getdoce0f3. 390KB Jun 04 2011 12:05:05 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b il i t y Vol. 16 (2011), Paper no. 18, pages 504–530.

Journal URL

http://www.math.washington.edu/~ejpecp/

Mirror coupling of reflecting Brownian motion

and an application to Chavel’s conjecture

∗

Mihai N. Pascu

∗

Faculty of Mathematics and Computer Science

Transilvania University of Bra¸sov

Bra¸sov – 500091, Romania

mihai.pascu@unitbv.ro

http://cs.unitbv.ro/~pascu

Abstract

In a series of papers, Burdzy et al. introduced themirror couplingof reflecting Brownian motions in a smooth bounded domainD_⊂Rd_{, and used it to prove certain properties of eigenvalues and}

eigenfunctions of the Neumann Laplaceian onD.

In the present paper we show that the construction of the mirror coupling can be extended to the case when the two Brownian motions live in different domainsD₁,D_2⊂Rd_.

As applications of the construction, we derive a unifying proof of the two main results concerning the validity of Chavel’s conjecture on the domain monotonicity of the Neumann heat kernel, due to I. Chavel ([12]), respectively W. S. Kendall ([16]), and a new proof of Chavel’s conjecture for domains satisfying the ball condition, such that the inner domain is star-shaped with respect to the center of the ball.

Key words:couplings, mirror coupling, reflecting Brownian motion, Chavel’s conjecture. AMS 2000 Subject Classification:Primary 60J65, 60H20; Secondary: 35K05, 60H3. Submitted to EJP on April 16, 2010, final version accepted January 26, 2011.

1

Introduction

The technique of coupling of reflecting Brownian motions is a useful tool, used by several authors in connection to the study of the Neumann heat kernel of the corresponding domain (see[2],[3],

[6],[11],[16],[17], etc).

In a series of paper, Krzysztof Burdzy et al. ([1],[2],[3],[6],[10],) introduced themirror coupling

of reflecting Brownian motions in a smooth domainD⊂Rdand used it in order to derive properties

of eigenvalues and eigenfunctions of the Neumann Laplaceian onD.

In the present paper, we show that the mirror coupling can be extended to the case when the two reflecting Brownian motions live in different domainsD₁,D_2⊂Rd.

The main difficulty in the extending the construction of the mirror coupling comes from the fact that the stochastic differential equation(s) describing the mirror coupling has a singularity at the

times when coupling occurs. In the case D1 = D2 = D considered by Burdzy et al. this problem

is not a major problem (although the technical details are quite involved, see[2]), since after the

coupling time the processes move together. In the case D_{1 6}= D₂ however, this is a major problem:

after the processes have coupled, it is possible for them to decouple (for example in the case when the processes are coupled and they hit the boundary of one of the domains).

It is worth mentioning that the method used for proving the existence of the solution is new, and it relies on the additional hypothesis that the smaller domain D₂ (or more generally D_1∩D₂) is a convex domain. This hypothesis allows us to construct an explicit set of solutions in a sequence of

approximating polygonal domains forD₂, which converge to the desired solution.

As applications of the construction, we derive a unifying proof of the two most important results on

the challenging Chavel’s conjecture on the domain monotonicity of the Neumann heat kernel ([12],

[16]), and a new proof of Chavel’s conjecture for domains satisfying the ball condition, such that the inner domain is star-shaped with respect to the center of the ball. This is also a possible new line of approach for Chavel’s conjecture (note that by the results in[4], Chavel’s conjecture does not hold in its full generality, but the additional hypotheses under which this conjecture holds are not known at the present moment).

The structure of the paper is as follows: in Section 2 we briefly describe the construction of Burdzy

et al. of the mirror coupling in a smooth bounded domainD_⊂Rd.

In Section 3, in Theorem 3.1, we give the main result which shows that the mirror coupling can be

extended to the case when D_{2 ⊂} D₁ are smooth bounded domains in Rd and D₂ is convex (some

extensions of the theorem are presented in Section 5).

Before proceeding with the proof of theorem, in Remark 3.4 we show that the proof can be reduced to the case whenD₁=Rd. Next, in Section 3.1, we show that in the caseD₂= (0,_∞)_⊂D₁=Rthe solution is essentially given by Tanaka’s formula (Remark 3.5), and then we give the proof of the main theorem in the 1-dimensional case (Proposition 3.6).

In Section 3.2, we first prove the existence of the mirror coupling in the case whenD2is a half-space

in Rd and D₁ = Rd (Lemma 3.8), and then we use this result in order to prove the existence of

the mirror coupling in the case whenD₂ is a polygonal domain inRd and D₁=Rd (Theorem 3.9).

In Proposition 3.10 we present some of the properties of the mirror coupling in the particular case whenD₂is a convex polygonal domain and D₁=Rd, which are essential for the construction of the general mirror coupling.

In Section 4 we give the proof of the main Theorem 3.1. The idea of the proof is to construct a sequence €Y_tn,Xt

Š

of mirror couplings in €Dn,Rd Š

, where Dn ր D2 is a sequence of convex

polygonal domains in Rd. Then, using the properties of the mirror coupling in convex polygonal

domains (Proposition 3.10), we show that the sequenceY_tn converges to a process Yt, which gives

the desired solution to the problem.

The last section of the paper (Section 5) is devoted to the applications and the extensions of the mirror coupling constructed in Theorem 3.1.

First, in Theorem 5.3 we use the mirror coupling in order to give a simple, unifying proof of the results of I. Chavel and W. S. Kendall on the domain monotonicity of the Neumann heat kernel (Chavel’s Conjecture 5.1). The proof is probabilistic in spirit, relying on the geometric properties of the mirror coupling.

Next, in Theorem 5.4 we show that Chavel’s conjecture also holds in the more general case when one can interpose a ball between the two domains, and the inner domain is star-shaped with respect to the center of the ball (instead of being convex). The analytic proof given here is parallel to the geometric proof of the previous theorem, and it can also serve as an alternate proof of it.

Without giving all the technical details, we discuss the extension of the mirror coupling to the case of

smooth bounded domainsD1,2⊂Rd with non-tangential boundaries, such that D1∩D2 is a convex

domain.

The paper concludes with a discussion of the non-uniqueness of the mirror coupling. The lack of uniqueness is due to the fact that after coupling the processes may decouple, not only on the boundary of the domain, but also when they are inside the domain.

The two basic solutions give rise to thesticky, respectivelynon-stickymirror coupling, and there is a whole range of intermediate possibilities. The stickiness refers to the fact that after coupling the processes “stick” to each other as long as possible (“sticky” mirror coupling, constructed in Theorem 3.1), or they can immediately split apart after coupling (“non-sticky” mirror coupling), the general case (weak/mildmirror coupling) being a mixture of these two basic behaviors.

We developed the extension of the mirror coupling having in mind the application to Chavel’s conjec-ture, for which the sticky mirror coupling is the “right” tool, but perhaps the other mirror couplings (the non-sticky and the mild mirror couplings) might prove useful in other applications.

2

Mirror couplings of reflecting Brownian motions

Reflecting Brownian motion in a smooth domainD⊂Rd_{can be defined as a solution of the}

stochas-tic differential equation

X_t= x+B_t+ Z t

0

ν_D X_sd L_sX, (2.1)

whereB_t is ad-dimensional Brownian motion,ν_Dis the inward unit normal vector field on∂Dand

LX_t is the boundary local time of Xt (the continuous non-decreasing process which increases only

whenX_t∈∂D).

