_gwithgˆs_a(b)2R_‡. (b) Ifa2I‡(F), then r_a¹(F)ˆF[sa(F).

Iffag ˆI₀(F), then r_a¹(F)ˆFandaˆb.

Ifa2I (F), then r_a¹(F)ˆ ;.

PROOF. (1) From the isometry property ofsa2O(N) we derive that for x2V andb2R‡

b:x ˆ s_a(b):s_a(x) s_a(b):x ˆ b:s_a(x) :

From Proposition 1.4 in [6] we know thats_a(R_‡n fag)ˆR_‡n fag. Both implications)and(then follow.

(2) (a) The proofs of both the first and the second assertions are straightforward. Leta2I (F),x2F andd2R‡. Then,

d:x ˆ sa(d):sa(x) ˆ sa(d):ra(x):

Therefore,

if dˆa; a:ra(x)>0 if dˆb; s_a(b):r_a(x)ˆ0 if d2I_‡(F); s_a(d):r_a(x)>0 if d2I (F)n fag; s_a(d):r_a(x)50

and using again Proposition 1.4 in [6] we get a new partition (I‡ˆsa(I‡(F))[ fag;I₀ˆ fsa(b)g;I ˆsa(I (F))n fag) corresponding to a new facetr_a(F):

(b) The proofs are straightforward or similar to the proofs in (a). p LEMMA4. (1) x2 [b2R‡Hb()p(x)2@C:

(2) If F2 F, there existsa2S such thatp(F)ˆF_a. (2) Fora2S,p ¹(F_a)ˆ [_F2FaFwhere we set

F_a:ˆ fF2 F :p(F)ˆF_ag:

PROOF. The first assertion is a consequence of (1) of Lemma 3 and of the propertyp(V)ˆC. We also obtain from (2.a) of Lemma 3 thatp(F)2 F and it follows there existsa2Ssuch thatp(F)ˆF_a. Fora2Swe deduce from (2.b) of Lemma 3 thatp ¹(F_a) is the union of facets fromF. Exactly all

facets inFaare involved. p

3. Random walk on a triangular lattice

TakeVto be the Euclidean planeR². To each (i;j)2Z²we associate a vertex in the plane with coordinates

x_(i;j) ˆi‡1

2 j y_(i;j) ˆ

3 p

2 j:

The simple random walk (Z_n)_n0 on this lattice T is a Markov chain with uniform transition probability p((i;j);(k;l))ˆ1=6 on the six nearest neighbors (k;l) of the vertex (i;j). We now consider the dihedral group D3 consisting of six orthogonal transformations that preserve T. We take aˆ(0;1) and bˆ( 

p3

=2; 1=2) to be the simple roots, and gˆ( 

p3

=2;1=2) to be the third positive root. The discrete positive Weyl chamber C_dis is given by f(i;j)2Z²: i>0;j>0g.

We note thatpˆrarbraˆrbrarb. It is easily seen that (p(Zn))_n0 is a Markov chain on the set of verticesf(i;j)2T:i0;j0gwith transition probability

q((i;j);(k;l))ˆp((i;j);(k;l)) ifi>0;j>0 q((i;0);(i;1))ˆq((i;0);(i 1;1))ˆ2q((i;0);(i1;0)) ˆ 1

3 ifi>0 q((0;j);(1;j))ˆq((0;j);(1;j 1))ˆ2q((0;j);(0;j1)) ˆ 1

3 ifj>0 q((0;0);(0;1))ˆq((0;0);(1;0))ˆ1

2 :

This is exactly what we should call the simple random walk normally re- flected on the walls of the Weyl chamberC_dis.

The examination of a simple random walk on a square or hexagonal lattice would lead us to the same conclusion.

