Random Homogenization and Convergence to Integrals with respect to the Rosenblatt Process

Transcription

1 andom Homogenization and Convergence to Integrals with respect to the osenblatt Process Yu Gu Guillaume Bal September 7, Abstract This paper concerns the random fluctuation theory of a one dimensional elliptic equation with highly oscillatory random coefficient. Theoretical studies show that the rescaled random corrector converges in distribution to a stochastic integral with respect to Brownian motion when the random coefficient has short-range correlation. When the random coefficient has long range correlation, it was shown for a large class of random processes that the random corrector converged to a stochastic integral with respect to fractional Brownian motion. In this paper, we construct a class of random coefficients for which the random corrector converges to a non-gaussian limit. More precisely, for this class of random coefficients with long-range correlation, the properly rescaled corrector converges in distribution to a stochastic integral with respect to a osenblatt process. Introduction The equation of interest in this paper is the following one-dimensional elliptic equation with highly oscillatory coefficients: d dx ax, ω d dx u x, ω = fx, x,. where the coefficient ax, ω is a stationary, bounded with bounded inverse, random potential. We are interested in the limiting behavior of the solution u x, ω when and more precisely in the size of the random fluctuations of u x, ω and of their limiting distribution after proper rescaling. Homogenization theory has been extensively studied in both periodic and random settings; see for instance [4, 6, 8, 9,, 4]. In the random setting, homogenization theory replace the random medium by a properly averaged effective, deterministic, medium. It is well known that u converges to the deterministic, homogenized solution ūx as. Fewer results are available concerning the theory of random fluctuations and in particular Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 7 yg54@columbia.edu;gb3@columbia.edu

2 the theory of the random corrector u ū. In the one-dimensional setting, it has been shown that the property of the corrector strongly depends on the correlation property of the random potential. Using the explicit expressions of the solution to., it has been shown in [6] that when the random potential had short-range correlation and satisfies certain mixing properties, then the corrector s amplitude is of order and, after rescaling, converges in distribution to a stochastic integral with respect to Brownian motion. In [3], the result has been extended to a large class of random potential with long-range correlation, where the corrector s amplitude is of order α/, with α, characterizing the decay of the correlation function of the random coefficient. Furthermore, the weak convergence limit of the rescaled corrector is then a stochastic integral with respect to fractional Brownian motion. No theories of correctors are available for elliptic equations of the form. in higher spatial dimension; see [4] for estimates of the size of the random fluctuations. Similar results to the ones described above have been obtained in higher dimensions for Schrödingertype equations with random potential. In [], homogenization and corrector theory has been developed for a large class of second order elliptic equations with short-range correlated potentials, and it has been generalized into the long-range correlation case in []. In all of these cases, the limiting distributions of the rescaled correctors are Gaussian random fields, which admit convenient representations as a stochastic integral with respect to Brownian motion or fractional Brownian motion. In this paper, we focus on. and follow the framework in [3] to obtain non-gaussian limits for the random fluctuations. We construct a class of random potential with longrange correlation for which the limiting distribution of the rescaled corrector is no longer Gaussian as, but rather the so-called osenblatt distribution, which is an element in the second-order Wiener chaos. The corrector s amplitude is α for α, /. Our results are closely related to classical examples of non-central limits in probability theory; see [, 7, ]. Our approach is based on the explicit expression for the solution to. and a careful analysis of the oscillatory integral. The paper is organized as follows. In section, we state the main theorem of the paper and compare it with previously established results. In section 3, we give a brief introduction to the osenblatt process and the corresponding stochastic integral. In section 4, we prove some key results concerning the weak convergence of oscillatory integral. In section 5, we prove the main theorem and briefly discuss possible extensions. Main results We formulate the problem as follows: d dx ax, ω d dx u x, ω = fx, u, ω =, u, ω = b x,, ω Ω,. where ax, ω [a, a ] for some positive a, and is a stationary ergodic random process associated with the probability space Ω, F, P. fx C[, ] and b. Classical theory

