A finite dimensional approximation of the effective diffusivity for a symmetric random walk in a random environment

Journal of Computational and Applied Mathematics 213 (2008) 186 – 204 www.elsevier.com/locate/cam

A finite dimensional approximation of the effective diffusivity for a symmetric random walk in a random environment夡 Małgorzata Cudnaa , Tomasz Komorowskia, b,∗ a Institute of Mathematics, UMCS pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland b Institute of Mathematics, PAN, ul. Sniadeckich ´ 8, 00-956 Warsaw, Poland

Received 16 January 2006; received in revised form 30 December 2006

Abstract We consider a nearest neighbor, symmetric random walk on a homogeneous, ergodic random lattice Zd . The jump rates of the walk are independent, identically Bernoulli distributed random variables indexed by the bonds of the lattice. A standard result from the homogenization theory, see [A. De Masi, P.A. Ferrari, S. Goldstein, W.D. Wick, An invariance principle for reversible Markov processes, Applications to random walks in random environments, J. Statist. Phys. 55(3/4) (1989) 787–855], asserts that the scaled trajectory of the particle satisfies the functional central limit theorem. The covariance matrix of the limiting normal distribution is called the effective diffusivity of the walk. We use the duality structure corresponding to the product Bernoulli measure to construct a numerical scheme that approximates this parameter when d  3. The estimates of the convergence rates are also provided. © 2007 Published by Elsevier B.V. MSC: Primary 65C35; 82C41; Secondary 65Z05 Keywords: Eandom walk on a random lattice; Corrector; Duality

1. Introduction Suppose that (e (x))x∈Zd , e ∈ Ed are certain positive valued, spatially homogenous, ergodic random fields given over a probability space (, F, P). Here Ed denotes the set of unit vectors of the lattice. Assume that for a “frozen” () random parameter  ∈  we are given a nearest neighbor, symmetric random walk on the lattice (Xt )t  0 whose jump rates are given by e (x; ). The random generator of the walk equals L f (x) := e∈Ed e (x; ) je f (x) for any bounded f . Here je f (x) := f (x + e) − f (x). We assume the symmetry of the jump rates, i.e., e (x; ) = −e (x + e; ), which, in turn, implies reversibility of the (infinite) counting measure on the lattice. Additionally, we suppose that the jump rates do not degenerate, i.e., there exist 0 < − < + < + ∞ such that e (x) ∈ [− , + ] for all x ∈ Zd , e ∈ Ed . Homogeneity and ergodicity of the environment, symmetry of the jump rates and their nondegeneracy together imply (see [5, Chapter 4]) that the trajectories of the random walk satisfy the central limit theorem. The above means that, () as t → +∞, the laws of t −1/2 Xt converge weakly, in P-probability with respect to , to the law of a Gaussian 夡

Research of both authors has been supported by the Polish Ministry of Science and Higher Education Grant N20104531.

∗ Corresponding author.

E-mail addresses: [email protected] (M. Cudna), [email protected] (T. Komorowski). URL: http://hektor.umcs.lublin.pl/∼komorow (T. Komorowski). 0377-0427/$ - see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.cam.2007.01.043

