Mirror coupling of reflecting Brownian motion and an application to Chavel's conjecture

In a series of papers, Burdzy et. al. introduced the \emph{mirror coupling} of reflecting Brownian motions in a smooth bounded domain $D\subset \mathbb{R}^{d}$, and used it to prove certain properties of eigenvalues and eigenfunctions of the Neumann Laplaceian on $D$. 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 domains $D_{1},D_{2}\subset \mathbb{R}^{d}$. As an application 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 (\cite{Chavel}), respectively W. S. Kendall (\cite{Kendall}).


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 the mirror coupling of reflecting Brownian motions in a smooth domain D ⊂ R d and used it in order to derive properties of eigenvalues and eigenfunctions of the Neumann Laplaceian on D.
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 domains D 1 , D 2 ⊂ R d .
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 D 1 = D 2 = D considered by Burdzy et. all. 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 = D 2 however, this is a major problem: after 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 D 2 (or more generally D 1 ∩ D 2 ) is a convex domain. This hypothesis allows us to construct an explicit set of solutions in a sequence of approximating polygonal domains for D 2 , which converge to the desired solution.
As an application of the construction, we will derive a unifying proof of the two most important results on the challenging Chavel's conjecture on the domain monotonicity Neumann heat kernel ( [12], [16]), which also gives a possible new line of approach for this 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 domain domain D ⊂ R d .
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 ⊂ D 1 are smooth bounded domains in R d and D 2 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 of the theorem can be reduced to the case when D 1 = R d . Next, in Section 3.1, we show that in the case D 2 = (0, ∞) ⊂ D 1 = R the 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 when D 2 is a half-space in R d and D 1 = R d (Lemma 3.8), and then we use this result in order to prove the existence of the mirror coupling in the case when D 2 is a convex polygonal domain in R d and D 1 = R d (Theorem 3.9). Some of the properties of coupling, essential for the extension to the general case are detailed in Proposition 3. 10.
In Section 4 we give the proof of the main Theorem 3.1. The idea of the proof is to construct a sequence (Y n t , X t ) of mirror couplings in (D n , D 1 ), where D n ր D 2 is a sequence of convex polygonal domains in R d , and to use the properties of the mirror couplings in polygonal domains (Proposition 3.10) in order to show that the sequence Y n t converges to a process Y t , which gives the desired solution to the problem.
The last section of the paper (Section 5) is devoted to discussing 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.
In Remark 5.5, we discuss the equivalent analytic counterpart of the proof Theorem 5.3, which might give a possible new line of approach for extending Chavel's conjecture to other classes of domains.
Without giving all the technical details, we discuss the extension of the mirror coupling to other classes of domains (smooth bounded domains D 1,2 ⊂ R d with non-tangential boundaries, such that D 1 ∩ D 2 is a convex domain).
The paper concludes with a discussion on the (non) uniqueness of the mirror coupling. It is shown here that the lack of uniqueness is due to the fact that after coupling, the processes might decouple, not only on the boundary of the domain, but even when they are inside of it.
The two basic solutions give rise to the sticky, respectively non-sticky mirror couplings, 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 (this is the coupling constructed in Theorem 3.1) hence the name "sticky" mirror coupling, or they can split apart immediately after coupling, in the case of "non-sticky" mirror coupling, the general case (weak/mild mirror couplings) being a mixture of these two basic behaviors.
We developed the extension of the mirror coupling having in mind the application to Chavel's conjecture, 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.

Mirror couplings of reflecting Brownian motions
Reflecting Brownian motion in a smooth domain D ⊂ R d can be defined as a solution of the stochastic differential equation where B t is a d-dimensional Brownian motion, ν D is the inward unit normal vector field on ∂D and L X t is the boundary local time of X t (the continuous non-decreasing process which increases only when X t ∈ ∂D).
In [1], the authors introduced the mirror coupling of reflecting Brownian motion in a smooth domain D ⊂ R d (piecewise C 2 domain in R 2 with a finite number of convex corners or a C 2 domain in R d , d ≥ 3).