3 of elliptic equations shows the existence of a unique solution u., ω H, P-a.s. It was shown that as the scale of the micro-structure, the solution u x, ω to. converges P-a.s. to the deterministic solution ūx of the following equation: d d dx a = fx, x,, dxūx. ū =, ū = b where a is the harmonic mean of ax, ω, i.e., a = E{a,.}. See e.g. [8, 9, ]. The analysis of the random corrector is based on the explicit expression for the solutions to. and.. If we denote a x = a x and F x = x fydy, we have: x x u x, ω = c ω a y, ω dy F y a y, ω dy, c ω = b + ūx = c x a x F y a dy, c = a b + F y a y,ω dy a dy.3 y,ω F ydy..4 We note that u x, ω ūx contains oscillatory integrals of the form a y, ω a hydy.5 for some function hy. Next we make some assumptions on the random process a y, ω a.. Assumptions on the random process Our goal is to analyze the statistical property of u ū as for a large class of random process ax, ω, and show the existence of a non-gaussian limiting corrector. To do this, we make the following assumptions on ax, ω. Let and assume that qx, ω = ax, ω a.6 qx, ω = Φgx, ω.7 for some function Φ and a random process gx, ω constructed explicitly as: gx = ξ [x+u].8 where {ξ k, k Z} is a centered stationary Gaussian sequence with unit variance, and its auto-correlation function rk = E{ξ ξ k } κ g k α for some κ g > and α,. U is uniformly distributed on [, ] and independent of {ξ k, k Z}. Through some elementary computation, we can show that gx defined in.8 is a centered stationary Gaussian process with unit variance and long-range correlation: g x = E{ggx} κ g x α. 3

4 ecall the definition of Hermite polynomials: H n x = n exp x dn dx n exp x,.9 assume E{Φgx } <, and define V n = E{H n gxφgx}, we make the following key assumptions on Φx: V = V =, V.. The smallest integer n such that V n is called the Hermite rank of Φ. So. states that the Hermite rank of Φ is. We will see later that V = is the key condition for the existence of a non-gaussian corrector. When V, by Theorem.3 below, the rescaled corrector converges in distribution to some stochastic integral with respect to fractional Brownian motion. The condition V = ensures that qx, ω is centered, and we have the following lemma concerning the correlation property of qx, ω. For simplicity, we denote it as qx from now on. Lemma.. Let x = E{qqx}, then where κ = V κ g. x κ x α. Proof. We have x = E{ΦgΦgx}, and by Hermite expansion Therefore, E{ΦgΦgx} = Since V = V =, we have n= Φgx = n= V n n! H ngx. V n n! E{H ngh n gx} = E{ΦgΦgx} = g x n= n= V n n! gx n.3 V n n! gx n.4 Because Vn n= n! < by assumption on Φ and g x κ g x α, we verify that The proof is complete. x = E{ΦgΦgx} V κ g x α..5 Since we assume α,, then x / L so that qx has long-range correlation and we have x M x α for some constant M. 4

5 . Main theorem Now we state the main theorem and compare it with the previous results. Theorem.. Let u and ū be the solutions in.3 and.4 and let qx, ω be a centered stationary random process of the form.7 with the Hermite rank of Φ being equal to. Then u ū is a random process in C[, ], and we have the following convergence in distribution in the space of continuous functions C[, ]: u x ūx α distribution Ūx.6 where Ūx = V κ g F x, yd D y.7 F x, y = c [,x] y F y [,x] y + xf y Here D y is a osenblatt process with D = α. F zdz a b [,] y..8 It should be contrasted with the convergence results for processes with long-range correlation and the Hermite rank equals to [3] or with short-range correlation [6]. Theorem.3. Let u and ū be the solutions in.3 and.4, and let qx, ω be a centered stationary random process of the form.7 with the Hermite rank of Φ being equal to. Then u ū is a random process in C[, ], and we have the following convergence in distribution in the space of continuous functions C[, ]: where u x ūx α/ κ g V Ūx = HH distribution Ūx.9 F x, ydb H y. Here F x, y is given by.8 and B H y is a fractional Brownian motion with Hurst index H = α. emark.4. In Theorem.3, we can assume α, instead of α, in Theorem.. Theorem.5. Let u and ū be the solutions in.3 and.4, and let qx, ω be a centered stationary random process of the form.7. If the correlation function g of g is integrable instead of being equivalent to x α at infinity, then is also integrable. The corrector u ū is a random process in C[, ] and we have the following convergence in distribution in the space of continuous functions C[, ]: u x ūx distribution Ūx. 5