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random vector with a nondegenerate covariance matrix, called the effective diffusivity tensor. This limiting procedure is sometimes called homogenization. To compute the effective diffusivity one needs to solve the corresponding cell problem, see (2.4) below. In the case when the environment is random, the cell problem is usually formulated in an appropriate infinite dimensional space. Finding an efficient approximating scheme for the effective diffusivity matrix is in general a difficult problem. It has been shown in [4], see also [15,3] for the continuum case, that the effective diffusivity can be approximated by the diffusivity matrices corresponding to a periodized medium. In the present paper we investigate a special model of a walk whose jump rates are independent, identically distributed random variables that are indexed by the bond lattice Bd . We assume furthermore that each rate can only take two possible, strictly positive values, either − , or + and − < + . Models of random walks in two phase media have been extensively investigated (especially in the context of percolation, i.e., when − = 0) in both mathematics and physics literature, see, e.g., [1,2,9,12,18,16,19]. Using duality structure corresponding to the probability measure under consideration we formulate the approximation scheme for the effective diffusivity. In this case the solution of the cell problem, called the corrector, can be expanded in a certain orthonormal base, the so-called duality base, indexed by all finite subsets of Bd , see Section 2.3 below. The Fourier coefficients of the corrector field satisfy an infinite dimensional system of linear equations, see (3.1). In our first result, see Theorem 3.1 below, we prove that the solution of the system can be approximated in an appropriate sense by the solutions of the systems truncated with respect to the cardinality N of the index set. The convergence rate is better than N −k for an arbitrary k 1, see (3.6). Unfortunately, the truncated system is still infinite. To obtain a finite dimensional approximation we periodize equations and regularize them by adding to the left-hand side the term −I , where  > 0. Our second result, see Theorem 4.2, gives an estimate of the convergence rate of the solutions of the finite dimensional system, see (4.7), to the solution of infinite dimensional one corresponding to the cell problem, provided that the dimension d 3. The key tool used in the proof of the theorem is a discrete analog of the estimates due to Yurinskii [20], that provide the convergence rates of the -regularized correctors to the solution of the cell problem in the case of random diffusions. We adapt the argument contained in [20] to the case of discrete, symmetric random walks on random lattices, see Section 5 below. The results obtained in the present article can possibly be generalized to environments given by other product probability measures that admit the duality structure, see, e.g., [13]. In the continuum case the analog is provided by the Hermite polynomials corresponding to a Gaussian measure. In fact, one can use this property of Gaussian measures to formulate an analogous approximation scheme for random diffusions, see, e.g., [10], for the time dependent case. 2. Preliminaries 2.1. The description of the model We denote by Bd the set of bonds on a d-dimensional lattice Zd , i.e., the set consisting of all unordered pairs {x, x +e}, where x ∈ Zd and e ∈ Ed . Here Ed = {±ei , i = 1, . . . , d} and e1 , . . . , ed is the canonical basis in Zd . Let  be a d compact metric space {0, 1}B , i.e., it consists of all functions  : Bd → {0, 1} equipped with the standard Tichonoff’s product topology. By B() we denote its Borel -algebra. Let B ,  ∈ [0, 1] be the Bernoulli probability measure on d {0, 1} given by B [{1}] = , B [{0}] = 1 −  and let  := B⊗B be the product measure defined on (, B()) with E the expectation with respect to . For any p ∈ [1, +∞] we denote by  · Lp () the respective Lp norm. Let us fix two numbers 0 < − < + < + ∞ and introduce a function  : {0, 1} → {− , + }, given by (0) = − , (1) = + . For a given e ∈ Ed define a random field e (x; ) := (({x, x + e})), x ∈ Zd ,  ∈ . With  ∈  and () y ∈ Zd fixed let us consider a continuous time nearest neighbor random walk (Xt (y))t  0 on Zd . This walk is a time homogeneous, Markov process, defined over a certain probability space ( , A, P ), that starts at y and at any given time performs a jump from its present location at x ∈ Zd to one of the neighboring sites x + e, e ∈ Ed at the rate e (x; ), e ∈ Ed . The generator of the process equals L f (x) :=

 e∈E

d

e (x; )je f (x) = −

d 

j∗p (p (x; )jp f (x)),

f ∈ C0 (Zd ),

(2.1)

p=1

where C0 (Zd ) is the space of compactly supported functions on Zd . Here jp f := jep f , j∗p f := j−ep f and p (x; ) := (n)

ep (x; ). The random walk on a random lattice is then defined as the process Xt (y; , ) := Xt (y; ), t 0, where

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(, ) ∈ × . It should be noted that, in contrast with the random walk in a frozen environment  ∈ , this process is not Markovian. In the particular case when y = 0 we shall drop this parameter from our notation. 2.2. The central limit theorem A question of considerable interest in the theory of random media is to describe the asymptotic behavior of Xt , as t → +∞. A convenient tool used for that purpose is the environment process. It is an -valued Markov process given by

t := TXt ().

(2.2)

Here the shift operator Tx ()(b) := (x + b) for any x ∈ Zd , b ∈ Bd and for b = {y, y + e}, y ∈ Zd , e ∈ Ed we let x + b := {x + y, x + y + e}. Intuitively speaking t describes the environment as seen at time t from the vintage point of the random walker. Its generator is given by LF () =

 e∈E

e ()De F () = −

d 

Dp∗ (p ()Dp F ()),

F ∈ C().

