Homogenization of a multivariate diffusion with semipermeable interfaces

We study the homogenization problem for a system of stochastic differential equation with local time terms that models a multivariate diffusion in presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.


Setting and the main results
The mathematical homogenization problem usually deals with the study of effective parameters of a system with rapidly varying spatial characteristics.Its original motivation comes from the analysis of composite materials with periodic structure.If the period of the structure is very small in comparison to the object's macroscopic size one can consider the material as a new homogeneous substance.A typical example here is the analysis of material's thermal conductivity.In mathematical terms, one solves a boundary value problem ´∇pApx{εq∇u ε pxqq " f in some domain G Ď R n with certain boundary conditions on BG.The matrix (thermal conductivity tensor) Ap¨q is assumed to be periodic in R n , and the parameter ε ą 0 is small.The solution u ε gives the temperature distribution in the domain G.The goal of homogenization consists in determining the effective thermal conductivity A such that u ε Ñ u as ε Ñ 0 where u is the solution of the homogenized equation A∆u " f .There is vast literature devoted to this subject see, e.g., Bensoussan et al. [1], Jikov et al. [16], and Berdichevsky et al. [2], Chechkin et al. [5] that comprises both analytic and stochastic methods.
In the present paper, we consider a different type of a stochastic homogenization problem: homogenization of a diffusion in the presence of narrowly located semipermeable hyperplane interfaces.In simple words, our model may remind of a dynamics of heat conduction in a foiled composite material consisting of a media interlaced with very thin plates of different permeability.In material science such models are referred to as reinforced materials like a glass wool reinforced by aluminium foil, as it was considered in Yüksel et al. [43].
To formulate the model rigorously, we introduce a small parameter ε ą 0 and assume that the state space R ˆRn , n ě 0, is sliced into layers by countably many hyperplane interfaces (membranes) such that the family ta ε k u Ď R has no accumulation points.We look for a n `1 dimensional continuous strong Markov process pX ε , Y ε q " pX ε , Y ε,1 , . . ., Y ε,n q, X ε P R, Y ε P R n that behaves as follows.
Between the membranes tta ε k u ˆRn u kPZ the process pX ε , Y ε q is a usual diffusion with the drift vector pb i q 0ďiďn and the diffusion matrix pσ i l q 0ďiďn,1ďlďm , m ě 1. Upon hitting a membrane ta ε k u ˆRn at a point pa ε k , yq at time τ , the process pX ε , Y ε q starts anew and 'leaves' the membrane in the direction ˘p1, θpa ε k , yqq with the so-called penetration probabilities 1  2 p1 ˘εβpa ε k , yqq.We look for such a process pX ε , Y ε q as a solution of the stochastic differential equation (SDE) where W " pW 1 , . . ., W m q is a standard m-dimensional Brownian motion and L a pX ε q is the symmetric local time of X ε at a P R (see the end of this Section for precise definitions).The case n " 0 corresponds to a one-dimensional diffusion X ε and point interfaces ta ε k u located on the real line.
The goal of this paper is to establish the existence and uniqueness of a weak solution of the system (1.2) and to study its homogenization limit as ε Ñ 0.
We make the following assumptions about the coefficients in the system (1.2) and the location of the membranes.By C k b , k ě 1, we denote the space of k times continuously differentiable bounded functions with bounded derivatives.Assumptions A: A coeff : β, b i , θ i , σ i l P C 2 b pR ˆRn , Rq, i " 0, . . ., n, l " 1, . . ., m. ( A a : The points ta ε k u satisfy a ε k " where d P C 1 b pR, Rq such that inf xPR dpxq ą 0. A Σ : The matrix Σ " pΣ ij q 0ďi,jďn with the entries is uniformly positive definite.
The main result of this paper is formulated in the following theorems.
Theorem 1.1.Let Assumptions A hold true.Then there is ε 0 P p0, 1s small enough such that for all ε P p0, ε 0 s the system (1.2) has a unique weak solution pX ε , Y ε q, which is a strong Markov process.
Remark 1.2.The condition that the parameter ε ą 0 is small is essential.First, since the values 1 2 p1 εβpx, yqq have the meaning of penetration probabilities through the membranes, we have to demand that ε}β} 8 ď 1; otherwise the solution pX ε , Y ε q does not exist even in the simplest one-dimensional case of a skew Brownian motion, see Harrison and Shepp [14].Another condition on the smallness of ε will be needed for the proof of uniqueness of the solution in a neighbourhood of each membrane a ε k , see Section 2. It is well known that uniqueness is necessary for the strong Markov property of the solution.
Theorem 1.3.Let Assumptions A hold true and let pX ε , Y ε q be unique weak solutions of (1.2).In the limit ε Ñ 0 the weak convergence pX ε , Y ε q ñ pX, Y q (1.6) in CpR `, R n`1 q holds true, where pX, Y q is a (unique) weak solution of the SDE i " 1, . . ., n. (1.7) As we see, the limiting process is a diffusion with the same diffusion matrix tσ i l u as the processes pX ε , Y ε q.The semipermeable membranes, i.e., the local time terms of (1.2), turn into the regular additional drift which can alter diffusion's behaviour significantly as it is demonstrated in the following example.
Example 1.4.Let n " 1.We consider a two-dimensional stochastic differential equation that describes, e.g., the velocity of a charged particle in a constant magnetic field subject to external stochastic electric field pW 1 , W 2 q, see Chechkin et al. [4].The random trajectory of a particle tends to make windings around 0 in the positive direction, see Fig. 1