6 where Ūx = τdτ F x, ydby. Here F x, y is given by.8 and By is a standard Brownian motion. We can see from the above theorems that the size of the corrector not only depends on the correlation property of the random process qx but also the Hermite rank of Φ, and the limiting distribution of the properly rescaled corrector can be non-gaussian. The rest of this paper is devoted to the proof of Theorem.. To do this, we first give a brief introduction of the osenblatt process and the stochastic integral with respect to it. 3 osenblatt process In this section, we briefly recall some general facts about the osenblatt process and the stochastic integral with respect to it [3]. 3. Non-central limit theorem and Wiener-Itô integral representation The following theorem gives rise to the osenblatt process []. Theorem 3.. Assume X n are centered stationary Gaussian sequence with unit variance rk = E{X X k } k D Lk, where D, / and Lk is a slowly varying function. Define with d N N D LN as N. Z N, t = d N [Nt] Xi Then the finite dimensional distributions of Z N, t converge to the corresponding finite dimensional distributions of the osenblatt process D t. By Theorem 3., we have the characteristic function of the osenblatt distribution D in a small neighborhood of the origin: { [ iθ n ]} expiθ D = exp n n= [,] n x x D x 3 x D... x x n D dx, 3. i= The osenblatt process has the following representation as a Wiener-Itô integral: t D t = cd s y +D + s y +D + dsdby dby. 3. The constant cd is chosen such that E{ D } =. From 3., we see that the osenblatt process lives in the second order Wiener chaos and is a self-similar process with stationary increments. The Hurst index H = D. 6

7 3. Stochastic integral with respect to osenblatt process The representation 3. is not very convenient to define stochastic integrals. ecall that for fraction Brownian motion, we have: B H t = t with B t, t [, T ] a standard Brownian motion and where t > s and K H t, sdbs 3.3 t K H t, s = c H s H u s H u H du 3.4 c H = There are similar results for osenblatt process: s HH β H, H. 3.5 Proposition 3.. Let K be the kernel in 3.4 and D t t [,T ] a osenblatt process with Hurst index H = D. Then it holds that t [ t ] t K H D t = dh y y u u, y KH u u, y du dby dby 3.6 where B t, t [, T ] is a standard Brownian motion, H = H+, and dh = H + H H. 3.7 By the representation 3.6, the stochastic integral with respect to osenblatt process can be defined as follows. We first rewrite D t = T T I [,t] y, y dby dby 3.8 where the operator I is defined on the set of functions f : [, T ], takes values in the set of functions g : [, T ] and it is given by T Ify, y = dh fu KH y y u If f is an element of the set E of step functions on [, T ] of the form u, y KH u u, y du. 3.9 n f = a i ti,t i+ ], = t < t <... < t n = T 3. i= 7

8 it is natural to define its stochastic integral with respect to D t as T n ftd D t = a i D t i+ D t i = i= T T Let H be the set of deterministic functions f such that It can be shown that Ify, y dby dby. 3. T T f H = HH fufv u v H dudv <. 3. f H = Therefore the mapping T T T Ify, y dy dy = E{ ftd D t }. 3.3 f T ftd D t 3.4 defines an isometry from E to L Ω and it can be extended by continuity to an isometry from H to L Ω because E is dense in H. We call this extension the Wiener integral of f H with respect to D t. 4 Analysis of oscillatory integrals From.3 and.4, we can see that the rescaled corrector u ū contains oscillatory integrals of the form: α α qx hxdx 4. for some compactly supported hx. results: The main goal of section is to prove the following Proposition 4.. Assume that hx is compactly supported in [, and continuous, then α g x hxdx distribution κ g hxd D x 4. where D x is the osenblatt process with D = α. Proof. Since gx = ξ [x+u], the LHS of 4. can be written as α [g x ]hxdx = k= A k, ξk 4.3 where A k, = k+ U k U hx dx. 4.4 α 8