(2.3)

p=1

d

Here C() denotes the space of all continuous functions on , De F := F ◦ Te − F and e () := e (0; ). In addition, Dp F := Dep F , p () := ep (), p = 1, . . . , d. The operator Dp∗ := D−ep is the adjoint of Dp with respect to the scalar product of L2(). One can show, see, e.g., [9, p. 1285], that  is an invariant and ergodic measure for the environment process, i.e., LF d = 0 for all F ∈ C() and the only bounded functions that satisfy LF = 0 are constant  a.s. With the help of the individual ergodic theorem one can conclude therefore that Xt = 0, t→+∞ t lim

√ both  ⊗ P -a.s. and in the L1 sense. Moreover (see [5, Theorem 4.5, p. 821]), the laws of Xt / t converge, as t → +∞, to the law of a zero mean, nondegenerate Gaussian random vector. The covariance matrix  = [ i,j ] of this vector is called the effective diffusivity of the random walk. In fact, note that  is invariant under any transformation induced by a matrix U ∈ U (Zd )—the space of all integer valued matrices with | det U| = 1. In consequence, the random walks Xt and UXt have the same laws, thus UUT =  for all U ∈ U (Zd ). Here UT denotes the transpose of U. The effective diffusivity must be therefore of the form  = I, where I is the identity matrix and  > 0, see [5, Theorem 4.6, p. 823]. To calculate  one needs to solve the following abstract cell problem: Lu∗ () = f∗ (),

(2.4)

f∗ () := −e1 () − e1 () = D1∗ 1 ().

(2.5)

where

Then, see [5, (4.18)], ⎛  = 2[¯ + (f∗ , u∗ )L2 () ] = 2 ⎝¯ − (2.3)

d  

⎞ p (Dp u∗ )2 d⎠ ,

(2.6)

p=1

where ¯ := + + (1 − )− is the average jump rate. A few comments are in order. Note first that the direction e1 in the definition of f∗ () can be replaced by any ep , p=2, . . . , d, due to exchangeability of the measure . The solution u∗ to (2.4) is called the corrector field. Unfortunately, the equation in question in general has no solution in C(), nor in L2 (). One has to expand the space L2 () in order to guarantee the existence of the corrector. The appropriate space H+ is defined as the completion of the subspace

  L20 () := F ∈ L2 () : F d = 0

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in the norm given by F 2+ :=

d  

(Dp F )2 d,

F ∈ L20 ().

(2.7)

p=1

Eq. (2.4) has then a unique solution in H+ . It is also obvious from (2.6) (recall that (LF, F )L2 () < 0 for F  = const) that  < 2¯—the diffusivity of the nearest neighbor random walk with a constant intensity equal to the averaged jump rate. Note that f∗ belongs to the Hilbert space H− , which consists of those elements F of L02 () for which F − :=

sup (F, G)L2 () < + ∞.

(2.8)

G+ =1

2.3. The duality structure on L2 () We adopt the notation of [17]. Let (b) −  b () := √ , (1 − )

b ∈ Bd ,  ∈ .

d Denote by Fn the family of all subsets of B of cardinality n with the convention that F0 consists only of an empty set ∅. Let F := n  0 Fn . For any A ∈ F we let

1 if A = ∅, A () :=  b () otherwise. b∈A

 2 The functions A form an orthonormal basis of L2 () and L2 () = +∞ n=0 Hn , where Hn is the L ()-closure of In analogy to a similar gradation existing for Gaussian measures (cf., e.g., [7]) we shall call the span{A : A ∈ Fn }. 2 elements of PN := N n=0 Hn polynomials of degree N. We denote by n the L () orthonormal projection onto Hn . For an element F ∈ L2 () we let Fn := n F , n0. We have then the Fourier series expansion  F= (2.9) Fˆ (A)A , A∈F

which defines the corresponding function Fˆ : F → R. The Plancherel equality guarantees that  F 2L2 () := Fˆ 2 (A) < + ∞.

(2.10)

A∈F

The space L2 () is therefore isometrically isomorphic with the space Lˆ 2 consisting of all functions Fˆ : F → R, ˆ ∈ Lˆ 2 we denote by (Fˆ , G) ˆ ˆ2 = for which the expression on the right-hand side of (2.10) is finite. For given Fˆ , G L  1/2 ˆ ˆ ˆ ˆ ˆ ˆ F (A) G(A) their scalar product and by  F  := ( F , F ) the respective norm. We shall identify H with the n ˆ 2 A L2

Lˆ

subspace of functions that are equal to 0 for all A ∈ / Fn . Let n : Lˆ 2 → Hˆ n be given by  0, A∈ / Fn , n Fˆ (A) := ˆ F (A), A ∈ Fn .