They considered the following system of stochastic differential equations: for t < ξ, where ξ = inf {s > 0 : X s = Y s } is the coupling time of the processes, after which the processes X and Y evolve together, i.e. X t = Y t for t ≥ ξ and In the notation of [1], considering the Skorokhod map Γ : we have X = Γ (x + W ) and Y = Γ (y + Z), and the above system reduces to where ξ = inf {t ≥ 0 : Γ (x + W s ) = Γ (y + Z s )}, for which the authors proved the pathwise uniqueness and the strong uniqueness of the process Z t (given the Brownian motion W t ).
In the above G : R d → M d×d denotes the function defined by where for a unitary vector m ∈ R d , H (m) represents the linear transformation given by the d × d matrix is the mirror image of v ∈ R d with respect to the hyperplane through the origin perpendicular to m (m ′ denotes the transpose of the vector m, vectors being considered as column vectors). The pair (X t , Y t ) t≥0 constructed above is called a mirror coupling of reflecting Brownian motions in D starting at x, y ∈D.
which shows that for t < ξ the increments of Z t are mirror images of the increments of W t with respect to the line of symmetry M t of X t and Y t , which justifies the name of mirror coupling.

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 D 1 and Y t is a reflecting Brownian motion in D 2 . 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 restrict to the case when one of the domains is strictly contained in the other one.
The main result is the following: Theorem 3.1. Let D 1,2 ⊂ R d be smooth bounded domains (piecewise C 2 -smooth boundary with convex corners in R 2 , or C 2 -smooth boundary in R d , d ≥ 3 will suffice) with D 2 ⊂ D 1 and D 2 convex domain, and let x ∈D 1 and y ∈D 2 be arbitrarily fixed points. Given a d-dimensional Brownian motion (W t ) t≥0 starting at 0 on a probability space (Ω, F , P ), there exists a strong solution of the following system of stochastic differential equations where Γ andΓ denote the corresponding Skorokhod maps which define the reflecting Brownian motion X = Γ (x + W ) in D 1 , respectively Y =Γ (y + Z) in D 2 , and G : R d → M d×d denotes the following modification of the function G defined in the previous section: Remark 3.2. As it will follow from the proof of the theorem, with the choice of G above, in the case D 1 = D 2 = D the solution of the equation (3.4) given by the theorem above is the same as the solution of the equation (2.5) considered by the authors in [1] (as pointed out by the authors, the choice of G (0) is irrelevant in this case), and therefore the above theorem is a natural generalization of their result to the case when the two processes live in different spaces. We will refer to a solution X t , Y t given by the above theorem as a mirror coupling of reflecting Brownian motions in D 1 , respectively D 2 , starting from (x, y) ∈ D 1 × D 2 with driving Brownian motion W t . As we will see in Section 5, without additional assumptions, the solution of (3.4) is not pathwise unique. This is due to the fact that the stochastic differential equation has a singularity at the origin (i.e. at times when the 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 processes X t and Y 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 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 (the sticky mirror coupling) to give a unifying proof of Chavel's conjecture on the domain monotonicity of the Neumann heat kernel for domains D 1,2 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 case D 1 = R d .
Remark 3.3. The main difference from the case when D 1 = D 2 = D considered by the authors in [1] is that after the coupling time ξ the processes X t and Y t may decouple. For example, if t ≥ ξ is a time when X t = Y t ∈ ∂D 2 , the process Y t being conditioned to stay in D 2 , receives a push in the direction of the inward unit normal to the boundary of D 2 , while the process X t behaves like a free Brownian motion near this point (we assumed that D 2 is strictly contained in D 1 ), and therefore the processes X and Y will drift apart, that is they will decouple. Also, as shown in Section 5, because the function G has a discontinuity at the origin, it is possible that the solutions decouple even inside the domain D 2 , so, 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 ε > 0 chosen small enough such that ε < dist (∂D 1 , ∂D 2 ), we consider the sequence (ξ n ) n≥1 of coupling times and the sequence (τ n ) n≥0 of times when the processes are ε-decoupled (ε-decoupling times, or simply decoupling times by an abuse of language) defined inductively by where τ 0 = 0 and ξ 1 = ξ is 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 solution Z of (3.4)) do not rely on the fact that D 1 = D 2 , hence the same arguments can be used to prove the existence of a strong solution of (3.4) on the time interval [0, Indeed, given W t , (3.1) has a strong solution which is pathwise unique (the reflecting Brownian motion X t in D 1 ), 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 = D 2 . Also note that as pointed by the authors, the value G (0) is irrelevant in their proof, since the problem is constructing the processes until they meet, that is for Y t − X t = 0, for which the definition of G coincides with (3.5).