9 Since hx is compactly supported, the sum contains finitely many terms. We compute the conditional characteristic function of 4.3 as follows: { } c θ =E exp iθ α gx hxdx U { } 4.5 =E exp iθ A k, ξk U k= Freeze U, then A k, are constants. If we assume k = m,..., n and n m + = N, then we have n c θ = exp iθ A π N Σ N k, x k exp N x Σ N x dx k=m = Σ N Σ N A 4.6 N, θ exp TrAN, θ = I N Σ N A N, θ exp TrAN, θ where Σ N is the covariance matrix of ξ m,..., ξ n, and A N, θ is the N N diagonal matrix where the diagonals are iθa k,, k = m,..., n. Let λ k, θ, k =,..., N be the eigenvalues of Σ N A N, θ, we claim that there exists δ >, such that if θ < δ, we have N N c θ = exp λ k, θ λ k, θ = exp k= k= N λ k, θ + ln λ k,θ k= = exp n n To see this, we only need to show that when θ < δ, then for every N, we have n= 4.7 N λ k, θ n. k= n n n= N λ k, θ n <. 4.8 k= Actually, λ k, θ M θ α N i,j= ri j for some constant M. Since N converges as, and by the result in [], N i,j= ri j ON α, the claim is proved. Next, we show that as, N k= λ k,θ n converges for each n. Since λ k, θ are the eigenvalues of Σ N A N, θ, we have N λ k, θ n = TrΣ N A N, θ n. 4.9 k= If we denote Σ N A N, θ ij = ρ ij, we can write the HS of 4.9 as follows: TrΣ N A N, θ n = N N... i = i = 9 N ρ i i ρ i i 3... ρ in i n ρ ini. 4. i n=

10 It is straightforward to check that By stationarity, we have ρ kj = r j k iθa j+m,. 4. TrΣ N A N, θ n =iθ n A i,a i,... A in,r i i r i 3 i... r i i n. 4. By 4.4, we have where i,...,i n=m,...,n A k, = N α N α B k,n = k+ U k U k+ U k U is an approximation to hx. Therefore, hxdx := N α N α B k,n 4.3 hxdx, k = m,..., n 4.4 TrΣ N A N, θ n =iθ n N n α i,...,i n=m,...,n N n B i,n... B in,nr i i N α... r i i n N α. 4.5 By assumption, rk κ g k α so that there exists some constant M independent of n and such that TrΣ N A N, θ n θ n M n 4.6 and TrΣ N A N, θ n iθκ g n hx hx... hx n n x x α x 3 x α... x x n α dx... dx n 4.7 as, where we have used the fact that hx is continuous. Therefore, if θ < δ, we have iθκ g n C n c θ exp 4.8 n where n= hx hx... hx n C n = n x x α x 3 x α... x x n α dx... dx n. 4.9 We have shown that the conditional characteristic function c θ converges in a small neighborhood of the origin, and we verify that izκ g n C n c z = exp 4. n n=

11 is analytic when z < δ. Therefore, c z agrees with a unique characteristic function for all real values of z. We still denote the characteristic function as c z. Take expectation of c θ in 4.5, by the Dominated Convergence Theorem, we have E { exp iθ α gx hxdx } c θ. 4. Therefore, the random variable g x α hxdx converges in distribution and we claim that the limit is κ g hxd Dx. To see this, we only have to prove that the characteristic function of κ g hxd Dx agrees with c θ when θ < δ. Let h x = N i= a,i t i,t i ] x, = t < t <... < t N = T be an approximation to hx in the sense that h x hx H 4. as. So h xd D x = N i= a,i D t i Dt i. We claim that exp iθ h iθ n h x h x... h x n xd D x = exp n n= n x x α x 3 x α... x x n α dx 4.3 when θ is sufficiently small. To see this, we consider h x h x... h x n n x x α x 3 x α... x x n α dx = N i,...,i n= [ n t i a,ik k= t i t in... t in x x α x 3 x α... x x n α dx ] 4.4 If we define θ,i = a,i a,i+ for i =,..., N and θ,n = a,n, then we have a,i = N k=i θ,k and 4.4 can be rewritten as h x h x... h x n n x x α x 3 x α... x x n α dx [ N n t ] j t jn = θ,jk... x x α x 3 x α... x x n α dx. j,...,j n= On the other hand, k= h xd D x = N k= 4.5 θ,k D t k. 4.6 By the results in [], the characteristic function of h xd D x in a small neighborhood of the origin is exp iθ h xd D x N n = exp iθn t j t jn θ,jk... n x x α x 3 x α... x x n α dx n= j,...,j n= k= 4.7