4. Brownian motion in a Weyl chamber

We consider a probability space (V;A;P) endowed with a right-con- tinuous filtration (At)_t0 and a V-valued Brownian motion X with de- composition

X_t ˆ X₀‡B_t :

We are interested in the continuous processp(X). It can be seen as aW- invariant Dunkl process of zero multiplicity function. As such it is a Markov process with values inCand semi-group density ([5], p. 216):

pt(x;y)ˆ 1 c₀t^N=2 exp

jxj²‡ jyj² 2t

X

w2W

exp 1

tx:w(y)

wherec₀is a normalizing constant. We remark that for any fixedy2Cthe functionp_t(x;y) is a solution to the heat equation with Neumann boundary conditions:

@_tp_tˆ1

2Dp_t inC

@p_t

@n ˆ0 on@C

where @p_t=@n is any outward normal derivative of p_t. Thus there is no surprise when asserting thatp(X) should be a Brownian motion normally reflected on the boundary of C. The aim of this section is to obtain a de- composition of the continuous semimartingalep(X). We first investigate the action of the operator ra on continuous V-valued semimartingales. To proceed, recall that ifZis a continuous real semimartingale with local time L(Z) at 0, then Tanaka formula shows thatZ is still a semimartingale that decomposes as:

Z_t ˆZ₀ Z^t

0

1_fZs0gdZs‡1 2Lt(Z): …4†

LEMMA5. Let Y be a continuous V-valued semimartingale with Doob decomposition

Y_tˆY₀‡C_t‡A_t

where C is a Brownian motion and A is a continuous process of finite variation, both vanishing at time 0. Then Y⁰ˆra(Y)has a similar de- composition

Y_t⁰ˆY₀⁰‡C⁰_t‡A⁰_t where Y₀⁰ ˆra(Y0), C⁰is a Brownian motion given by

C_t⁰ˆ Z^t

0

O⁰_s:dC_s

with O⁰ a O(N)-valued process, and the finite variation process A⁰ is given by

A⁰_t ˆ At 2 Z^t

0

1fa:Ys0gd(a:As)a ‡Lt(a:Y)a:

PROOF. We combine (3) with (4). Then

ra(Yt) ˆ Yt‡2(a:Y0) a 2 Z^t

0

1_fa:Ys0gd(a:Ys)a‡Lt(a:Y)a

ˆ r_a(Y₀)‡C_t 2 Z^t

0

1_fa:Ys0gd(a:C_s)a‡A⁰_t:

The processO⁰defined by

O⁰_s:uˆu 21_fa:Ys0g(a:u)a foru2V satisfies

(O⁰_s:u):(O⁰_s:v)ˆu:v for anyu;v2V. It follows ([7], Ch. IV, Ex. 3.22) that

C_t 2 Z^t

0

1_fa:Ys0gd(a:C_s)aˆ Z^t

0

O⁰_s:dC_s

defines aV-valued Brownian motion. p

We are now in a position to prove our main result. It states that a Brownian motion projected onto the closed Weyl chamber is another Brownian motion normally reflected on the faces of the chamber.

THEOREM6. Let X_tˆX₀‡B_tbe a V-valued Brownian motion. Then

p(Xt)ˆp(X0)‡ Z^t

0

Os:dBs‡1 2

X

a2S

Lt(a:p(X))a

where O is a O(N)-valued process.

PROOF. a) Let pˆr_al r_a1. We proceed by iteration, setting X_t⁰:ˆX_t;B⁰_t :ˆB_t;A⁰_t :ˆ0 and X^j_t :ˆr_aj(X_t^{j 1}) for jˆ1;. . .;l. If we de-

compose

X_t^jˆX₀^j‡B^j_t‡A^j_t; then Lemma 5 yields

X₀^j ˆ r_aj(X₀^{j 1})

B^j_t ˆ Z^t

0

O_s^j:dB_s^{j 1}

A^j_t ˆ A^j_t ¹ 2 Z^t

0

1_faj:X^{j 1}

s 0gd(a_j:A^j_s ¹)a_j‡Lt(a_j:X^{j 1})a_j;

…5†

withO^jaO(N)-valued process. Setting O:ˆO^l O¹

we obtain the Brownian term in the decomposition ofp(X)ˆX^l.

b) From the recurrence formula (5) we deduce thatdA^l_tis supported by [^l_jˆ1fX^j_t ¹2H_ajgwhich is included infX_t 2 [_b2R‡H_bgby (1) of Lemma 3.