(left).
Let us add to the model spatial membranes (lines) located equidistantly at x-points a ε k " kε, k P Z, so that dpxq " 1. Assume that in some large ball around the origin the permeability characteristics β and θ of the membranes are given by βpx, yq " 2y 3 γ `y2 , θpx, yq " ´x3 y pγ 2 `x2 qpγ `y2 q , γ " 10 ´2. (1.9) Outside this ball we define them in such a way that they satisfy the assumptions A. Then in the limit ε Ñ 0, the diffusion with local times pX ε , Y ε q converges to a diffusion pX, Y q which solves the SDE with additional interface-induced drift: (1.10) In a large ball around the origin, the new drift approximately equals pY, ´Xq, i.e., the rotation direction of the homogenized particle is now negative, see Fig. 1 (right).
The novelty of this paper, besides the stochastic homogenization Theorem 1.3, consists in the proof of the existence and uniqueness Theorem 1.1 in a multivariate setting.
In dimension one, the first result on a existence and uniqueness for an SDE with local time was obtained by [14].They proved that a skew Brownian motion is a unique strong solution to an SDE X t " x `Wt p2β ´1qL 0 t pXq for |β| ď 1.In the proof, by using the speed measure the initial equation was transformed regularly into an equivalent one with the locally constant coefficients and without local times.Later, Le Gall [20] generalized this approach and applied it to the general one-dimensional SDEs with local times of the unknown process.The resulting SDE has possibly discontinuous coefficients but does not involve local times.The general theory was outlined in Engelbert and Schmidt [7], see also Lejay [21].
Sample paths of the "underlying" diffusion pX 0 , Y 0 q without membranes (left) and of the limiting homogenized diffusion pX, Y q (right).The membranes with penetration probabilities 1  2 `ε 2 βpx, yq and penetration directions p1, θpx, yqq generate an additional drift that reverses the diffusion's rotation direction.Both samples start at the point p2, 2q.
In higher dimensions, diffusion processes with reflection (i.e., with εβpx, yq " 1) are best studied.The existence problem for them is usually formulated in terms of a martingale problem (see, e.g., Stroock and Varadhan [38]) or the Skorokhod problem (see, e.g., Tanaka [39], and Lions and Sznitman [22]).Zaitseva [44] considered an SDE for a multidimensional Brownian motion with oblique skewness at a hyperplane provided that the skewing coefficient and the reflection direction are constant.Under this condition, the tangential coordinates do not depend on the normal one.This allows to obtain the strong existence and uniqueness of a solution by standard arguments.Unfortunately, this method is inapplicable in the case of oblique reflection since the coordinates are necessarily dependent.
The general case of an oblique skew diffusion with non-constant coefficients is much more complicated and the question of existence and uniqueness is still open.Portenko [33,34] formulated the problem of existence and uniqueness of a diffusion process with semipermeable membranes in term of a parabolic conjugation boundary problem.With the help of potential theory methods it was proved that the solution of such a problem defines the semigroup of operators that corresponds to a process which diffusion characteristics exist in the sense of generalized functions.Portenko and Kopytko [32] obtained the same result for a diffusion with a semipermeable membrane on a hyperplane with oblique reflection.However it does not follow immediately from these works that the constructed process is a (unique) solution of an SDE with local time terms.