Sometimes we shall also write Fˆn := n Fˆ . Denote by Lˆ 20 the subspace of Lˆ 2 consisting of functions of zero mean, i.e., those Fˆ for which Fˆ0 = 0. For any Fˆ : F → R we let je Fˆ (A) := Fˆ (A + e) − Fˆ (A), A ∈ F and jp Fˆ (A) := je Fˆ (A), p = 1, . . . , d. Here, p

ˆ + with the space of for a given A = {b1 , . . . , bn }, we let A + x := {b1 + x, . . . , bn + x}. We can identify the space H 1/2 ˆ ˆ ˆ ˆ ˆ functions F , for which F (∅) = 0 and F + := (F , F )+ < + ∞, where the scalar product (·, ·)+ is given by ˆ + := (Fˆ , G)

d   p=1 A∈F

ˆ jp Fˆ (A)jp G(A),

ˆ +. ˆ ∈H Fˆ , G

(2.11)

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ˆ − introduced in such way is therefore isometrically ˆ ˆ 2 : G ˆ + =1]. The space H We can also define Fˆ − := sup[(Fˆ , G) L isomorphic with the previously defined space H− . Note that the requirement Fˆ − < + ∞ implies, in particular, that of the respective Hˆ n ’s. The respective orthogonal Fˆ has zero mean. We introduce the subspaces Hˆ + n as the H+ closures N + ˆ := ˆ+ projection operators will be denoted by n . Let also P n=1 H n be the space of all polynomials of degree at N ˆ + , Lˆ 2 , or H ˆ − we denote by F its isometric image in the respective most N. For any Fˆ belonging to any of the spaces H 2 ˆ space H+ , L (), or H− . The image of L in H+ can be then described as follows: ˆ Fˆ )n (A) = Bn,n−1 Fˆn−1 (A) + Bn,n Fˆn (A) + Bn,n+1 Fˆn+1 (A), (L

A ∈ Fn ,

ˆ + , n0. Here where Fˆ ∈ H

  Ac (e)j−e Fˆn+1 (A ∪ {e}), Bn,n+1 Fˆn+1 (A) :=  (1 − ) Bn,n Fˆn (A) :=

 e∈E

(2.12)

e∈Ed

{− + [ + (1 − 2)A (e)]}j−e Fˆn (A)

(2.13)

d

for n0 and

  Bn,n−1 Fˆn−1 (A) :=  (1 − ) A (e)j−e Fˆn−1 (A\{e})

(2.14)

e∈Ed

ˆ H ˆ + ) ⊂ Lˆ 2 . for n1. We adopt the convention that B0,−1 = 0 and Fˆ−1 ≡ 0. Note that L( 0 c Here  := + − − , the symbol A denotes a complement of A and  1 if e ∈ A, A (e) := 0 if e ∈ / A. To see formulas (2.12)–(2.14), let F =

∞

n=1 Fn ,

where Fn =



A∈Fn

Fˆ (A)A . Note that

 e () = + (e) + − (1 − (e)) =  (1 − )e +   + −

(2.15)

and (2.15)

LFn = (  + − )

  e∈Ed A∈Fn

Since

   j−e Fˆ (A)A +  (1 − ) j−e Fˆ (A)A e . e∈Ed A∈Fn

 1 − 2 A + Ac (e)A∪{e} . e A = A (e) A\{e} + √ (1 − )

We obtain LFn = (  + − )

  e∈Ed A∈Fn

(2.16)

 

 A (e)j−e Fˆ (A)A\{e} j−e Fˆ (A)A +  (1 − ) e∈Ed A∈Fn

1 − 2 A (e)j−e Fˆ (A)A + Ac (e)j−e Fˆ (A)A∪{e} +√ (1 − )



and (2.12)–(2.14) follows. By a direct calculation, using formulas (2.12)–(2.14), we conclude the following result. ˆ ∈ Lˆ 2 we have Proposition 2.1. For any n 0 and Fˆ , G ˆ ˆ 2 = (Fˆ , Bn,n G) ˆ ˆ2 (Bn,n Fˆ , G) L L