Moreover, since one-points sets are polar sets for the planar Brownian motion, we remark that for any t>0, the Brownian motion X does not meet facets withCardfI0g>1. Therefore

1_{fXt2[b2R‡Hbg}ˆX

F2F

1_fXt2Fg a.s.

and we only need to study the sequence (A^j)jˆ1;...;l on the setsfXt2Fg for anyF2 F. SetF0:ˆFandFj:ˆraj(Fj 1) forjˆ1;. . .;l. LetHb_j be the support of the facet F_j. We will prove by induction that for any jˆ0;. . .;l

1_fXt2FgdA^j_tˆdL^j_tb_j …6†

whereL^jis an increasing process. This is true forjˆ0 withL⁰_t ˆ0. Assume this is true forj 1 and use (5). Then

1_fXt2FgdA^j_t

ˆ1fXt2Fg[dL^j_t ¹b_{j 1} 21_faj:X^{j 1}

s 0g(aj:b_{j 1})dL^j_t ¹aj‡dLt(aj:X^{j 1})aj]:

We apply Lemma 3.

Ifaj:x>0 forx2Fj 1, thenFjˆFj 1,b_j ˆb_{j 1} 6ˆajand 1fXt2FgdA^j_t ˆdL^j_t ¹b_{j 1}ˆdL^j_t ¹b_j: Ifa_j:xˆ0 forx2F_{j 1}, thenF_jˆF_{j 1},b_j ˆb_{j 1}ˆa_jand

1_fXt2FgdA^j_tˆ[ dL^j_t ¹‡1_fXt2FgdLt(a_j:X^{j 1})]b_j : Ifa_j:x50 forx2F_{j 1}, thena_j6ˆb_{j 1},b_jˆs_aj(b_{j 1}) and

1_fXt2FgdA^j_tˆdL^j_t ¹b_j:

In the first and third casesL^jˆL^{j 1} is increasing from the induction as- sumption. In the second case we use Tanaka formula again. As

a_j:X_t^jˆa_j:r_aj(X_t^{j 1})0 we get

a_j:X_t^j ˆ (a_j:X_t^j)^‡

ˆ (a_j:X₀^j)^‡‡ Z^t

0

1_fa

j:Xs^j>0gd(a_j:X_s^j)‡1

2 L_t(a_j:X^j) and therefore

dL^j_t ˆ 1_fXt2Fgd(a_j:A^j_t)

ˆ 1_fXt2Fg(1_faj:X^j

t>0gd(a_j:A^j_t)‡1

2 dLt(a_j:X^j))

ˆ 1

21_fXt2FgdL_t(a_j:X^j); which proves thatL^jis an increasing process.

c) We have obtained that for each F2 F there exists an increasing processL^lsuch that

1_fXt2FgdA^l_tˆdL^l_tb_l:

From Lemma 4 we deduce thatFlˆp(F)ˆFa for somea2S, which entailsb_l ˆa. TakeL^ato be the sum of theL^l's for allF2 Fa. ClearlydL^a_t is supported by

[_F2FafX_t2Fg ˆ fX_t2 [_F2FaFg

ˆ fp(Xt)2Fag

where the last equality derives from (3) in Lemma 4. Summarizing

we get

dA^l_t ˆ 1_fXt2[F2FgdA^l_t

ˆ X

a2S

X

F2Fa

1_fXt2FgdA^l_t

ˆ X

a2S

dL^a_t a:

d) It remains to identify the boundary processesL^a. We use the method in b) again. As

a:p(Xt)0 we get

a:p(X_t) ˆ (a:p(X_t))^‡

ˆ (a:p(X0))^‡‡ Z^t

0

1_fa:p(Xs)>0gd(a:p(Xs))‡1

2Lt(a:p(X)): On the one hand,

1_fa:p(Xt)ˆ0gd(a:p(Xt)) ˆ 1_fa:p(Xt)ˆ0g(a:(Os:dBs)‡X

b2S

dL^b_t a:b)

ˆ dL^a_t

since the set fa:p(Xt)ˆ0g has zero Lebesgue measure anddL^b_t is sup- ported byfp(Xt)2Fbgforb6ˆa. On the other hand,

1_fa:p(Xt)ˆ0gd(a:p(Xt))^‡ˆ1

2dLt(a:p(X))

and we are done. p

5. Complements on the boundary process

We have already seen in the previous section that on each faceFaof the Weyl chamber the boundary process is

L^a_t ˆ1

2dLt(a:p(X))

From Corollary VI.1.9 in [7] we know that this is a.s.

lime#0

1 2e

Z^t

0

1_[0;e)(a:p(X_s))ds

for everyt. In the sequel, we seek for an expression ofL^a involving the original processXrather than the reflected onep(X). Forx2VandAV we shall use the standard notation

d(x;A):ˆinffjx yj: y2Ag and fora2Sdefine

K_a:ˆp ¹(F_a)ˆ [_w2Ww(F_a):