Next, assuming the existence of a strong solution to (3.4) on [ξ 1 , τ 1 ] (and therefore on [0, τ 1 ]), since at time τ 1 the processes are ε > 0 units apart, we can apply again the results in [1] (withB t = B t+τ1 − B τ1 and starting points X τ1 and Y τ1 ) in order to obtain a strong solution of (3.4) on the time interval [τ 1 , ξ 2 ], and therefore by patching we obtain the existence of a strong solution of (3.4) on the time interval [0, For an arbitrarily fixed t > 0, since only a finite number of coupling/decoupling times ξ n and τ n can occur in the time interval [0, t] (we use here the fact that D 2 is strictly contained in D 1 ), it follows that there exists a strong solution to (3.4) on [0, t] for any t > 0 (and therefore on [0, ∞)), provided we show the existence of a strong solution of (3.4) on [ξ n , τ n ], n ≥ 1.
In order to prove this claim, it suffices therefore to show that for any starting points x = y ∈D 2 of the mirror coupling, there exists a strong solution to (3.4) until the ε-decoupling time τ 1 . Since ε < dist (∂D 1 , ∂D 2 ), it follows that the process X t cannot reach the boundary ∂D 1 before the ε-decoupling time τ 1 , and therefore we can consider that X t is a free Brownian motion in R d , that is we can reduce the proof of Theorem 3.1 to the case when D 1 = R d .
We will first give the proof of the in the 1-dimensional case, then we will extend the construction to polygonal domains in R d , 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 consider D 1 = R and D 2 = (0, a) and to construct a strong solution for t ≤ τ 1 = inf {s > 0 : |X s − Y s | > ε} of the following system: for an arbitrary choice x ∈ [0, a] of the starting point of the mirror coupling, where ε ∈ (0, a) is arbitrarily small, (W t ) t≥0 is a 1-dimensional Brownian motion starting at W 0 = 0 and the function G : R → M 1×1 ≡ R is given in this case by Remark 3.5. Before proceeding with the proof, it is worth mentioning that the heart of the construction is Tanaka's formula. To see this, consider for the moment a = ∞, and note that Tanaka formula gives a representation of the reflecting Brownian motion |x + W t | in which the increments of the martingale part of |x from the definition of the function G it follows that the above can be written in the form which shows that a strong solution to (3.6) -(3.8) above (in the case a = ∞) is given explicitly by We have the following: Proof. Since ε < a, it follows that for t ≤ τ 1 we have X t = x + W t ∈ (−a, 2a), and therefore .
Applying the Tanaka-Itô formula to the function f (z) = |a − |z − a|| and to the Brownian motion X t = x + W t , for t ≤ τ 1 we obtain + are the local times of x + W t at 0, respectively at a, and ν D2 (0) = +1, ν D2 (a) = −1.
From (3.9) and the definition of G we have and therefore the previous formula can be written equivalently and L Y t = L 0 t + L a t is a continuous nondecreasing process which increases only when x + W t ∈ {0, a}, that is only when Y t ∈ ∂D 2 , which concludes the proof.
3.2. The case of polygonal domains. In this section we will consider the case when D 2 ⊂ D 1 ⊂ R d are convex polygonal domains (convex domains bounded by hyperplanes in R d ). From Remark 3.4 it follows that we can consider D 1 = R d and therefore it suffices to prove the existence of a strong solution to where W t is a d-dimensional Brownian motion starting at W 0 = 0 and X 0 = Y 0 ∈ D 2 . The construction relies on the skew product representation of Brownian motion in spherical coordinates, that is where R t = |X t | ∈ BES (d) is a Bessel process of order d and Θ t ∈ BM S d−1 is an independent Brownian motion on the unit sphere S d−1 in R d , run at speed which depends only on R t .