In[1], the authors introduced themirror couplingof reflecting Brownian motion in a smooth domain

D_⊂Rd (piecewiseC2 domain inR2 with a finite number of convex corners or aC2 domain inRd,

They considered the following system of stochastic differential equations:

Xt = x+Wt+ Z t

0

ν_D Xs

d L_sX (2.2)

Y_t = y+Z_t+ Z t

0

ν_D X_sd L_sY (2.3)

Z_t = W_t−2

Z t

0

Ys−Xs

Y_s−X_s2

Y_s−X_s·dW_s (2.4)

for t < ξ, whereξ = infs>0 :Xs=Ys is the coupling time of the processes, after which the

processesX andY evolve together, i.e.X_t=Y_t andZ_t=W_t+Z_ξ−W_ξfort_≥ξ. In the notation of[1], considering the Skorokhod map

Γ:C€[0,∞):RdŠ_→C€[0,∞):DŠ,

we haveX = Γ (x+W),Y = Γ y+Z, and therefore the above system is equivalent to

Z_t= Z t∧ξ

0

G€Γ y+Z_s−Γ (x+W)_sŠdW_s+1_t≥ξ€W_t−W_ξŠ, (2.5)

whereξ=inf¦s>0 :Γ (x+W)_s= Γ y+Z_s©. In[1]the authors proved the pathwise uniqueness and the strong existence of the processZ_t in (2.5) (given the Brownian motionW_t).

In the aboveG:Rd_{→ Md×d} denotes the function defined by

G(z) = (

H z

kzk

, ifz₆=0

0, ifz=0 , (2.6)

where for a unitary vector m∈Rd, H(m) represents the linear transformation given by the d×d

matrix

H(m) =I₋2m m′, (2.7)

that is

H(m)v=v−2(m·v)m (2.8)

is the mirror image ofv∈Rd with respect to the hyperplane through the origin perpendicular tom

(m′denotes the transpose of the vectorm, vectors being considered as column vectors).

The pair X_t,Y_tt≥0 constructed above is called amirror couplingof reflecting Brownian motions in

Dstarting at(x,y)_∈D_×D.

Remark2.1. The relation (2.4) can be written in the equivalent form

d Z_t=G X_t−Y_tdW_t,

which shows that for t < ξ the increments of Zt are mirror images of the increments ofWt with

3

Extension of the mirror coupling

The main contribution of the author is the observation that the mirror coupling introduced above can be extended to the case when the two reflecting Brownian motion have different state spaces,

that is when X_t is a reflecting Brownian motion in a domain D₁ and Y_t is a reflecting Brownian

motion in a domain D₂. Although the construction can be carried out in a more general setup (see

the concluding remarks in Section 5), in the present section we will consider the case when one of the domains is strictly contained in the other.

The main result is the following:

Theorem 3.1. Let D_1,2⊂Rd be smooth bounded domains (piecewise C2-smooth boundary with convex corners inR2, or C2-smooth boundary inRd, d_≥3will suffice) with D_2⊂D₁and D₂ convex domain, and let x∈D1and y∈D2 be arbitrarily fixed points.

Given a d-dimensional Brownian motion Wt

t≥0starting at0on a probability space(Ω,F,P), there exists a strong solution of the following system of stochastic differential equations

Xt = x+Wt+ Z t

0

νD1 Xs

d L_sX (3.1)

Y_t = y+Z_t+ Z t

0

ν_D

2 Ys

d LY_s (3.2)

Zt = Z t

0

G Ys−Xs

dWs (3.3)

or equivalent

Z_t= Z t

0

G€eΓ y+Z_s−Γ (x+W)_sŠdW_s, (3.4)

whereΓandΓedenote the corresponding Skorokhod maps which define the reflecting Brownian motion X = Γ (x+W) in D1, respectively Y =eΓ y+Z

in D2, and G :Rd → Md×d denotes the following

modification of the function G defined in the previous section:

G(z) = (

H

z

kzk

, if z6=0

I, if z=0 . (3.5)

Remark3.2. As it will follow from the proof of the theorem, with the choice ofG above, the solution

of the equation (3.4) in the case D₁ = D₂ = D is the same as the solution of the equation (2.5)

considered by the authors in[1](as also pointed out by the authors, the choice ofG(0)is irrelevant in this case).

Therefore, the above theorem is a natural generalization of the mirror coupling to the case when the two processes live in different spaces. We will refer to a solution(Xt,Yt)given by the above theorem

as amirror couplingof reflecting Brownian motions in(D₁,D₂)starting from x,y_∈D_1×D₂, with

driving Brownian motionW_t.

As indicated in Section 5, the solution of (3.4) is not pathwise unique, due to the fact that the stochastic differential equation has a singularity at the times when coupling occurs. The general mirror coupling can be thought as depending on a parameter which is a measure of the stickiness of

the coupling: once the processesX_tandY_t have coupled, they can either move together until one of them hits the boundary (sticky mirror coupling- this is in fact the solution constructed in the above theorem), or they can immediately split apart after coupling (non-sticky mirror coupling), and there is a whole range of intermediate possibilities (see the discussion at the end of Section 5).

As an application, in Section 5 we will use the former mirror coupling to give a unifying proof

of Chavel’s conjecture on the domain monotonicity of the Neumann heat kernel for domains D1,2

satisfying the ball condition, although the other possible choices for the mirror coupling might prove useful in other contexts.

Before carrying out the proof, we begin with some preliminary remarks which will allow us to reduce the proof of the above theorem to the caseD₁=Rd.

Remark3.3. The main difference from the case whenD₁=D₂=Dconsidered by the authors in[1]

is that after the coupling timeξthe processes Xt andYt may decouple. For example, ift ≥ξis a

time whenX_t =Y_t∈∂D2, the processYt (reflecting Brownian motion inD2) receives a push in the

direction of the inward unit normal to the boundary of D₂, while the processX_t behaves like a free Brownian motion near this point (we assumed thatD2is strictly contained inD1), and therefore the

processes X andY will drift apart, that is they willdecouple. Also, as shown in Section 5, because the functionGhas a discontinuity at the origin, it is possible that the solutions decouple even when

they are inside the domainD₂. This shows that without additional assumptions, the mirror coupling

is not uniquely determined (there is no pathwise uniqueness of (3.4)).

Remark 3.4. To fix ideas, for an arbitrarily fixed ǫ > 0 chosen small enough such that ǫ < dist ∂D₁,∂D₂, we consider the sequence ξ_nn≥1 of coupling times and the sequence τ_nn≥0

of times when the processes areǫ-decoupled (ǫ-decoupling times, or simplydecoupling timesby an

abuse of language) defined inductively by

ξ_n = inft> τ_n−1:Xt=Yt , n≥1,

τ_n = inft> ξ_n:_kX_t−Y_tk> ǫ , n_≥1, whereτ₀=0 andξ₁=ξis the first coupling time.

To construct the general mirror coupling (that is, to prove the existence of a solution to (3.1) – (3.3) above, or equivalent to (3.4)), we proceed as follows.

First note that on the time interval [0,ξ], the arguments used in the proof of Theorem 2 in [1]

(pathwise uniqueness and the existence of a strong solutionZ of (3.4)) do not rely on the fact that

D₁=D₂, hence the same arguments can be used to prove the existence of a strong solution of (3.4) on the time interval[0,ξ₁] = [0,ξ]. Indeed, givenW_t, (3.1) has a strong solution which is pathwise

unique (the reflecting Brownian motionXt inD1), and therefore the proof of pathwise uniqueness

and the existence of a strong solution of (3.4) is the same as in[1]considering D= D₂. Also note that as also pointed out by the authors, the valueG(0)is irrelevant in their proof, since the problem is constructing the processes until they meet, that is forYt−Xt 6=0, for which their definition ofG

is the same as in (3.5).