We also mention the works by Ouknine et al. [28] and Ramirez [35] who investigated diffusions with infinitely many interfaces in dimensions one and two.
Concerning the homogenization problem, besides the works cited in the beginning of the paper, the following authors studied homogenization of (regular) diffusions by analytic and stochastic methods: Pardoux [29], Pavliotis and Stuart [30], Hairer and Pardoux [13], Makhno [23].
Results on homogenization for diffusions with interfaces are rather sparse.Weinryb [41] studied the periodic homogenization of planar diffusions with permeable membranes on lines or circles under the assumption on existence and uniqueness of solution to the corresponding martingale problem.Hairer and Manson [11,12,10] studied periodic diffusion homogenization with one interface.Limit theorems for onedimensional diffusions with interfaces were obtained by Makhno [24,25] and Krykun [18].
Sketch of the proof.The main idea of the proof is based on the decomposition of the solution pX ε , Y ε q into segments that live on random time intervals between the subsequent hittings of the neighbouring membranes.The existence and uniqueness result (Theorem 1.1) for the system (1.2) will be obtained with the help of a multivariate coordinate transformation pU ε , V ε q " pF ε pX ε , Y ε q, GpX ε , Y ε qq that transforms the system (1.2) into an equivalent system of SDEs with discontinuous coefficients without local times in a neighbourhood of each membrane.The uniqueness will follow from the result by Gao [8] and the global solution is obtained by glueing these solutions together (see Section 2).
To prove the convergence Theorem 1.3 we note that the essential dynamics of the diffusion with local times pX ε , Y ε q is catched by the embedded Markov chain that comprises the values pX ε , Y ε q at the hitting times of the membranes.
To obtain the formula for the infinitesimal operator L of the limit diffusion we study the pseudocharacteristic operator of the embedded Markov chain where τ ε is the first hitting time of a neighbouring membrane.Let for brevity n " 1. Writing the Taylor formula for a function We will be able to calculate the fine asymptotics of the average exit time from the current strip Eτ ε , which will be of the order ε 2 as well as the moments E x,y pX ε τ ε ´xq, E x,y pY ε τ ε ´yq, etc, with accuracy Opε 3´δ q, δ P p0, 1q (see Lemma 3.5 and Section 3.4).These estimates will be derived by approximating the transformed diffusion without local times by a diffusion with coefficients "frozen" at the initial point px, yq.
The limit second order generator L that determines the limit diffusion pX, Y q, see equation (1.7), is a pointwise limit of the sequence tL ε u as ε Ñ 0.
The proof of the convergence Theorem 1.3 consists of a) the standard step of establishing the weak relative compactness of the family pX ε , Y ε q (Section 4.1) and b) of verification that any limit process pX, Y q satisfies the (well-posed) martingale problem Notation.In this paper, | ¨| denotes the Euclidiean distance in R n , n ě 1 as well as the Frobeinus norm of a matrix A, i.e., |A| " p ř i,j a 2 ij q 1{2 ; }f } 8 " sup x |f pxq| is the supremum norm of a real-, vector-or matrix-valued function f ; x `" maxtx, 0u.The Jacobian matrix of a vector-valued function f is denoted by Df .In particular, for a smooth f we have |f pxq ´f pyq| ď }Df } 8 |x ´y|.