Remark 3.7. One way to construct the Brownian motion Θ t = Θ d−1 t on the unit sphere S d−1 ⊂ R d is to proceed inductively on d ≥ 2, using the skew product representation of Brownian motion on the sphere Θ d−1 t ∈ S d−1 (see [15]) Therefore, considering independent processes To construct the solution we first consider first the case when D 2 is a half-space Given an angle ϕ ∈ R, we introduce the rotation matrix R (ϕ) ∈ M d×d which leaves invariant the first d − 2 coordinates and rotates clockwise by the angle α the remaining 2 coordinates, that is We have the following: given by (3.14) -(3.16) above, that is: and L 0 t θd−1 , L π t θd−1 represent the local times ofθ d−1 at 0, respectively at π.
A strong solution to (3.10) -(3.12) above is explicitly given by where ξ = inf {t > 0 : X t = Y t } is the coupling time, the rotation angle ϕ t is given by Note that the composition R • G (a symmetry followed by a rotation) is a symmetry, and since |Y t | = |X t | 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 G it follows that we have Also note that when L 0 s θd−1 increases, Y s ∈ ∂D 2 and we have and if L π s θ d−1 increases, Y s ∈ ∂D 2 and we have It follows that the relation (3.20) above can be written in the form θd−1 is a continuous non-decreasing process which increases only when Y t ∈ ∂D 2 , and therefore Y t given by (3.19) is a strong solution to (3.10) -(3.12) for t ≤ ξ.
. . , X d t , and similar to the 1dimensional case, by Tanaka formula we obtain: is a continuous non-decreasing process which increases only when Y t ∈ ∂D 2 (we denoted by L 0 t X d the local time at 0 of the last cartesian coordinate X d of X), which shows that Y t also solves (3.10) -(3.12) for t ≥ ξ, and therefore Y t is a strong solution to (3.10) -(3.12) for t ≥ 0, concluding the proof.
Consider now the case of a general polygonal domain D 2 ⊂ R d . We will show that a strong solution to (3.10) -(3.12) can be constructed from the previous lemma, by choosing the appropriate coordinate system.
If X 0 = Y 0 ∈ ∂D 2 belong to a certain bounding hyperplane of D 2 , we can chose the coordinate system so that this hyperplane is H d = z 1 , . . . , z d ∈ R d : z d = 0 and D 2 ⊂ H + d , and we let H d be any bounding hyperplane of D 2 otherwise. Then, on the time interval [σ 0 , σ 1 ), the strong solution to (3.10) -(3.12) is given explicitly by (3.19) in Lemma 3.8.
If σ 1 < ∞, we distinguish two cases: X σ1 = Y σ1 and X σ1 = Y σ1 . Let H denote the bounding hyperplane of D which contains Y σ1 , and let ν H denote the unit normal to H pointing inside D 2 .
If X σ1 = Y σ1 ∈ H, choosing again the coordinate system conveniently, we may assume that H is the hyperplane is H d = z 1 , . . . , z d ∈ R d : z d = 0 , and on the time interval [σ 1 , σ 2 ) the coupling (X σ1+t , Y σ1+t ) t∈[0,σ2−σ1) is given again by Lemma 3.8. If X σ1 = Y σ1 ∈ H, in order to apply the lemma, we have to show that we can choose the coordinate system so that the condition (3.18) holds. If Y σ1 − X σ1 is a vector perpendicular to H, by choosing the coordinate system so that H = H d = z 1 , . . . , z d ∈ R d : z d = 0 , the problem reduces to the 1-dimensional case (the first d − 1 coordinates of X and Y are the same), and it can be handled as in Proposition 3.6 by the Tanaka formula. The proof being similar, we omit it.