We obtain therefore the existence of a strong solutionZ_tto (3.4) on the time interval[0,ξ₁]. By this we understand that the process Z verifies (3.4) for all t _≤ξ₁ andZ_t is_Ft measurable for t _≤ξ₁, where(_Ft)_t≥0 denotes the corresponding filtration of the driving Brownian motionWt.

[0,T]as follows. Considerξ₁T =ξ_1∧T, and note that ifZsolves (3.4), then

Z_ξT

1+t−Zξ

T

1 = Z ξT

1+t

ξT₁

G€Γe y+Z_s−Γ (x+W)_sŠdW_s

= Z t 0 G e

Γ y+Z_ξT

1+s−Γ (x+W)ξ

T

1+s

dW_ξT

1+s.

By the uniqueness results on the Skorokhod map (in the deterministic sense), we have

e

Γ y+Z_ξT

1+s=eΓ

e

Γ(y+Z)_ξT

1 −Zξ

T

1 +Zξ

T

1+·

s

and

Γ (x+W)_ξT

1+s= Γ

Γ(x+W)_ξT

1−Wξ

T

1 +Wξ

T

1+·

s

fors≥0.

It is known thatWf_s =W_ξT

1+s−Wξ

T

1 is a Brownian motion starting at the origin, with corresponding

filtration_Fse =σ

B_ξT

1+u−Bξ

T

1 : 0≤u≤s

independent of_FξT

1.

Setting eZ_t=Z_ξT

1+t−ZξT1 and combining the above equations we obtain e

Z_t= Z t 0 G e Γ e

Γ y+Z_ξT

1 +Ze

s−Γ

Γ (x+W)_ξT

1 +Wf

s

dWf_s, (3.6)

which is the same as the equation (3.4) for eZ, with the initial points x,y of the coupling replaced byY_ξT

1 =Γ(e y+Z)ξT1, respectivelyXξ1T = Γ(x+W)ξ1T, and the Brownian motionW replaced byWf.

If we assume the existence of a strong solution Ze_t of (3.6) until the first ǫ-decoupling time, by

patchingZ andeZwe obtain that

Z_t1_t≤ξT

1 +eZt−ξ

T

11ξ

T

1≤t≤τ

T

1

is a strong solution to (3.4) on the time interval[0,τT₁], whereτ₁T =τ1∧T.

Ifτ₁T = T, we are done. Otherwise, since at time τT₁ the processes X and Y areǫ units apart, we can apply again the results in[1](with the Brownian motionW_τT

1+t−WτT1 instead ofWt, and the

starting points of the couplingX_τT

1 andYτ

T

1 instead of x and y) in order to obtain a strong solution

of (3.4) until the first coupling time. By patching we obtain the existence of a strong solution of (3.4) on the time interval”0,ξT₂—.

Proceeding inductively as indicated above, since only a finite number of coupling/decoupling times

ξ_n and τ_n can occur in the time interval[0,T], we can construct a strong solution Z to (3.4) on the time interval[0,T]for anyT>0 (and therefore on[0,_∞)), provided we show the existence of strong solutions of equations of type (3.6) until the firstǫ-decoupling time.

In order to prove this claim, since Γ(e y +Z)_ξT

1 and Γ (x+W)ξ

T

1 are Fξ

T

1 measurable and the

σ-algebra _FξT

1 is independent of the filtrationFe = (Fte)t≥0 of the driving Brownian motionWft, it

suffices to show that for any starting points x= y ∈D2 of the mirror coupling, there exists a strong

solution of (3.4) until the firstǫ-decoupling timeτ1. Sinceǫ <dist ∂D1,∂D2

processX_t cannot reach the boundary∂D₁ before the firstǫ-decoupling timeτ₁, and therefore we

can consider thatXt is a free Brownian motion inRd, that is, we can reduce the proof of Theorem

3.1 to the case when D1=Rd.

We will give the proof of the Theorem 3.1 first in the 1-dimensional case, then we will extend it to the case of polygonal domains inRd, and we will conclude with the proof in the general case.

3.1

The

1-dimensional case

From Remark 3.4 it follows that in order to construct the mirror coupling in the 1-dimensional case, it suffices to considerD1=RandD2= (0,a), and to show that for an arbitrary choice x ∈[0,a]of

the starting point of the mirror coupling,ǫ_∈(0,a)sufficiently small and W_tt≥0 a 1-dimensional Brownian motion starting atW₀=0, we can construct a strong solution on[0,τ₁]of the following system

X_t = x+W_t (3.7)

Y_t = x+Z_t+LY_t (3.8)

Z_t = Z t

0

G Y_s−X_sdW_s (3.9)

where τ₁ = infs>0 :_|X_s−Y_s|> ǫ is the first ǫ-decoupling time and the function G :

R_{→ M1×1≡}Ris given in this case by

G(x) = ¨

−1, if x 6=0

+1, if x =0 . (3.10)

Remark 3.5. Before proceeding with the proof, it is worth mentioning that the heart of the

con-struction is Tanaka’s formula. To see this, consider for the momenta = _∞, and note that Tanaka

formula

x+W_t=x+ Z t

0

sgn x+W_sdW_s+L0_t(x+W)

gives a representation of the reflecting Brownian motionx+Wt

in which the increments of the

martingale part ofx+Wt

are the increments ofWtwhenx+Wt∈[0,∞), respectively the opposite

(minus) of the increments ofW_t in the other case (L0_t(x+W) denotes here the local time at 0 of

x+W_t).

Since x +W_{t ∈}[0,_∞) is the same asx+W_t= x +W_t, from the definition of the function G it follows that the above can be written in the form

_x+Wt

=x+

Z t

0

G€x+Ws

_{− x}+Ws

Š

dWs+Lxt+W,

which shows that a strong solution to (3.7) – (3.9) above (in the casea=_∞) is given explicitly by

X_t=x+W_t,Y_t=x+W_tandZ_t=R₀tsgn x+W_sdW_s. We have the following:

Proposition 3.6. Given a 1-dimensional Brownian motion W_tt≥0 starting at W₀ = 0, a strong solution on[0,τ₁]of the system (3.7) – (3.9) is given by

  

X_t=x+W_t Yt=

a−x+Wt−a

Z_t=Rt

0sgn Ws

sgn a₋W_sdW_s

,

whereτ1=inf

¦

s>0 :X_s−Y_s> ǫ©and

sgn(x) = ¨

+1, if x≥0

−1, if x<0 .

Proof. Sinceǫ <a, it follows that fort_≤τ₁we haveX_t= x+W_t∈(₋a, 2a), and therefore

Y_t=a₋x+W_t−a=   

− x+W_t, x+W_t∈(₋a, 0)

x+Wt, x+Wt∈[0,a]

2a−x−W_t, x+W_t∈(a, 2a)

. (3.11)

Applying the Tanaka-Itô formula to the function f(z) =_|a_{− |}z₋a_|| and to the Brownian motion

Xt=x+Wt, for t≤τ1we obtain Y_t = x+

Z t

0

sgn x+W_ssgn a₋x₋W_sd x+W_s+L0_{t −}La_t

= x+ Z t

0

sgn x+W_ssgn a₋x₋W_sdW_s+

Z t

0

ν_D

2 Ys

d€L_s0+L_saŠ,

where L0_t = sup_s≤t x+W_s− and La_t = sup_s≤t x+W_s−a+ are the local times of x +W_t at 0, respectively ata, andν_D

2(0) = +1,νD2(a) =−1.