ˆ ˆ 2 = (Fˆ , Bn+1,n G) ˆ ˆ2. and (Bn,n+1 Fˆ , G) L L

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191

3. The truncation of the degree of the corrector field 3.1. The corrector equation in Fourier coordinates With the help of the duality base we can rewrite Eq. (2.4), using an analog of the matrix/column vector notation for operators and elements of the respective space, in the following form: ⎡B 0 0 · · · ⎤ ⎡ uˆ ∗,1 ⎤ ⎡ fˆ∗ ⎤ 1,1 B1,2 0 · · · ⎥ ⎢ uˆ ∗,2 ⎥ ⎢ 0 ⎥ ⎢ B2,1 B2,2 B2,3 ⎢ ⎥⎢ ⎥ ⎢ ⎥ B3,2 B3,3 B3,4 · · · ⎥ ⎢ uˆ ∗,3 ⎥ = ⎢ 0 ⎥ , ⎢ 0 (3.1) ⎢ ⎥⎢ ⎥ ⎢ ⎥ 0 B4,3 B4,4 · · · ⎦ ⎣ uˆ ∗,4 ⎦ ⎣ 0 ⎦ ⎣ 0 .. .. .. .. .. .. .. . . . . . . . √ where fˆ∗ ∈ Hˆ 1 is given by fˆ∗ (±e1 ) := ∓  (1 − ) and equals zero otherwise. The effective diffusivity equals  (3.2)  = 2[¯ +  (1 − )j1 uˆ ∗ (−e1 )].

3.2. Truncated corrector equation Our first result deals with the issue of truncation of the infinite matrix appearing in (3.1), with respect to the degree (N) of the polynomial N. More specifically, we will be looking for the solution uˆ ∗ of the following equation: ⎡ (N) ⎤ ⎡ ⎤ ⎡B 0 0 ··· 0 ⎤ uˆ ∗,1 fˆ∗ 1,1 B1,2 (N) ⎥ u ˆ ⎢ ⎥ ⎥⎢ ⎢ ∗,2 ⎥ 0 ··· 0 ⎥⎢ ⎢ B2,1 B2,2 B2,3 0⎥ ⎥ ⎢ ⎢ ⎥ ⎥⎢ ⎢ ⎥ ⎥ ⎢ uˆ (N) ⎥ ⎢ ⎥ ⎢ B3,2 B3,3 B3,4 · · · 0 ⎥⎢ ⎢ ⎢ 0 ∗,3 ⎥ ⎢0⎥ ⎥ ⎥⎢ ⎢ (3.3) ⎥. ⎥ ⎢ (N) ⎥ ⎢ ⎥=⎢ 0 B4,3 B4,4 · · · 0 ⎥⎢ ⎢ ⎥ ⎢ 0 0 u ˆ ∗,4 ⎥ ⎢ ⎥ ⎥⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ . ⎥ ⎢ .. ⎥ .. .. .. .. .. ⎥ ⎥⎢ ⎢ . . ⎢ . . . . . ⎦ ⎣ .. ⎥ ⎣ . ⎦ ⎣ . ⎦ 0

0

0

0

· · · BN,N

(N)

uˆ ∗,N

0

ˆ N the operator on the left-hand side of (3.3). For any Fˆ , G ˆ ∈ P+ we have Denote by L N ˆ Fˆ , G) ˆ N Fˆ , G) ˆ ˆ 2 | = |(L ˆ ˆ 2 | = |(LF, G)L2 () | |(L L L    d     ˆ + =  p (Dp F )(Dp G) d + F + G+ = + Fˆ + G p=1 

(3.4)

and ˆ N Fˆ , Fˆ ) ˆ 2 = −(LF, F )L2 () = −(L L

d  

p (Dp F )2 d− F 2+ = − Fˆ 2+ .

(3.5)

p=1

The announced approximation result can be formulated as follows. (N)

Theorem 3.1. Eq. (3.3) has a unique solution uˆ ∗ ˆ ∗ +  uˆ (N) ∗ −u

ˆ + . Moreover, for each integer k 1 we have in P N

Cˆ . Nk

The constant Cˆ is given by    1  C∗ + +  (1 − )  (1 − ), Cˆ = − 2

(3.6)

(3.7)

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where  1/2 2nk1 1 1 C∗ = + · · · + 2k 12 − n1 √

and

 8ek  n1 =

 (1 − ) + 1. −

(3.8)

Proof. The coercivity property (3.5) implies uniqueness of solutions to (3.3). The existence part is a simple consequence of the Riesz representation theorem applied in the Hilbert space P+ N . We shall focus therefore only on proving the ˆ ∈ Lˆ 2 : G(∅) ˆ estimate (3.6). Let us introduce first some additional notation. Suppose that Fˆ ∈ Lˆ 20 := [G = 0] and k is +∞ 2k +∞ 2k 2 2 2 2 ˆ ˆ ˆ := n  F  and ? F ? := n  F  . We shall use the following two a positive integer. Let ? Fˆ ? n + n − +,k −,k n=1 n=1 results. Lemma 3.2. Suppose that k is a positive integer and ? fˆ? −,k < + ∞. Then equation ⎡B

1,1

⎢ B2,1 ⎢ ⎢ 0 ⎢ ⎣ 0 .. .