If X σ1 = Y σ1 ∈ H and Y σ1 − X σ1 is not orthogonal to H, considerX σ1 = pr H X σ1 the projection of X σ1 onto H, and thereforeX σ1 = Y σ1 . The plane of symmetry of X σ1 and Y σ1 intersects the line determined byX σ1 and Y σ1 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 vectors Y σ1 − X σ1 and Y σ1 −X σ1 were parallel, which is impossible since then Y σ1 − X σ1 , Y σ1 −X σ1 and Y σ1 −X σ1 , X σ1 −X σ1 were perpendicular pairs of vectors, contradictingX σ1 = Y σ1 -see Figure 2).  Figure 2). Note that by the construction, the vectors e 1 , . . . , e d−2 are orthogonal to the 2-dimensional hyperplane containing the origin and the points X σ1 and Y σ1 , and therefore X σ1 and Y σ1 have the same (zero) first d − 2 coordinates; also, since X σ1 and Y σ1 are at the same distance from the origin, it follows that Y σ1 can be obtained from X σ1 by a rotation which leaves invariant the first d − 2 coordinates, which shows that the condition (3.18) of Lemma 3.8 is satisfied.
We summarize the above discussion in the following: there exists a strong solution to (3.10) -(3.12) above.
Moreover, between successive hits of different bounding hyperplanes of D 2 (i.e. on each time interval [σ n , σ n+1 ) in the notation above) and for an appropriately chosen coordinate system, the solution is given by Lemma 3.8.
We will refer to the solution (X t , Y t ) t≥0 constructed in the previous theorem as a mirror coupling of reflecting Brownian motions in R d , D 2 with starting point If X t = Y t , the hyperplane M t of symmetry between X t and Y t , passing through , will be referred to as the mirror of the coupling. For definiteness, when X t = Y t we let M t denote any hyperplane passing through X t = Y t , for example we choose M t such that it is a left continuous function with respect to t.
Some of the properties of the mirror coupling are contained in the following: , the mirror coupling given by the previous theorem has the following properties: i) If the reflection takes place in the boundary hyperplane H of D 2 with inward unitary normal ν H , then the angle ∠(m t ; ν H ) decreases monotonically to zero. ii) When processes are not coupled, the mirror M t lies outside D 2 . iii) Coupling can take place precisely when X t ∈ ∂D 2 . Moreover, if X t ∈ D 2 , then X t = Y t . iv) If D α ⊂ D β are two polygonal domains and (Y α t ; X t ), (Y β t ; X t ) are the corresponding mirror coupling starting from x ∈ D α , for any t > 0 we have Proof. i) In the notation of Theorem 3.9, on the time interval [σ 0 , σ 1 ) we have Y t = X t so ∠ (m t , ν H ) = 0, and the claim is verified in this case.
On an arbitrary time interval [σ n , σ n+1 ), in an appropriately chosen coordinate system, Y t is given by Lemma 3.8. For t < ξ, Y t is given by the rotation R (ϕ t ) of G (Y 0 − X 0 ) X t which leaves invariant the first (d − 2) coordinates, and therefore which proves the claim in this case (note that before the coupling time ξ only one of the non-decreasing processes L 0 t and L π t is not identically zero). Since for t ≥ ξ we have Y t = X 1 t , . . . , X d t , we have ∠ (m t , ν H ) = 0 which concludes the proof of the claim.
ii) On the time interval [σ 0 , σ 1 ) the processes are coupled, so there is nothing to prove in this case.
On the time interval [σ 1 , σ 2 ), in an appropriately chosen coordinate system we have Y t = X 1 t , . . . , X d t , thus the mirror M t coincides with the boundary hyperplane H d = z 1 , . . . , z d ∈ R d : z d = 0 of D 2 where the reflection takes place, thus M t ∩ D 2 = ∅ in this case.
Inductively, assume the claim is true for t < σ n . By continuity, M σn ∩ D 2 = ∅, By construction, the driving Brownian motion Z n t of Y n t satisfies Consider the process and note that since Y is F W -adapted and ||G|| = 1, by Lévy's characterization of Brownian motion, Z t is a free d-dimensional Brownian motion starting at Z 0 = 0, also adapted to the filtration F W . We will show that Z is the driving process of the reflecting Brownian motion Y t , i.e. we have Note that the mapping z −→ G (z) is continuous with respect to the norm If Y s = B s ∈ D, since D n ր D, there exists N ≥ 1 such that B s ∈ D N , hence B s ∈ D n for all n ≥ N . By Proposition 3.10, it follows that Y n s = B s for all n ≥ N , hence in this case we also have G If Y s = B s ∈ ∂D, since D n ⊂ D we have Y n s − B s = 0, and therefore by the definition of G we have: since Y t is a reflecting Brownian motion in D, and therefore it spends zero Lebesgue time on the boundary of D.