From (3.11) and the definition (3.10) of the functionG we obtain

sgn x+Ws

sgn a−x−Ws

=

  

−1, x+W_s∈(₋a, 0) +1, x+W_s∈[0,a] −1, x+W_s∈(a, 2a) =

¨

+1, X_s=Y_s

−1, Xs6=Ys = G Y_s−X_s, and therefore the previous formula can be written equivalently

Y_t=x+Z_t+ Z t

0

ν_D

2 Ys

d L_sY, where

Zt = Z t

0

G Ys−Xs

dWs

andLY_t = L0_t+L_tais a continuous nondecreasing process which increases only whenx+W_t∈ {0,a}, that is only whenY_t∈∂D₂.

3.2

The case of polygonal domains

In this section we will consider the case when D_{2 ⊂} D_{1 ⊂} Rd are polygonal domains (domains

bounded by hyperplanes in Rd). From Remark 3.4 it follows that we can consider D1 = Rd and

therefore it suffices to prove the existence of a strong solution of the following system

Xt = X0+Wt (3.12)

Y_t = Y₀+Z_t+ Z t

0

ν_D2 Y_sd LY_s (3.13)

Z_t = Z t

0

G Y_s−X_sdW_s (3.14)

or equivalently of the equation

Zt= Z t

0

G€Γe Y0+Z

s−X0−Ws Š

dWs, (3.15)

whereW_t is ad-dimensional Brownian motion starting atW₀=0 andX₀=Y₀∈D₂.

The construction relies on the following skew product representation of Brownian motion in spher-ical coordinates:

X_t=R_tΘ_σ

t, (3.16)

where R_t = _kXtk ∈ BES(d) is a Bessel process of order d andΘ_{t ∈}BM€Sd−1Š is an independent Brownian motion on the unit sphereSd−1inRd, run at speed

σ_t= Z t

0

1

R2_sds, (3.17)

which depends only onR_t.

Remark3.7. One way to construct the Brownian motionΘ_t = Θ_td−1 on the unit sphereSd−1 _⊂Rd

is to proceed inductively on d _≥2, using the following skew product representation of Brownian

motion on the sphereΘd_t−1_∈Sd−1(see[15]):

Θd_t−1=cosθ_t1, sinθ_t1Θd_α−2

t

,

where θ1 ∈ LEG(d−1) is a Legendre process of order d−1 on [0,π], and Θd−2

t ∈Sd−2 is an

independent Brownian motion onSd−2, run at speed

α_t= Z t

0

1 sin2θ1

s

ds.

Therefore, ifθ_t1, . . .θ_td−1are independent processes, withθi_∈LEG(d₋i)on[0,π]fori=1, . . . ,d₋

2, andθ_td−1is a 1-dimensional Brownian (note thatΘ1_t =€cosθ_t1, sinθ_t1Š_∈S1is a Brownian motion onS1), Brownian motionΘd_t−1on the unit sphereSd−1_⊂Rd is given by

or by

Θ_td−1=€θ_t1, . . . ,θ_td−2,θ_td−1Š (3.18)

in spherical coordinates.

To construct the solution of (3.12) – (3.14), we first consider the case when D₂ is a half-space

H_d+=¦€z1, . . . ,zdŠ∈Rd_:zd>0©.

Given an angleϕ_∈R, we introduce the rotation matrixR ϕ_{∈ Md×d} which leaves invariant the

firstd−2 coordinates and rotates clockwise by the angleαthe remaining 2 coordinates, that is

R(α) =        

1 0 0 0

... _{· · · · · ·}

0 1 0 0

0 · · · 0 cosϕ −sinϕ 0 _{· · ·} 0 sinϕ cosϕ

       

. (3.19)

We have the following: Lemma 3.8. Let D2=H_d+=

¦€

z1, . . . ,zdŠ∈Rd:zd>0©and assume that Y0=R ϕ0

X0 (3.20)

for someϕ_0∈R.

Consider the reflecting Brownian motionθe_td−1 on[0,π]with driving Brownian motion θ_td−1, where

θ_td−1is the(d−1)spherical coordinate of G Y0−X0

Xt, given by (3.16) – (3.18) above, that is: e

θ_td−1=θ_td−1+L0_t€θed−1Š₋Lπ_t €θed−1Š, t_≥0,

and L0_t€θed−1Š, Lπ_t €θed−1Šrepresent the local times ofθed−1 at0, respectively atπ. A strong solution of the system (3.12) – (3.14) is explicitly given by

Y_t= ¨

R ϕ_tG Y₀₋X₀X_t, t< ξ

X_td, t≥ξ (3.21)

whereξ=inft>0 :X_t=Y_t is the coupling time, the rotation angleϕ_t is given by

ϕ_t=L0_t€θed−1Š−Lπ_t €θed−1Š, t≥0,

and for z=€z1,z2. . . ,zdŠ_∈Rd we denoted by_|z_|d =€z1,z2, . . . ,zdŠ.

Proof. Recall that form∈Rd_{− {}0_},G(m)v denotes the mirror image ofv∈Rd with respect to the

hyperplane through the origin perpendicular tom.

By Itô formula, we have

Y_t∧ξ=Y₀+ Z t∧ξ

0

R ϕ_sG Y₀₋X₀d X_s+ Z t∧ξ

0 R



ϕ_s+π

2

‹

Xt

Yt=R(ϕt)G(Y0−X0)X_t

M0 Mt

X0 Y0

G(Y0−X0)Xt

H

+_d

νH+ d

m0 mt

R(ϕt)

Figure 1: The mirror coupling of a free Brownian motionX_t and a reflecting Brownian motionY_t in

the half-space_Hd+.

Note that the composition R◦G (a symmetry followed by a rotation) is a symmetry, and since

kY_tk = _kX_tk for all t _≥ 0, it follows that X_t and Y_t are symmetric with respect to a hyperplane

passing through the origin for all t _≤ξ. Therefore, from the definition (3.5) of the function G it

follows that we haveYt=G Yt−Xt

Xt for allt≤ξ.

Also note that whenL_s0€θed−1Šincreases,Ys∈∂D2 and we have R



ϕ_s+π

2

‹

G Y₀₋X₀X_s=R

_π

2

‹

Y_s=ν_D

2 Ys

, and ifL_sπ€θed−1Šincreases,Y_s∈∂D₂and we have

R



ϕ_s+π

2

‹

G Y₀₋X₀X_s=R

_π

2

‹

Y_s=₋ν_D2 Y_s. It follows that the relation (3.22) can be written in the equivalent form

Y_t∧ξ=Y₀+ Z t∧ξ

0

G Y_s−X_sd X_s+ Z t∧ξ

0

ν_D

2 Ys

d L_sY,

where LY_t = L0_t€θed−1Š+Lπ_t €θed−1Š is a continuous non-decreasing process which increases only whenY_{t ∈}∂D₂, and thereforeY_t given by (3.21) is a strong solution of the system (3.12) – (3.14) fort≤ξ.

For t≥ξ, we haveYt = _Xt

d = €

case, by Tanaka formula we obtain:

Y_t∨ξ = Yξ+ Z t∨ξ

ξ €

1, . . . , 1, sgn€X_sdŠŠd X_s+ Z t∨ξ

ξ

(0, . . . , 0, 1)L0_t€XdŠ (3.23)

= Y_ξ+ Z t∨ξ

ξ

G Y_s−X_sd X_s+ Z t∨ξ

ξ

ν_D

2 Ys

LY_t, since in this case

G Y_s−X_s = ¨

(1, . . . , 1,+1), X_s=Y_s

(1, . . . , 1,₋1), X_s6=Y_s

= ¨

(1, . . . , 1,+1), X_sd≥0

(1, . . . , 1,₋1), X_sd<0

= €1, . . . , 1, sgn€X_sdŠŠ.