Sometimes the constant C ą 0 denotes a generic constant that does not depend on ε; its value may vary within the same chain of inequalities.
We write that f pεq À gpεq if |f pεq| ď C|gpεq| for small ε ą 0. We also will write We also recall that the symmetric local time L a pXq at a P R of a continuous real-valued semimartingale X is the unique non-decreasing process satisfying where see, e.g., Chapter VI in Revuz and Yor [36].The local time L a pXq satisfies 2 Existence and uniqueness.Proof of Theorem 1.1 First, we consider the system (1.2) in a neighbourhood of one membrane.Without loss of generality we fix k " 0 so that a ε 0 " 0. Since outside the membrane pX ε , Y ε q is a diffusion with regular coefficients, we assume that the initial values are x " 0 and y P R n .With some abuse of notation we denote βpyq :" βp0, yq, θpyq :" θp0, yq and look for the solution pX ε , Y ε q of the following SDE with one membrane: We construct the transformation of pX ε , Y ε q into a diffusion without the local time terms.Let ε ą 0 be small enough such that ε}β} 8 ď 1{2, and let We have ( Define the functions Gpx, yq :" y ´xθpyq. ( It is clear that the transformation px, yq Þ Ñ pF ε px, yq, Gpx, yqq does not have to be a one-to-one bijection of R ˆRn .However, this map is injection in a "thin" strip r´Aε, Aεs ˆRn , for any A ą 0 fixed and ε ą 0 small enough.
The proof will consist in the application of the Hadamard's global inverse function theorem, see Theorem 6.2.4 in Krantz and Parks [17].
Proposition 2.3.The process pX ε , Y ε q tďτ ε is a (weak) solution of (2.1) if and only if pU ε , V ε q tďτ ε is a (weak) solution of (2.17 where xθ i y j pyqσ j l px, yq, i " 1, . . ., n, l " 1, . . ., m. (2.18) Proof.Recall the Tanaka formula Hence the application of the Itô formula to B ε pY ε q ´1 and the product Itô formula yields that and the representation (2.17) follows immediately.Note that all the coefficients of the system (2.17) are smooth on the half-spaces R ˘ˆR n and is discontinuous on the hyperplane U " 0. To transform the system (2.17) into (2.1)we apply the Itô formula with local times as proven by Peskir [31].
Proof of Theorem 1.1.We consider the system (2.17) in some strip r´A 2 ε, A 2 εsˆR n extend all the coefficients in (2.17) such that they are bounded, smooth functions on the half-spaces tu ă 0u and tu ą 0u that may have a discontinuity on tu " 0u.Note that the diffusion matrix of pU ε , V ε q is bounded, uniformly elliptic on R ˆRn , continuous on the half-spaces tu ă 0u and tu ą 0u but may have a discontinuity on the hyperplane t0u ˆRn .Hence the system (2.17) has a weak solution by Krylov [19].In dimension one, i.e., for n " 0, uniqueness follows by means of the theorems by Nakao [27] and Yamada and Watanabe [42].In dimension two, i.e., for n " 1, uniqueness follows again from Krylov [19].For n ě 2, uniqueness follows from Theorem 1.1 in Gao [8].Hence, this solution is also uniquely defined up to the exit from the strip and this solution is independent of extension of coefficients to the whole space.Since all the coefficients are bounded, the process V ε cannot explode in finite time with probability 1. Hence the process pX ε , Y ε q is also uniquely defined up to the moment of exiting from some strip around the membrane.A global solution is obtained by gluing together the solutions in each strip.By uniqueness, this solution is strong Markov.
Eventually we have to check that the solution does not blow up in a finite time.This will be shown in Lemma 4.2 later.l