Since ||G|| = 1, using the above and the bounded convergence theorem we obtain and therefore by Doob's inequality it follows that for any t ≥ 0, which shows that Z n t converges uniformly on compact sets to From the construction, Z n t is the driving Brownian motion for Y n t , that is and passing to the limit with n → ∞ we obtain where A t = lim n→∞ t 0 ν Dn (Y n s ) dL Yn s . It remains to show that A t is a process of bounded variation. For an arbitrary partition 0 = t 0 < t 1 < .
where σ 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 Dn (t, x, y) (see the remarks preceding Theorem 2.1 and the proof of Theorem 2.4 in [7]). From the above it follows that A t = Y t − x − Z t is a continuous, F W -adapted process (Y t , Z t are continuous, F W -adapted processes) of bounded variation.
By the uniqueness in the Doob-Meyer semimartingale decomposition of Y treflecting Brownian motion in D, it follows that where L Y is the local time of Y on the boundary ∂D, and therefore the reflecting Brownian motion Y t in D constructed above is a strong solution to concluding the proof of Theorem 3.1.

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, y) of a smooth bounded domain D ⊂ R d we have the monotonicity in the case of the Neumann heat kernel should be reversed. The above observation was conjectured by Isaac Chavel ([12]), as follows: Conjecture 5.1 (Chavel's conjecture, [12]). Let D 1,2 ⊂ R d be smooth bounded convex domains in R d , d ≥ 1, and let p D1 (t, x, y), p D2 (t, x, y) denote the Neumann heat kernels in D 1 , respectively D 2 . If D 2 ⊂ D 1 , then for any t ≥ 0 and x, y ∈ D 1 .
Remark 5.2. The smoothness assumption in the above conjecture is meant to insure the a.e. existence the inward unit normal to the boundaries of D 1 and D 2 , that is the boundary should have locally a differentiable parametrization. Requiring that the boundary of the domain is of class C 1,α (0 < α < 1) is a convenient hypothesis on the smoothness of the domains D 1,2 .
In order to simplify the proof, we will assume that D 1,2 are smooth C 2 domains (the proof can be extended to a more general setup, by approximating D 1,2 by less smooth domains).
Among the positive results on Chavel conjecture, the most general known results (and perhaps the easiest to use in practice) are due to I. Chavel and W. Kendall (see [12], [16]), and they show that if there exists a ball B centered at either x or y such that D 2 ⊂ B ⊂ D 1 , then the inequality (5.1) in Chavel's conjecture holds true for any t > 0.
While there are also other positive results which suggest that Chavel's conjecture is true (see for example [11], [14]), in [4] R. Bass and K. Burdzy showed that Chavel's conjecture does not hold in its full generality (i.e. without additional hypotheses).
We note that both the proof of Chavel (the case when D 1 is a ball centered at either x or y) and Kendall (the case when D 2 is a ball centered at either x or y) relies in an essential way that one of the domains is a ball: the first uses an integration by parts technique, while the later uses a coupling argument of the radial parts of Brownian motion, and none of them can be applied to the other case.
Using the mirror coupling, we can derive a simple, unifying proof of these two important results, as follows: Theorem 5.3. Let D 2 ⊂ D 1 ⊂ R d be smooth bounded domains and assume that D 2 is convex. If for x, y ∈ D 2 there exists a ball B centered at either x or y such that D 2 ⊂ B ⊂ D 1 , then for all t ≥ 0 we have (5.2) p D1 (t, x, y) ≤ p D2 (t, x, y) .
Proof. Consider x, y ∈ D 2 fixed and assume without loss of generality that D 2 ⊂ B = B (y, R) ⊂ D 1 for some R > 0.
Consider a mirror coupling (X t , Y t ) of reflecting Brownian motions in (D 1 , D 2 ) starting at y ∈ D 2 .
The idea of the proof is to show that at all times Y t is at a distance from y smaller than (or equal) to that of X t from y.