The process LY_t = L0_t€XdŠ in (3.23) is a continuous non-decreasing process which increases only whenYt ∈∂D2 (L0t

€

XdŠ represents the local time at 0 of the last cartesian coordinateXd ofX), which shows that Y_t also solves (3.12) – (3.14) for t _≥ξ, and thereforeY_t is a strong solution of (3.12) – (3.14) fort_≥0, concluding the proof.

Consider now the case of a general polygonal domain D2 ⊂ Rd. We will show that a strong

so-lution of the system (3.12) – (3.14) can be constructed from the previous lemma by choosing the appropriate coordinate system.

Consider the times σn

n≥0 at which the solution Yt hits different bounding hyperplanes of∂D2,

that isσ₀=infs_≥0 :Y_s∈∂D₂ and inductively σn+1=inf

¨

t≥σn:

Y_t∈∂_D2andY_t,Y_σn belong to different1

bounding hyperplanes of∂D₂

«

, n≥0. (3.24)

If X₀ = Y_{0 ∈} ∂D₂ belong to a certain bounding hyperplane of D₂, we can chose the coordinate

system so that this hyperplane isHd =¦€z1, . . . ,zdŠ∈Rd:zd=0©andD2⊂ H_d+, and we letHd

be any bounding hyperplane ofD₂ otherwise.

By Lemma 3.8 it follows that on the time interval[σ₀,σ₁), the strong solution of (3.12) – (3.14) is given explicitly by (3.21).

If σ₁ < _∞, we distinguish two cases: X_σ

1 = Yσ1 and Xσ1 6= Yσ1. Let H denote the bounding

hyperplane ofDwhich containsYσ1, and letνH denote the unit normal toH pointing inside D2.

If X_σ1 = Y_{σ1 ∈ H}, choosing again the coordinate system conveniently, we may assume that_H is

the hyperplane is_Hd =¦€z1, . . . ,zdŠ_∈Rd :zd=0©, and on the time interval[σ₁,σ₂)the coupling

€

X_σ

1+t,Yσ1+t Š

t∈[0,σ2−σ1)is given again by Lemma 3.8.

IfXσ16=Yσ1∈ H, in order to apply Lemma 3.8 we have to show that we can choose the coordinate

system so that the condition (3.20) holds. IfYσ1−Xσ1 is a vector perpendicular toH, by choosing 1_{Since 2-dimensional Brownian motion does not hit points a.s., thed-dimensional Brownian motionY}

t does not hit

the coordinate system so that_H =_Hd=¦€z1, . . . ,zdŠ_∈Rd:zd=0©, the problem reduces to the

1-dimensional case (the firstd−1 coordinates of X andY are the same), and it can be handled as

in Proposition 3.6 by the Tanaka formula. The proof being similar, we omit it.

IfXσ16=Yσ1∈ H andYσ1−Xσ1 is not orthogonal toH, considerXeσ1 =prH Xσ1 the projection of

Xσ1 ontoH, and therefore Xeσ1 6= Yσ1. The plane of symmetry of Xσ1 and Yσ1 intersects the line

determined byXe_σ

1 and Yσ1 at a point, and we consider this point as the origin of the coordinate

system (note that the intersection cannot be empty, for otherwise the vectorsY_σ1−X_σ1andY_σ1−Xe_σ1

were parallel, which is impossible since then Yσ1−Xσ1,Yσ1−Xeσ1 and Yσ1−Xeσ1,Xσ1−Xeσ1 were

perpendicular pairs of vectors, contradictingXeσ16=Yσ1 – see Figure 2).

Yσ

1

Xσ

1

˜

X

σ1

H

0

Mσ

1

ed=_νH

ed−₁

{e₁, . . . , ed−₂}

Figure 2: Construction of the appropriate coordinate system.

Choose an orthonormal basis e₁, . . . ,e_d in Rd such that e_d = ν_H is the normal vector to _H

pointing inside D₂, e_d−1 = 1

kYσ₁−Xσ₁k

€

Y_σ

1−Xσ1 Š

is a unit vector lying in the 2-dimensional plane

determined by the origin and the vectors e_d and Y_σ

1−Xσ1, and

e₁, . . . ,e_d−2 is a completion of

e_d−1,ed to an orthonormal basis inRd (see Figure 2).

Note that by the construction, the vectors e1, . . . ,ed−2 are orthogonal to the 2-dimensional

hyper-plane containing the origin and the pointsX_σ1 andY_σ1, and therefore X_σ1 and Y_σ1 have the same (zero) first d₋2 coordinates; also, since X_σ

1 and Yσ1 are at the same distance from the origin,

it follows that Yσ1 can be obtained from Xσ1 by a rotation which leaves invariant the first d−2

coordinates, which shows that the condition (3.20) of Lemma 3.8 is satisfied.

Since by construction the bounding hyperplane _H of D₂ at Y_σ1 is given by _Hd =

¦€

z1, . . . ,zdŠ∈Rd :zd =0© and D2 ⊂ H_d+ =

¦€

z1, . . . ,zdŠ∈Rd:zd>0©, we can apply Lemma

3.8 to deduce that on the time interval [σ1,σ2) a solution of (3.12) – (3.14) is given by

€

X_σ1+t,Y_σ1+tŠ

t∈[0,σ2−σ1).

Repeating the above argument we can construct inductively (in the appropriate coordinate systems) the solution of (3.12) – (3.14) on any time interval [σ_n,σ_n+1), n≥1, and therefore we obtain a

strong solution of (3.12) – (3.14) defined for t_≥0. We summarize the above discussion in the following:

Theorem 3.9. If D2⊂Rd is a polygonal domain, for any X0=Y0∈D2, there exists a strong solution of the system (3.12) - (3.14).

Moreover, between successive hits of different bounding hyperplanes of D₂ (i.e. on each time interval

[σ_n,σ_n+1) in the notation above), the solution is given by Lemma 3.8 in the appropriately chosen coordinate system.

We will refer to the solution X_t,Y_tt≥0constructed in the previous theorem as amirror couplingof reflecting Brownian motions in€Rd,D₂Šwith starting pointX₀=Y_0∈D₂.

If X_{t 6}= Y_t, the hyperplane M_t of symmetry between X_t and Y_t (the hyperplane passing through

Xt+Yt

2 with normal mt =

1

kYt−Xtk Yt−Xt

) will be referred to as the mirror of the coupling. For definiteness, whenXt=Yt we letMt denote any hyperplane passing through Xt=Yt, for example

we can chooseM_t such that it is a left continuous function with respect to t.

In the particular case of a convex polygonal domainD₂, some of the properties of the mirror coupling are contained in the following:

Proposition 3.10. If D_2⊂Rd is a convex polygonal domain, for any X₀=Y_0∈D₂, the mirror coupling given by the previous theorem has the following properties:

i) If the reflection takes place in the bounding hyperplane_H of D₂ with inward unitary normal

ν_H, then the angle∠(mt;νH)decreases monotonically to zero.

ii) When processes are not coupled, the mirror M_t lies outside D₂.

iii) Coupling can take place precisely when X_t∈∂D₂. Moreover, if X_t∈D₂, then X_t=Y_t. iv) If Dα ⊂ Dβ are two polygonal domains and (Ytα;Xt), (Y

β

t ;Xt) are the corresponding mirror

coupling starting from x_∈D_α, for any t>0we have

sup

s≤tk

Y_sα₋Y_sβ_{k ≤}Dist€Dα,DβŠ:= max

xα∈∂Dα,xβ∈∂Dβ

(xβ−xα)·νDα(xα)≤0

kx_α−x_βk. (3.25)

Proof. i) In the notation of Theorem 3.9, on the time interval [σ₀,σ₁) we have Y_t = X_t, so

∠ mt,νH

=0 and therefore the claim is verified in this case.