B1,2 B2,2 B3,2 0 .. .

0 B2,3 B3,3 B4,3 .. .

0 0 B3,4 B4,4 .. .

· · · ⎤ ⎡ uˆ 1 ⎤ ⎡ fˆ1 ⎤ · · · ⎥ ⎢ uˆ 2 ⎥ ⎢ fˆ2 ⎥ ⎥⎢ ⎥ ⎢ ⎥ · · · ⎥ ⎢ uˆ 3 ⎥ = ⎢ fˆ3 ⎥ ⎥⎢ ⎥ ⎢ ⎥ · · · ⎦ ⎣ uˆ 4 ⎦ ⎣ fˆ4 ⎦ .. .. .. . . .

(3.9)

ˆ + . In addition, we have has a unique solution uˆ in H ? u? ˆ +,k C∗ ? fˆ? −,k

(3.10)

and the constant C∗ is the same as in the statement of Theorem 3.1. ˆ ∈ Lˆ 2 we have the following estimates: Lemma 3.3. For any n1 and Fˆ , G  ˆ ˆ 2 | 1  (1 − )Fˆn + G ˆ n−1 + , |(Fˆ , Bn,n−1 G) L

2

 ˆ ˆ 2 | 1  (1 − )Fˆn + G ˆ n−1 + . |(Bn−1,n Fˆ , G) 2 L

(3.11) (3.12)

For the clarity sake, we postpone for the moment the demonstration of the above lemmas and apply them first to finish the proof of estimate (3.6). According to Lemma 3.2 for any k 1 we can find a constant such that +∞ 

n2k uˆ ∗,n 2+ C∗2 fˆ∗ 2− .

n=N+1

√ Since fˆ∗ − =  (1 − ) we obtain that +∞ 

uˆ ∗,n 2+ C∗2 N −2k ( )2 (1 − ).

(3.13)

n=N+1 (N)

(N)

We can write therefore uˆ ∗ = vˆ ∗ + rˆN , where vˆ ∗ notation we rewrite (3.1) in the form ˆ N vˆ (N) ˆ rN . ˆ L ˆ ∗,N − Lˆ ∗ = f∗ − BN+1,N u

=

N

ˆ ∗,n n=1 u

√ and ˆrN + C∗ N −k  (1 − ). Using the above (3.14)

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From (3.14) and (3.3) we obtain ˆ N (vˆ (N) ˆ rN . L ˆ (N) ˆ ∗,N − Lˆ ∗ −u ∗ ) = −BN+1,N u

(3.15)

(N) (N) Taking the Lˆ 2 scalar product of both sides of (3.15) against vˆ ∗ − uˆ ∗ we obtain (N) ˆ vˆ (N) ˆ rN , vˆ (N) (L( ˆ (N) ˆ (N) ˆ ∗,N , vˆ (N) ˆ (N) ˆ (N) ∗ ), vˆ ∗ − u ∗ )Lˆ 2 = (BN+1,N u ∗ −u ∗ )Lˆ 2 − (Lˆ ∗ −u ∗ )Lˆ 2 . ∗ −u

Hence, using (3.12) and the coercivity estimate (3.5) we get  2 ˆ (N) rN + + 21  (1 − )uˆ ∗,N + )vˆ (N) ˆ (N) − vˆ (N) ∗ −u ∗ + (+ ˆ ∗ −u ∗ + .

(3.16) √

Note, however, that in light of (3.13) we have that both ˆrN + and uˆ ∗,N + can be estimated by C∗ N −k  (1 − ), hence   −k 1 (3.17) − vˆ (N) ˆ (N) ∗ + C∗ N (+ + 2  (1 − ))  (1 − ). ∗ −u The estimate (3.6) follows now easily from (3.17).



ˆ − the proof of the existence and uniqueness of uˆ follows straightforProof of Lemma 3.2. Since, in particular fˆ ∈ H ˆ To show (3.10) we adapt the argument wardly from coercivity and symmetry of the bilinear form corresponding to L. contained in the proof of Theorem 5.1 of [17]. Let n2 > n1 1 be some positive integers  that shall be further deterˆ n , uˆ ∈ Lˆ 2 with mined later on. For a given k 0 define a bounded linear operator T : Lˆ 2 → Lˆ 2 by T uˆ := +∞ n=1 t (n)u k k k ˆ t (n) := n1 ∨ (n ∧ n2 ). Obviously T extends to a linear isomorphism of H+ . After a simple calculation, we obtain that ˆ uˆ = [T , L]