To prove the claim, consider a time t 0 ≥ 0 when the processes are at the same distance from y, that is |Y t0 − y| = |X t0 − x|. If X t0 = Y t0 , for t ≥ t 0 the distances from X t and Y t to y will remain equal until the time t 1 when the processes hit the boundary of D 2 , and Y t receives a push in the direction of the inward unit normal to the boundary of D 2 . Since D 2 is convex, this decreases the distance of Y t from y, and the claim follows in this case.
If the processes are decoupled and |Y t0 − y| = |X t0 − x|, the hyperplane M t0 of symmetry between X t0 and Y t0 passes through y, and the ball condition shows that we cannot have X t0 ∈ ∂D 1 . Therefore for t ≥ t 0 , the processes X t and Y t will remain at the same distance from y until Y t ∈ ∂D 2 , when the distance of Y t from y is again decreased by the local push received as in the previous case, concluding the proof of the claim.
Therefore, for any ε > 0 we have and dividing by the volume of the ball B (y, ε) and passing to the limit with ε ց 0, from the continuity of the transition density of the reflecting Brownian motion in the space variable we obtain for any t ≥ 0, concluding the proof of the theorem.
Remark 5.4. As also pointed out by Kendall in [16] (the case when D 2 is a ball), we note that the convexity of the larger domain D 1 is not needed in the above proof in order to derive the validity of condition (5.1) in Chavel's conjecture.
Remark 5.5. We also note that the above proof uses only geometric considerations on the relative position of the reflecting Brownian motions coupled by mirror coupling. Analytically, the above proof reduces to showing that R t = |X αt | 2 − |Y αt | 2 ≥ 0 for all 0 ≤ t < ξ = inf {s > 0 : X s = Y s }, where R t is the solution of the following stochastic differential equation where B t = αt 0 Xs−Ys is the inverse of the non-decreasing process A t defined by Perhaps a better understanding of the mirror coupling, based on the analysis of the local times L X and L Y spent by X t and Y t on the boundaries of D 1 , respectively D 2 , in connection to the geometry of the boundaries ∂D 1 and ∂D 2 could give a proof of Chavel's conjecture for some new classes of convex domains, but so far we were unable to implement it.
We have chosen to carry out the construction of the mirror coupling in the case of smooth domains with D 2 ⊂ D 1 and D 2 convex, having in mind the application to Chavel's conjecture. However, although 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 when D 1 and D 2 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 ⊂ D 1 (needed in order to reduce by a localization argument the construction to the case D 1 = R d ) and the hypothesis on the convexity of the inner domain D 2 (which allowed us to construct a solution of the equation of the mirror coupling in the case D 1 = R d ).
Replacing the first hypothesis by the condition that the boundaries ∂D 1 and ∂D 2 are not tangential (needed for the localization of the construction of the mirror coupling) and the second one by condition that D 1 ∩ D 2 is 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 (D 1 , D 2 ). 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 when D 2 = (0, ∞) ⊂ D 1 = R.
Fixing x ∈ (0, ∞) as starting point of the mirror coupling (X t , Y t ) in (D 1 , D 2 ), the equations of the mirror coupling are (5.6) where in this case Until the hitting time τ = {s > 0 : Y s ∈ ∂D 2 } of the boundary of ∂D 2 we have L Y t ≡ 0, and with the substitution U t = − 1 2 (Y t − X t ), the stochastic differential for Y t becomes 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 function σ is N = {0} and σ −2 is locally integrable on R.
In fact, more can be said about the solutions of (5.7) in this case. It is immediate that both U t ≡ 0 and U t = W t are solutions to 5.7, 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 time τ of the boundary, we obtain as solutions and an intermediate range of solutions, which agree with (5.8) for some time, then switch to (5.9) (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 (5.8)), or they immediately split apart after coupling (as in (5.9)), and there is a whole range of intermediate possibilities. The first case can be referred to as sticky mirror coupling, the second as non-sticky mirror coupling, and the intermediate possibilities as weak/mild sticky mirror coupling.
The same situation occurs in the general setup in R d , and it is the cause of lack uniqueness of the stochastic differential equations which define 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 constructions and of the applications, although the other types of mirror coupling might prove useful in other applications.