PROPOSITION7. For anyx2V anda2S, a:p(x)ˆd(x; Ka): …7†

PROOF. For anye>0 there existsy2p ¹(F_a) such that e‡d(x;Ka) jx yj

jp(x) p(y)j a:(p(x) p(y))

ˆ a:p(x);

where we have used the contraction property ofp. In fact it is easy to check that each rb is contracting and so is p by iteration, which proves that d(x;K_a)a:p(x). Conversely, let x2V and let w2W be such that w(x)ˆp(x)2C. Let us consider the facetw ¹(F_a). Using thata:b0 for anya;b2Switha6ˆb([6]) we check that

x (a:w(x))w ¹(a)2w ¹(F_a)K_a and therefore

d(x;Ka)a:w(x)ˆa:p(x):

p We obtain a new expression for the boundary process:

L^a_t ˆlim

e#0

1 2e

Z^t

0

1_fd(Xs;Ka)5egds: …8†

A little more can be said in a particular case.

PROPOSITION8. Ifa2Sis the only simple root in its orbitRa:ˆW:a, then

Kaˆ [g2Ra\R‡Hg

and L^a_t ˆ X

g2Ra\R‡

Lt(g:X):

PROOF. Letw2WandF2 FsatisfyFˆw(Fa). LetG2 Fhave same support H_g (g2R_‡) as F. There exist b2S and w⁰2W such that Gˆw⁰(F_b). Then from Lemma 1

gˆ w(a)ˆ w⁰(b):

andaandb are conjugate. It follows from the hypothesis thatbˆa, and actually the whole hyperplaneHgbelongs toKa. MoreoverKais the union of all hyperplanesH_gwithga positive root in the orbit ofa. Now the difference

X

g2Ra\R‡

1_fjg:Xsj5eg 1_{fming2Ra\R‡jg:Xsj5eg}

is nonnegative and bounded above by a finite sum of terms of the form 1_{fjg1:Xsj5e;jg2:Xsj5eg}

where g₁ and g₂ are independent unit vectors. Let Y be the orthogonal projection ofXonto the two-dimensional space generated byg₁andg₂. Then there existsc>0 such that

1_{fjg1:Xsj5e;jg2:Xsj5eg}1_fjYsj5ceg

and since the two-dimensional Bessel processjYjhas local time zero at 0 it follows that a.s.

lime#0

1 2e

Z^t

0

1_{fjg1:Xsj5e;jg2:Xsj5eg}dsˆ0:

Thus

L^a_t ˆlim

e#0

1 2e

Z^t

0

X

g2Ra\R‡

1_fjg:Xsj5egdsˆ X

g2Ra\R‡

L_t(g:X):

p We now give two illustrative examples.

DIHEDRAL GROUP OF ORDER8. This group consists of the orthogonal transformations which preserve a square centered at the origin in the plane V ˆR². There are two simple rootsaˆ 1

p2(e₁ e₂);bˆe₂which lie in two different orbits. The claim of the previous proposition for the simple

rootais illustrated by the following picture:

DIHEDRAL GROUP OF ORDER6. We come back to the example considered in Section3and identifyR²with the complex planeC. There are six two- dimensional roots represented in complex notations as

fe ^ip=2e^imp=3;mˆ1;2;3g

and there is only one orbit so that the hypothesis of the proposition breaks down. The reflection group Wˆ D3 contains three rotations of angles 2mp=3;m2 f1;2;3gand three reflectionsC3z7!ze^2imp=3. When choosing Sˆ fa;bg ˆ fe^ip=2;e ^ip=6g, C is a wedge of anglep=3in the positive quad- rant of the plane and

fd(x;Ka)5eg …9†

is the hachured part of the following picture:

It follows that

L^b_t ˆ Z^t

0

1_feip=3:Xsi0gdL_s(b:X)‡ Z^t

0

1_feip:Xs0gdL_s(a:X)

‡ Z^t

0

1_fei5p=3:Xs0gdL_s(g:X):

Acknowledgments. We thank J. P. Anker and P. Bougerol for helpful information and discussions on reflection groups. The research of the first author was partially supported by Agence Nationale de la recherche grant ANR-09-BLAN-0084-01.

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Manoscritto pervenuto in redazione il 5 gennaio 2011.