+∞ 

(t (n + 1)) − t (n))Bn+1,n uˆ n +

n=1

+∞ 

(t (n − 1) − t (n))Bn−1,n uˆ n .

n=2

Hence, +∞   |([T , L]u, ˆ T u) ˆ L2 () | =  t (n + 1)(t (n + 1) − t (n))(Bn+1,n uˆ n , uˆ n+1 )Lˆ 2  n=1 +∞ 

+

n=2

   t (n − 1)(t (n − 1) − t (n))(Bn−1,n uˆ n , uˆ n−1 )Lˆ 2  

+∞   t (n + 1) − t (n) T uˆ n + T uˆ n+1 + . 2  (1 − ) t (n)

(3.18)

n=1

Notice that for any k < n1 n n2 we have     k   k k   t (n + 1) − t (n) 1 k j k  1 ek 1 k k −j n  − 1   1+  . j t (n) n j ! n1 n1 j ! n1 j =1

j =1

j =1

√ Hence the utmost right-hand side of (3.18) can be estimated by 2n−1 ˆ 2+ . Applying the operator T 1 ek  (1 − )T u to both sides of (3.9) and performing scalar multiplication of its both sides by T uˆ we get ˆ u, − T u ˆ 2+  − (T fˆ, T u) ˆ Lˆ 2 + ([T , L] ˆ T u) ˆ Lˆ 2 . The right-hand side of (3.19) can be estimated with the help of Schwartz inequality and (3.18) by  T fˆ− T u ˆ + + 2ekn−1 ˆ 2+ 1  (1 − )T u   1 ˆ 2+ + 2ekn−1  T fˆ2− + − T u ˆ 2+ . 1  (1 − )T u 2− 2

(3.19)

194

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Let us choose n1 as in (3.8). Substituting into (3.19) we obtain, after simple calculation, that   n2  2n2k 1 2 1 2 2k 2 2 2 1 ˆ fˆ? n uˆ n + T u ˆ +  2 T f −  2 + · · · + 2k ? −,k . 12   n − − 1 n=1 Estimate (3.10) is obtained upon passing to the limit with n2 → +∞.

(3.20)



Proof of Lemma 3.3. It suffices only to prove (3.11). Estimate (3.12) is then a consequence of Proposition 2.1. For ˆ ∈ Lˆ 2 we have (see (2.14)) any Fˆ , G    ˆ ˆ 2 =  (1 − ) ˆ (Fˆ , Bn,n−1 G) (3.21) A (e)Fˆ (A)j−e G(A\{e}). L A∈Fn e∈Ed

Observe that     ˆ ˆ − = A (e)Fˆ (A − e)j−e G(A\{e}) A+e (e)Fˆ (A)je G(A\{−e}) A∈Fn e∈Ed

A∈Fn e∈Ed

=

 

ˆ A (−e)Fˆ (A)je G(A\{−e})

A∈Fn e∈Ed

=

 

A∈Fn e∈Ed

ˆ A (e)Fˆ (A)j−e G(A\{e}).

Hence, the right-hand side of (3.21) equals   1  ˆ ˆ − A (e)Fˆ (A − e)j−e G(A\{e})] [A (e)Fˆ (A)j−e G(A\{e})

 (1 − ) 2 d A∈Fn e∈E

  1  ˆ = −  (1 − ) A (e)j−e Fˆ (A)j−e G(A\{e}) 2 d A∈Fn e∈E

and its absolute value is clearly estimated by the expression on the right-hand side of (3.11).