On an arbitrary time interval[σn,σn+1), in the appropriately chosen coordinate system,Yt is given

by Lemma 3.8. Fort< ξ,Y_tis given by the rotationR ϕ_tofG Y₀₋X₀X_t which leaves invariant the first(d₋2)coordinates, and therefore

∠ _mt,ν_H=∠ _m0,ν_H+ L

0

t −L π t

2 ,

which proves the claim in this case (note that before the coupling time ξ only one of the

non-decreasing processesL0_t and Lπ_t is not identically zero).

Since fort_≥ξwe haveY_t=€X_t1, . . . ,X_tdŠ, we have∠ m_t,ν_H=0 which concludes the proof of the claim.

ii) On the time interval[σ0,σ1)the processes are coupled, so there is nothing to prove in this case.

On the time interval [σ1,σ2), in the appropriately chosen coordinate system we have

¦€

z1, . . . ,zdŠ_∈Rd :zd =0© of D₂ where the reflection takes place, and therefore M_{t ∩}D₂ = ∅

in this case.

Inductively, assume the claim is true for t < σ_n. By continuity, Mσn∩D2=∅, thusD2 lies on one side ofMσn. By the previous proof, the angle∠ mt,νH

betweenm_t and the inward unit normal

ν_H to bounding hyperplane_H ofD₂where the reflection takes place decreases to zero. Since D₂ is

a convex domain, it follows that on the time interval[σ_n,σ_n+1)we haveMt∩D2=∅, concluding

the proof.

iii) The first part of the claim follows from the previous proof (when the processes are not coupled, the mirror (henceX_t) lies outsideD₂; by continuity, it follows that at the coupling timeξwe must haveX_ξ=Y_ξ∈∂D₂).

To prove the second part of the claim, consider an arbitrary time interval[σ_n,σ_n+1)between two successive hits ofYtto different bounding hyperplanes ofD2. In the appropriately chosen coordinate

system,Y_t is given by Lemma 3.8. After the coupling timeξ,Y_t is given byY_t=€X_t1, . . . ,X_tdŠ, and therefore ifX_t∈D₂(thusXd_{t ≥}0) we haveY_t=€X1_t, . . . ,X_tdŠ=X_t, concluding the proof.

iv) LetM_tα andM_tβ denote the mirrors of the coupling inDα, respectivelyDβ, with the same driving Brownian motionX_t.

SinceY_tα andX_t are symmetric with respect to M_tα, and Y_tβ and X_t are symmetric with respect to

Mβ_t, it follows that Y_tβ is obtained from Y_tβ by a rotation which leaves invariant the hyperplane

Mα_t ∩M_tβ, or by a translation by a vector orthogonal to M_tα (in the case when M_tα and M_tβ are parallel).

The angle of rotation (respectively the vector of translation) is altered only when either Y_tα or

Y_tβ are on the boundary of Dα, respectively Dβ. Since Dα ⊂ Dβ are convex domains, the angle

of rotation (respectively the vector of translation) decreases when Y_tβ _∈ D_β or when Y_tα _∈ ∂D_α

and

Y_tβ−Y_tα

·ν_Dα€Y_tአ> 0 (in these cases Y_tβ and Y_tα receive a push such that the distance

kY_tα₋Y_tβ_kis decreased), thus the maximum distance _kY_tα₋Y_tβ_k is attained whenY_tα _∈∂D_α and

Y_tβ₋Y_tα

·ν_Dα€Y_tαŠ_≤0, and the formula follows.

4

The proof of Theorem 3.1

By Remark 3.4, it suffices to consider the case when D₁ = Rd and D_{2 ⊂} Rd is a convex bounded

domain with smooth boundary. To simplify the notation, we will drop the index and writeDforD2

in the sequel.

Let D_nn≥1 be an increasing sequence of convex polygonal domains in Rd with D_{n ⊂} D_n+1 and

∪n≥1Dn=D.

Consider€Y_tn,X_tŠ

t≥0 a sequence of mirror couplings in

€

D_n,RdŠ with starting point x _∈ D₁ and

driving Brownian motion W_tt≥0 withW₀=0, given by Theorem 3.9.

By Proposition 3.10, for anyt>0 we have sup

s≤t

Y_sm−Y_sn≤Dist D_n,D_m= max

xn∈∂Dn,xm∈∂Dm

(xm−xn)·νDn(xn)≤0

x_n−x_mn,m→

henceY_tn converges a.s. in the uniform topology to a continuous processY_t.

Since(Yn)_n≥1 are reflecting Brownian motions in D_nn≥1 and D_{n ր}D, the law of Y_t is that of a

reflecting Brownian motion in D, that is Yt is a reflecting Brownian motion in Dstarting at x ∈D

(see [8]). Also note that since Y_tn are adapted to the filtration FW _{= Ft}

t≥0 generated by the

Brownian motionW_t, so isY_t.

By construction, the driving Brownian motionZ_tn ofY_tnsatisfies

Z_tn= Z t

0

G€Y_tn₋X_tŠdW_t, t_≥0.

Consider the process

Zt= Z t

0

G Ys−Xs

dWs,

and note that sinceY is_FW-adapted and_||G||=1, by Lévy’s characterization of Brownian motion,

Zt is a freed-dimensional Brownian motion starting atZ0=0, also adapted to the filtrationFW.

We will show that Z is the driving process of the reflecting Brownian motion Yt, that is, we will

show that

Yt=x+Zt+LYt =x+ Z t

0

G Ys−Xs

dWs+LYt, t ≥0.

Note that the mapping z _7−→ G(z) is continuous with respect to the norm _||A_|| =

€a_{i j}Š

=

Pd i,j=1a

2

i j of d ×d matrices at all points z ∈ R

d

− {0_}, hence G€Y_sn−Xs Š

→

n→∞ G Ys−Xs

if

Y_s−X_s6=0. IfY_s−X_s=0, then eitherY_s=X_s∈DorY_s=X_s∈∂D.

If Y_s = X_{s ∈} D, since D_{n ր} D, there exists N _≥ 1 such that X_{s ∈} D_N, hence X_{s ∈} D_n for all

n_≥N. By Proposition 3.10, it follows that Y_sn =X_s for all n_≥N, hence in this case we also have

G€Y_sn₋X_sŠ=G(0) _→

n→∞G(0) =G Ys−Xs

.

have: Z t 0

G€Y_sn₋X_sŠ₋G Y_s−X_s

2

1_Y

s=Xs∈∂Dds

= Z t 0 H

Y_sn−X_s

_Yn

s −Xs ! −I 2

1_Ys=Xs∈∂Dds

= Z t 0 I− 2 Y n s −Xs

Yn s −Xs

Y_sn₋X_s

Yn

s −Xs !_′ −I 2 1_Y

s=Xs∈∂Dds

= Z t 0 2

Y_sn₋X_s

_Yn

s −Xs

Y_sn₋X_s

_Yn

s −Xs !′ 2 1_Y

s=Xs∈∂Dds

= 4

Z t

0

1_Y

s=Xs∈∂Dds

≤ 4

Z t

0

1_∂D Ys

ds

= 0,

since Yt is a reflecting Brownian motion in D, and therefore it spends zero Lebesgue time on the

boundary ofD.