12 Therefore, 4.3 is proved. As, iθ n h x h x... h x n exp n n= n x x α x 3 x α... x x n α dx iθ n hx hx... hx n exp n n x x α x 3 x α... x x n α dx. n= 4.8 Since h xd D x hxd Dx in L Ω by the definition of stochastic integral with respect to osenblatt process, we have iθ n hx hx... hx n exp iθ hxd D x = exp n n= n x x α x 3 x α... x x n α dx 4.9 in a small neighborhood of the origin. This completes our proof. Proposition 4.. Under the same assumption as in Proposition 4., we have α qx hxdx distribution V κ g hxd D x. 4.3 Proof. By Hermite expansion, We claim that in probability. q x = Φgx = n= V n n! H ng x. 4.3 α qx V gx hxdx 4.3 Actually, we have { E α qx V } gx V n hxdx = n=3 α n! g x y n hxhydxdy 4.33 and α g x y n hxhydxdy M α hxhy dxdy + M nα 4.34 α x y nα hxhy dxdy x y <M x y >M for some constant M. Since α, and n 3, we show that the HS of 4.34 is uniformly bounded in n and converges to as.

13 Because Vn n=3 n! <, by the Dominated Convergence Theorem, the LHS of 4.33 converges to as. By Proposition 4., V α gx hxdx V κ g hxd D x 4.35 which completes the proof. emark 4.3. Although we assume that hx is continuous in Propositions 4. and 4., we see from the above proof that hx can be allowed to have finitely many jump discontinuities, and we will use this fact later. 5 Proof of the main theorem ecalling.3 and.4, we have where and u x ūx = with ρ the remainder term. Define x q y F ydy + c c x x a + c q y dy + r x 5. r x = c c c c = a F y U x = x x q y dy 5. F zdz a bq y dy + ρ 5.3 q y F ydy + c c ρ x a + c x q y dy 5.4 so that x u x ūx = U x + r x + ρ a 5.5 The proof of Theorem. contains two steps. First, we prove the weak convergence of x U α x as a process in C[, ]. Then we control the remainder term r x + ρ a. We use the notation a b when there exists a constant M such that a Mb. 5. Weak convergence in C[, ] ewrite U x = c [,x] y F y [,x] yq y dy + F y F zdz a bq y dy x. 5.6 Define F x, y = c [,x] y F y [,x] y + xf y F zdz a b [,] y so that α U x = α F x, yq y dy

14 Lemma 5.. Let Then in C[, ]. Ūx = V κ g F x, yd D y. 5.8 α U x distribution Ūx 5.9 Proof. We first prove the weak convergence of finite dimensional distributions and then prove tightness. x, x,..., x n [, ] and c, c,..., c n, consider n i= c i α U x i = α n i= c i F x i, yq y dy. 5. We see that n i= c if x i, y is compactly supported and has only finitely many discontinuities. Then by Proposition 4. we have n c i α U x i distribution i= n c i Ūx i. 5. i= Therefore, we have proved the weak convergence of the finite dimensional distributions. To prove tightness, we apply the Kolmogorov criteria [5]. Note that U =, so we only need to show that there exist δ, β, C > such that E{ α U x α U y β } C x y +δ. 5. Define F y = c F y and F y = F y F zdz a b. Then for y < x, we have α U x U y = α x y F zq z dz + x y α F zq z dz. 5.3 So E{ α U x α U y } α F z F z z z dz dz [y,x] + x y F α z F z z z dz [,] :=I + II. 5.4 F is bounded, α, and by Lemma., we have For I, we distinguish the cases y x < and y x. II x y