Dynamics around one membrane
We consider the dynamics of the process pX ε , Y ε q inside the strips pa ε k´1 , a ε k`1 q, k P Z.For definiteness we assume in this section that k " 0, a ε 0 " 0 and denote a ε ˘" ˘aε ˘1 ą 0, d :" dp0q, d 1 :" d 1 p0q.Moreover we omit the argument x " 0 in all functions, i.e., we write βpyq :" βp0, yq, b i pyq :" b i p0, yq etc.
The aim of this section is to obtain accurate moment estimates for expectations that appear in (1.12).This will allow us to show that the operator L ε defined in (1.11) approximates in some sense generator of the limit diffusion.

Lemma 3.1.
Let A ą 0 be fixed.For ε ą 0 small enough the following first order expansion holds true in a strip r´Aε, Aεs ˆRn : and Using the previous estimate we obtain (3.17)

Rough estimates of the exit time from the strip
´, a ε `qu. (3.18) To study the exit from the strip, we use the transformation (2.20) where U ε , V ε are Itô processes without the local time terms.
We extend the processes pU ε t , V ε t q tďτ ε obtained in (2.20) to t ě 0 by setting With the help of (3.5) we have (3.20) Recall constants q A, p A from (3.3), (3.4), and define the stopping times Consequently, sup Proof. 1.We use (3.22) and show that E 0,y e γ p τ ε {ε 2 ď A for all ε ą 0 small.Consider the Lyapunov function For α ą p A 2 , h ε pt, xq ě 0 on x P r´p Aε, p Aεs.We have The Itô formula yields for every N ě 1: (3.30) for some γ ą 0 small and all ε ą 0 small.Hence and the statement follows for p τ ε by the monotone convergence theorem.Using that x k ď e x k! we get and the lower bound for E 0,y τ ε follows.
Lemma 3.3.For each k ě 1 and ε ą 0 small enough In particular, for any δ P p0, 1q and any n ě 1 With the help of the Doob inequality and (2.20), we estimate for each i " 1, . . ., n and k ě 1: (3.37)By Markov's inequality, for any k ě 1 and δ P p0, 1q Choosing k large enough, the order of the r.h.s.can be made arbitrarily small.The estimates for Y ε follow from Lemma 3.1.

Accurate estimates for X ε
Recall (3.19) and decompose the process U ε into the sum where the terms B x b 0 psq, B y i b 0 psq, etc. are bounded by Assumption A. In the representation (3.39), the process R is a Brownian motion with drift, and A ε and M ε are "small" on t P r0, τ ε s as shown in the next Lemma.
Lemma 3.4.For ε ą 0 small enough we have Proof.To obtain these estimates, we use Remark 2.4, Lemmas 3.2 and 3.3 as well as the Cauchy-Schwarz inequality: To estimate the martingale M ε we apply the Doob inequality: (3.45) We will prove that the exit probabilities of X ε from p´a ε ´, a ε `q coincide with the exit probabilities of the Brownian motion with drift R from the modified interval p´a ε ´, a ε `p1 ´2εβpyqqq up to the terms of the order Opε 2´δ q and calculate the moments of X ε τ ε with accuracy of the order Opε 3´δ q.Note that the local time terms that disappeared upon transition to the process U ε contribute now to the asymmetry of the exit interval.
Lemma 3.5.For any δ P p0, 1q and for ε ą 0 small enough we have Proof. 1.On the one hand we have (3.50) The Taylor formula for the mapping pε, Y q Þ Ñ B ε pY q yields where the remainder is estimated by |rpε, Y ´yq| ď Cpε 2 `ε|Y ´y| `ε|Y ´y| 2 q (3.52) for some C ą 0. Hence we get and for δ P p0, 1q with the help of Lemma 3.3 we estimate (3.54) Hence using (3.1) we get (3.55) 2. On the other hand, taking into account (3.39) and (3.41) we estimate where ) . (3.57) To calculate E 0,y R ρ ε , use the explicit formula 3.0.4(a) on p. 309 from Borodin and Salminen [3]: that gives us the asymptotics The mean value E 0,y ρ ε is obtained with the help of the formula 3.0.1 on p. 309 from Borodin and Salminen [3]: Hence by Lemma 3.6 (below) we get the estimate Lemma 3.6.For ε ą 0 small enough we have so that h 1 puq " ´2u `r a ε `´a ε ´, h 2 puq " ´2.The Itô formula on the event tρ ε ď τ ε u yields where we used that ř m l"1 σ 0 l px, yq 2 ě δ 1 ą 0, |b 0 px, yq `ϕε,0 px, yq| ď δ 2 , and |U ε t | ď Cε, for some C ą 0 fixed and ε ą 0 sufficiently small.We get We also have The second factor in the latter formula is of the order Opεq, and by Lemma 3.4 we have (3.69) 2. On the event tτ ε ď ρ ε u we have analogously: and the estimates on hpU ε τ ε q and A ε τ ε , and M ε τ ε apply as in the previous step.Combining the estimates in steps 1 and 2 yields (3.63).