Since_||G_||=1, using the above and the bounded convergence theorem we obtain

lim n→∞ Z t 0

G€Y_sn₋X_sŠ₋G Y_s−X_s

2

ds=0, and therefore by Doob’s inequality it follows that

Esup

s≤t k

Z_sn−Zsk2≤c EkZ_tn−Ztk2≤c E

Z t

0

kG€Y_sn−X_sŠ−G Y_s−X_sk2ds_n→

→∞0,

for anyt_≥0, which shows thatZ_tnconverges uniformly on compact sets toZ_t=R₀tG Y_s−X_sdW_s. By construction,Z_tn is the driving Brownian motion forY_tn, that is

Y_tn=x+Z_tn+ Z t

0

ν_D n

€

Y_snŠd LYn

s ,

and passing to the limit withn_{→ ∞}we obtain

Y_t=x+Z_t+A_t =x+ Z t

0

G Y_s−X_sdW_s+A_t, t_≥0,

whereA_t=lim_n→∞Rt

0νDn

€

Y_snŠd LYn

It remains to show thatA_t is a process of bounded variation. For an arbitrary partition 0=t₀<t₁< . . .tl=t of[0,t]we have

E

l X

i=1

kA_t

i−Ati−1k = _nlim_→∞E l X

i=1

Z ti

ti−1

ν_D n

€

Y_snŠd LYn

s ≤ lim supE LYn

t

= lim sup

Z t

0

Z ∂Dn

p_D

n s,x,y

σ_n d yds

≤ cpt,

whereσ_n is the surface measure on∂D_n, and the last inequality above follows from the estimates

in[5]on the Neumann heat kernels p_D

n t,x,y

(see the remarks preceding Theorem 2.1 and the proof of Theorem 2.4 in[7]).

From the above it follows thatAt =Yt−x−Zt is a continuous,FW-adapted process (since Yt, Zt

are continuous,_FW-adapted processes) of bounded variation.

By the uniqueness in the Doob-Meyer semimartingale decomposition of the reflecting Brownian motionYt inD, it follows that

A_t= Z t

0

ν_D Y_sd L_sY, t_≥0,

where LY is the local time ofY on the boundary∂D. It follows that the reflecting Brownian motion

Y_t inDconstructed above is a strong solution to

Yt=x+ Z t

0

G Ys−Xs

dWs+ Z t

0

ν_D Ys

d L_sY, t≥0,

or equivalent, the driving Brownian motionZt=

Rt

0 G Ys−Xs

dWs ofYt is a strong solution to

Z_t= Z t

0

G€Γe y+Z_s−X_sŠdW_s, t_≥0, concluding the proof of Theorem 3.1.

5

Extensions and applications

As an application of the construction of mirror coupling, we will present a unifying proof of the two most important results on Chavel’s conjecture.

It is not difficult to prove that the Dirichlet heat kernel is an increasing function with respect to the

domain. Since for the Neumann heat kernel p_D t,x,yof a smooth bounded domain D_⊂Rd we

have

lim

t→∞pD t,x,y

= 1

vol(D),

the monotonicity in the case of the Neumann heat kernel should be reversed. The above observation was conjectured by Isaac Chavel ([12]), as follows:

Proof. We will present an analytic proof which parallels the geometric proof of the previous theorem. Considerx,y _∈D₂ arbitrarily fixed and assume thatD_2⊂B=B y,R_⊂D₁ for someR>0 andD₂ is a star-shaped domain with respect to y.

By eventually approximatingD2 with star-shaped polygonal domains, it suffices to prove the claim in the case whenD2 is a polygonal star-shaped domain.

Let Xt,Yt

be a mirror coupling of reflecting Brownian motions in D1,D2

starting atx ∈D2. The idea of the proof is to show that for all timest _≥0,Y_t is at a distance from y is no greater than the distance fromX_t to y.

We can reduce the proof to the case whenD₁=Rd as follows. Consider the sequences of stopping times(ξ_n)_n≥1 and(τ_n)_n≥1defined inductively by

τ₀ = 0,

ξ_n = inft> τ_n−1:X_t∈∂D₁ , n_≥1,

τ_n = inft> ξ_n:_kX_t−y_k=_kY_t−y_k , n_≥1.

Note that by the ball condition we have _kXξn − yk > kYξn − yk for any n ≥ 1, and therefore

kXt− yk ≥ kYt−ykfor anyn≥1 and anyt∈

ξn,τn

. In order to prove that the same inequality holds on the intervalsτ_n,ξ_n+1

forn_≥0, we proceed as follows. On the set_{τ_n<_∞}, the pair(Xe_t,Ye_t) =€X_τn+t,Yτn+t

Š

defined fort_≤ξ_n+1−τnis a mirror coupling in(Rd,D2)with driving Brownian motionWft =Wτn+t−Wτn (and Zet= Zτn+t−Zτn), and starting

points(Xe₀,Ye₀) = (X_τ

n,Yτn)independent of the filtration of Bet (see Remark 3.4). In order to prove

the claim it suffices therefore to show that for any pointsu_∈Rd andv_∈D₂with_ku₋y_k=_kv₋y_k, the mirror coupling(X_t,Y_t)in(Rd,D₂)with starting points(X₀,Y₀) = (u,v)verifies

kX_t−y_{k ≥ k}Y_t− y_k, t_≥0. (5.4)

Consider therefore a mirror coupling (X_t,Yt) in (Rd,D2) with starting points(X0,Y0) = (u,v) ∈ Rd_×D2satisfyingku−yk=kv−yk.

If u= v, from the construction of the mirror coupling it follows that X_t = Y_t until the process Y_t hits the boundary of D₂, and therefore the inequality in (5.4) holds for these values of t. After the process Yt hits a bounding hyperplane of D2, by Lemma 3.8 it follows that in an appropriate coordinate systemYt is given byYt= (X1t, . . . ,Xtd−1,|Xtd|), until the timeσwhen the processY hits a different bounding hyperplane of D₂, and therefore the inequality in (5.4) is again verified for the corresponding values oft (in the chosen coordinate system we must have y = (y1, . . . ,yd)with

yd > 0, and therefore kXt−yk2− kYt− yk2 =2yd

€

|Xd_t| −Xd_tŠ≥ 0). If at timeσ the processes are coupled (i.e.X_σ=Y_σ∈∂D₂), we can apply the above argument inductively, and find a timeσ₁ when the processes are decoupled and_kX_t−y_{k ≥ k}Y_t−y_kfor allt_≤σ₁.

The above discussion shows that without loss of generality we may further reduce the proof of the claim to the case when(u,v)_∈Rd_×D2withu=6 vandku−yk ≥ kv−yk. Also, the above discussion shows that it is enough to prove (5.4) for all values of t _≤ζ, whereζ=inf_{s>0 :X_s=Y_s}is the first coupling time.

The mirror coupling defined by (3.1) – (3.3) becomes in the case

X_t = u+W_t (5.5)

Y_t = v+Z_t+

Z t 0

ν_D2 Y_sd L_sY (5.6)

Z_t =

Z t 0

G Y_s−X_sdW_s (5.7)

whereG is given by (3.5). In order to prove the claim we will show that

Rt=kXt−yk2− kYt−yk2≥0, t ≤ζ, (5.8) whereζis the first coupling time.