15 If y x <, since F and are both bounded, we have I x y α. If y x, by Lemma., we have I z z α dz dz Choose δ = α. We have [y,x] x y x y for some constant C. The proof is completed. t α dt x y α. 5.6 E{ α U x α U y } C x y +δ The remainder term To analyze the remainder term, we first write which gives ρ = c c = a a y dy [ F yq y dy a b + a dy + y We have the following lemma: Lemma 5.. b + a F ydy F ydy q y dy for some constant M. Furthermore, we have in C[, ]. Proof. a dy y a, 5.8 F yq y ] dy q y dy. 5.9 E{ ρ } + sup E{ r x } M α 5. x [,] α ρ x a + r x probability 5. { E q y } dy = y z dydz. 5. [,] By lemma., y z κ α y z α, so we have { E q y } dy α

16 By the Cauchy-Schwartz inequality, we can show in the same way that Since { E F yq y dy q y } dy α. 5.4 a y dy is bounded from below, we have E{ ρ } α. For r x = c c x q y dy, we write it in two parts: r x = a F y =rx + rx. F zdz a bq y x dy q y dy + ρ x q y dy 5.5 By Cauchy-Schwartz, E{ r x } α and the constant does not depend on x. qy is bounded and since E{ ρ } α, we also have sup x [,] E{ r x } α. Thus, we have proved 5., and we have sup x [,] α E{ ρ x a + r x }. 5.6 So we have the weak convergence of finite dimensional distribution. Now we prove tightness. We have r =, and r x r x = a F y Following the proof of 5.4, we have F zdz a bq y x dy q y dy + ρ x x x q y dy. 5.7 E{ r x r x } x x α. 5.8 Therefore, x E{ ρ a ρ x a + r x r x } C x x α 5.9 for some constant C. Thus α x ρ a + r x converges in distribution to as, so it converges in probability to, which completes the proof. ecall that u x ūx α = α U x + α r x x + ρ. 5.3 a We only need to combine lemma 5. and 5. to complete the proof of Theorem.. 6

17 6 Conclusions and further discussion We considered the homogenization and corrector random fluctuation theory of a one dimensional elliptic equation with highly oscillatory coefficients. For a certain class of random coefficients with long range correlations, we were able to show that the properly rescaled corrector converges in distribution in the space of continuous function to a stochastic integral with respect to the osenblatt process. Moreover, the corrector s amplitude is of order α and α, / such that x κ x α. Therefore, the longer the range of the correlations, the larger is the amplitude of the corrector. The appearance of the osenblatt process is due to the fact that the Hermite rank of Φ is. It is natural to ask what would happen if the Hermite rank of Φ was greater than. In [7, ], the non-central limit theorems for functionals of Gaussian fields of arbitrary Hermite rank was proved. When the Hermite rank is greater than, the limit is the socalled Hermite process. In that case, we expect the properly rescaled corrector to converge in distribution to some stochastic integral with respect to the Hermite process although we have not carried out the calculations in detail. It would also be interesting to generalize proposition 4. and 4. to the case of elliptic equations in higher dimensions, at least in the setting considered in []. To do this, we would have to first find the counterpart of the osenblatt process in higher dimensions, which is a nontrivial problem. eferences [] G. Bal, Central limits and homogenization in random media, Multiscale Model. Simul., 7 8, pp [] G. Bal, J. Garnier, Y. Gu, and W. Jing, Corrector theory for elliptic equations with long-range correlated random potential, to appear in Asymptot. Anal.,., 7 [3] G. Bal, J. Garnier, S. Motsch, and V. Perrier, andom integrals and correctors in homogenization, Asymptot. Anal., 59 8, pp. 6., 5 [4] A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, es. Inst. Math. Sci., Kyoto Univ., 5 979, pp [5] P. Billingsley, Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, second ed., 999. A Wiley-Interscience Publication. 4 [6] A. Bourgeat and A. Piatnitski, Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator, Asymptot. Anal., 999, pp ,, 5 7

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