Proof of Theorem 1.3
Let pX ε , Y ε q, ε P p0, ε 0 s, be weak solutions of (1.2) and pX, Y q be a weak solution of the limit SDE (1.7).
Recall that all these processes are strong Markov.The proof of the weak convergence pX ε , Y ε q ñ pX, Y q in Cpr0, 8q, R n q consists, as usual, in the proof of the weak relative compactness of the family pX ε , Y ε q and the proof of the convergence of finite dimensional distributions.
Denote by S ε pa ε j q " pa ε j´1 , a ε j`1 q the stripe around the j-th membrane.Let be the hitting times of the neighbouring membranes.Let ν ε " pν ε t q be a counting process of visits to ta ε k u, i.e., is a martingale difference and By the strong Markov property for any λ P r0, T s we have: for K ě 2{C 1 .By Markov's inequality which vanishes in the limit ε Ñ 0.

Weak relative compactness
The weak relative compactness of the family pX ε , Y ε q in CpR `, R n`1 q will follow from the its weak relative compactness in the Skorokhod space DpR `, R n`1 q guaranteed by the compact containment condition and Aldous' criterion (e.g., see conditions 3.21i and 4.4, and Theorem 4.5 in Chapter VI in Jacod and Shiryaev [15]), and the continuity of the paths of pX ε , Y ε q.
Lemma 4.2 (compact containment).For any T ą 0 and δ ą 0 and there is R " R δ,T ą 0 such that uniformly over x P R and y P R n belonging to compacts.
Proof. 1. Recall that τ ε 0 " inftt ě 0 : X t P ta ε k uu is the first hitting time of a membrane.Since the process pX ε , Y ε q is a regular diffusion on the interval r0, τ ε 0 s it is easy to see that sup px,yqPR n`1 Hence, in order to prove (4.8) it suffices to consider starting points px, yq P ta ε k u ˆRn .2. Let δ P p0, 1q.By Lemma 4.1, there is K ą 0 and ε 0 pδq ą 0 such that for all ε ď ε 0 pδq 3. With abuse of notation, denote now by p∆ ε k q kě1 the martingale difference Due to the strong Markov property and (3.47) we have so that due to (3.48) we get for some constants C 1 , C 2 ą 0. By Doob's inequality for ε ą 0 small enough and R ą 0 large enough we get The process is uniformly bounded by }d} 8 }θ} 8 ε.
, and on t P rτ ε k , τ ε k`1 s the process V ε has the representation where the process I ε is a diffusion  The estimates for the process Y ε are obtained analogously.

Convergence of finite dimensional distributions
We prove the convergence of finite dimensional distributions by the martingale problem method.Let pX, Y q be the diffusion with the generator L, X 0 " x P R, Y 0 " y P R n .
Let f P C 3 b pR n`1 , Rq, and let of the process pX, Y q, which is defined by equation (1.7).We have to show that for any l ě 1, h 1 , . . .h l P C b pR n`1 , Rq, and any 0 ď s 1 ă ¨¨¨ă s l ă s ă t E x,y "´f pX ε t , Y ε t q ´f pX ε s , Y ε s q ´ż t s Af pX ε u , Y ε u q du ¯l ź j"1 This convergence will essentially follow from the next lemma.
uniformly over x P ta ε k u and y P R n .

0
Ip|X s ´a| ď δq dxXy s .(1.16) Acknowledgments: O.A. and I.P. thank the German Research Council (grant Nr.PA 2123/7-1) and the VolkswagenStiftung (grant Nr. 9B946) for financial support.O.A. and A.P. were partially supported by the Alexander von Humboldt Foundation within the Research Group Linkage cooperation Singular diffusions: analytic and stochastic approaches between the University of Potsdam and the Institute of Mathematics of the National Academy of Sciences of Ukraine.The authors are grateful to the anonymous referees for their helpful reports.