Using the Itô formula it can be shown that the processR_tverifies the stochastic differential equation Rt=R0−2

Z t 0

Rs

Y_s−X_s

kYs−Xsk2

·dWs−2

Z t 0

(Y_s−y)_·νD2 Ys

d L_sY, t≤ζ. (5.9) The process B_t = ₋2R₀t Ys−Xs

kYs−Xsk2 ·dWs is a continuous local martingale on [0,ζ), with quadratic

variation

At=4 d

X

i=1

Z t∧ζ

0

(Y_si₋X_si)2

kY_s−X_sk4ds=

Z t∧ζ

0

4

kYs−Xsk2

ds, t ≥0, (5.10)

and therefore by Lévy’s characterization of Brownian motion it follows that eB_t = B_α

t∧ζ is a

1-dimensional Brownian motion (possibly stopped at time ζ, if Aζ < ∞), where the time change

α_t=inf_{s_≥0 :A_s>t_}is the inverse of the nondecreasing processA_t. SettingXe_t=X_αt∧ζ,Ye_t=Y_αt∧ζ,eR_t=R_αt∧ζandeLY_t =L_αY

t∧ζ, from (5.9) we obtain

e

Rt =eR0+

Z t 0

e

RsdBes−

Z t 0

(Yes−y)·νD2 €

e

Ys

Š

deL_sY, t≥0. (5.11) Since the polygonal domain D2 is assumed star-shaped with respect to the point y, geometric con-siderations show that

(z₋ y)_·ν_D

2(z)≤0, (5.12)

for all the pointsz∈∂D2for which the inside pointing normalνD2(z)at the boundary pointzofD2

is defined, that is for all pointsz∈∂D2 not lying on the intersection of two bounding hyperplanes ofD₂. Since the reflecting Brownian motionY_t does not hit the set of these exceptional points with positive probability, we may assume that the above condition is satisfied for all points, and therefore (Yes−y)·νD2(Yes)≤0 a.s, (5.13)

for all timess_≥0 whenYe_s∈∂D₂.

Since eLY_t is a nondecreasing process of t _≥ 0, a standard comparison argument for solutions of stochastic differential equations shows that the solutioneR_t of (5.11) satisfiesRe_{t ≥}ρ_t for all t_≥0, whereρt is the solution of the stochastic differential equation

ρ_t=Re₀+

Z t 0

The last equation has the explicit solutionρ_t=R₀eBet−12t, and since by hypothesisR

0=ku− yk2−

kv₋ y_k2_≥0, we obtain

Rαt∧ζ=Ret≥ρt=Re0e

e

Bt−12t≥0, t≥0, (5.15)

and thereforeRt=kXt− yk2− kYt−yk2≥0 for allt≤ζ, concluding the proof of the claim. By the initial remarks, it follows that if(X_t,Yt)is a mirror coupling in(D1,D2)with starting point X0=Y0= x, then

kX_t−y_{k ≥ k}Y_t− y_k, t_≥0. (5.16)

As in the proof of the last theorem, this shows that p_D1 t,x,y_≤ p_D2 t,x,y for all t _≥0, con-cluding the proof.

We have chosen to carry out the construction of the mirror coupling in the case of smooth domains withD2 ⊂ D1 and D2 convex, having in mind the application to Chavel’s conjecture. However, al-though the technical details can be considerably longer, it is possible to construct the mirror coupling in a more general setup.

For example, in the case whenD₁ andD₂ are disjoint domains, none of the difficulties encountered in the construction of the mirror coupling occur (the possibility of coupling/decoupling), so the constructions extends immediately to this case.

The two key ingredients in our construction of the mirror coupling were the hypothesis D_{2 ⊂} D₁ (needed in order to reduce by a localization argument the construction to the case D1 =Rd)and the hypothesis on the convexity of the inner domainD₂(which allowed us to construct a solution of the equation of the mirror coupling in the case D₁=Rd).

Replacing the first hypothesis by the condition that the boundaries∂D₁ and∂D₂ are not tangential (needed for the localization of the construction of the mirror coupling) and the second one by condition thatD1∩D2is a convex domain, the arguments in the present construction can be modified in order to give rise to a mirror coupling of reflecting Brownian motion in D₁,D₂(see Figure 3).

D₁

D2

Figure 3: Generic smooth domains D_{1,2 ⊂}Rd for the mirror coupling: D₁,D₂ have non-tangential boundaries andD1∩D2is a convex domain.

Remark5.5. Even though the construction of the mirror coupling was carried out without the ad-ditional assumption on the convexity of the inner domain D2 in the case when D2 is a polygonal

domain (see Theorem 3.9), we cannot extend the construction of the mirror coupling to the general case of smooth domainsD2⊂D1.

This is due to the fact that the stochastic differential equation which defines the mirror coupling has a singularity (discontinuity) when the processes couple, and we cannot prove the convergence of solutions in the approximating domains (as in the proof of Theorem 3.1). The convexity of the inner domain is an essential argument for this proof, which allowed us to handle the discontinuity of the stochastic differential equation which defines the mirror coupling: considering an increasing sequence of approximating domains D_{n ր} D₂, the convexity of D₂ was used to show that if the coupling occurred in the case of the mirror coupling in(Rd,D_N), then coupling also occurred in the case of the mirror coupling in(Rd_,D

n), for alln≥N.

It is easy to construct an example of a non-convex domain D2 and a sequence of approximating domains D_{n ր} D₂ such that the mirror coupling (X_t,Y_tn) in (Rd,D_n) does not have the above-mentioned property, and therefore we cannot prove the existence of the mirror coupling using the same ideas as in Theorem 3.1. However, this does not imply that the mirror coupling cannot be constructed by other methods in a more general setup.

We conclude with some remarks on the non-uniqueness of the mirror coupling in general domains. To simplify the ideas, we will restrict to the 1-dimensional case whenD₂= (0,_∞)_⊂D₁=R. Fixing x _∈(0,_∞)as starting point of the mirror coupling X_t,Y_tin D₁,D₂, the equations of the mirror coupling are

X_t = x+W_t (5.17)

Y_t = x+Z_t+LY_t (5.18)

Z_t =

Z t 0

G Y_s−X_sdW_s (5.19)

where in this case

G(z) =

¨

−1, ifz₆=0

+1, ifz=0 .

Until the hitting timeτ=s>0 :Y_s∈∂D₂ of the boundary of∂D₂ we haveLY_{t ≡}0, and with the substitutionU_t=₋1

2 Yt−Xt

, the stochastic differential forY_t becomes

U_t=

Z t 0

1−G Y_s−X_s

2 dWs=

Z t 0

σ U_sdW_s, (5.20)

where

σ(z) = 1−G(z)

2 =

¨

1, ifz₆=0 0, ifz=0 .

By a result of Engelbert and Schmidt ([13]) the solution of the above problem is not even weakly unique, for in this case the set of zeroes of the functionσisN =_{0_}andσ−2 is locally integrable onR.

In fact, more can be said about the solutions of (5.20) in this case. It is immediate that bothU_t≡0 andUt =Wt are solutions to 5.20, and it can be shown that an arbitrary solution can be obtained

from W_t by delaying it when it reaches the origin (sticky Brownian motion with sticky point the origin).

Therefore, until the hitting timeτof the boundary, we obtain as solutions

Y_t=X_t= x+W_t (5.21)

and

Y_t=X_t−2W_t= x₋W_t, (5.22)

and an intermediate range of solutions, which agree with (5.21) for some time, then switch to (5.22) (see[18]).

Correspondingly, this gives rise to mirror couplings of reflecting Brownian motions for which the solutions stick to each other after they have coupled (as in (5.21)), or they immediately split apart after coupling (as in (5.22)), and there is a whole range of intermediate possibilities. The first case can be referred to asstickymirror coupling, the second asnon-stickymirror coupling, and the intermediate possibilities asweak/mild stickymirror coupling.

The same situation occurs in the general setup inRd, and it is the cause of lack uniqueness of the stochastic differential equations which defines the mirror coupling. In the present paper we detailed the construction of the sticky mirror coupling, which we considered to be the most interesting, both from the point of view of the construction and of the applications, although the other types of mirror coupling might prove useful in other applications.

Acknowledgements

I would like to thank Krzysztof Burdzy and Wilfried S. Kendall for the helpful discussions and the encouragement to undertake the task of the present project.

I would also like to thank the anonymous referee for carefully reading the manuscript and for the suggestion of extending the result in Theorem 5.3 to the more general case in Theorem 5.4.

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