4. Finite dimensional approximation of the corrector field 4.1. Periodization Let M 1 be an integer. Denote by M the subset of Bd consisting of those bonds b = {x, y} ∈ Bd , for which |x|∞ , |y|∞ M and min{xi , yi } < M for each i = 1, . . . , d. Here x = (x1 , . . . , xd ), y = (y1 , . . . , yd ) ∈ Zd and |x|∞ := max{|x1 |, . . . , |xd |}. By 0M we denote the interior of M , i.e., the set consisting of those bonds, for which |x|∞ , |y|∞ < M. Suppose now that b, b ∈ Bd . We say that b and b are equivalent modulo 2M, and write then b≡M b , if there exists a vector x ∈ Zd with all coordinates divisible by 2M such that b = b + x. It is easy to see that ≡M defines an equivalence relation on Bd . Moreover for each b ∈ Bd , there exists a unique element b ∈ M such that b ∈ [b]≡M . We can identify therefore the set of equivalence classes of relation ≡M with M . Define by M the canonical projection of Bd onto M . d For any b ∈ M and x ∈ Z we introduce b⊕M x := M (b + x). Let dFn (M) := [A ∈ Fn : A ⊂ M ] and F(M) := n  0 Fn (M). For a given A = {b1 , . . . , bn } ∈ Fn (M) and x ∈ Z we let A⊕M x := {b1 ⊕M x, . . . , bn ⊕M x}. Let B(M) be the space of all functions Fˆ : F → R that satisfy Fˆ (∅) = 0 and Fˆ (A) = 0 for all A ∈ / F(M). We denote also by PN (M) the subspace of polynomials of at most Nth degree belonging to B(M). Define the periodic gradient (M) (M) / F(M) and jx Fˆ (A) := Fˆ (A⊕M x) − Fˆ (A), in the direction x of a function Fˆ ∈ B(M) as jx Fˆ (A) := 0 for A ∈ (M) (M) (M) if otherwise. We shall write jp to denote jep , p = 1, . . . , d. Note that je Fˆ (A) = je Fˆ (A) for each A ⊂ 0M  (M) and e ∈ Ed . Let Fˆ 2 := dp=1 jp Fˆ 2 2 . Let CM denote the cardinality of M . For any N CM we define +,M

Lˆ

M. Cudna, T. Komorowski / Journal of Computational and Applied Mathematics 213 (2008) 186 – 204

195

ˆ N -as follows: ˆ (M) : B(M) → B(M)-the periodization of the operator L L N ⎡ B (M) 1,1 ⎢ B (M) ⎢ 2,1

⎢ ⎢ 0 (M) ˆ ˆ LN F := ⎢ ⎢ 0 ⎢ ⎢ . ⎣ . . 0

(M)

0

0

···

0

(M)

B2,3

(M)

0

···

0

B3,2

(M)

B3,3

(M)

B3,4

(M)

···

0

0 .. . 0

(M) B4,3

(M) B4,4

.. . 0

.. . 0

··· 0 .. .. . . (M) · · · BN,N

B1,2

B2,2

⎤

⎡ ˆ ⎤ F ⎥ ˆ1 ⎥ ⎢ F2 ⎥ ⎥⎢ ˆ ⎥ ⎥ ⎢ F3 ⎥ ⎥⎢ ˆ ⎥ ⎥ ⎢ F4 ⎥ ⎥⎢ . ⎥ ⎥⎣ . ⎦ . ⎦ FˆN

(4.1)

(M) (M) (M) for any Fˆ ∈ B(M). The operators Bn−1,n , Bn,n , Bn+1,n are given by formulas (2.12)–(2.14), where the gradient (M)

operator je is replaced by the periodic gradient je . In the special case when N = CM we drop the subscript N from ˆ (M) . our notation and call the respective operator L Proposition 4.1. We have ˆ (M) Fˆ , Fˆ ) ˆ 2 − Fˆ 2 −(L +,M N L

(4.2)

ˆ (M) Fˆ , G) ˆ ˆ 2 | + F +,M G+,M |(L N L

(4.3)

and

ˆ N (M). ˆ ∈P for all Fˆ , G Proof. The proofs of both estimates are similar. We only show (4.2). The same calculations as in the proof of Lemma 3.3 show that for all Fˆ ∈ B(M) we have 1  (M) (Bn−1,n Fˆ , Fˆ )Lˆ 2 = −  (1 − ) 2 1  (M) (Bn+1,n Fˆ , Fˆ )Lˆ 2 = −  (1 − ) 2 and (M) ˆ ˆ (Bn,n F , F )Lˆ 2





A∈Fn (M) e∈Ed





A∈Fn (M) e∈Ed

(M) (M) A (e)j−e Fˆ (A)j−e Fˆ (A\{e}),

(4.4)

(M) (M) Ac (e)j−e Fˆ (A)j−e Fˆ (A ∪ {e})

(4.5)

⎧ ⎨   1 (M) (M) [j−e Fˆ (A)]2 −   Ac (e)[j−e Fˆ (A)]2 ⎩ 2 A∈Fn (M) e∈Ed A∈Fn (M) e∈Ed ⎫ ⎬   (M) +(1 − ) A (e)[j−e Fˆ (A)]2 . ⎭ d

1 = − − 2





A∈Fn (M) e∈E

Combining (4.4)–(4.6) we conclude that 1 ˆ (M) ˆ ˆ −(L N F , F )Lˆ 2 = 2 N





n=1 A∈Fn (M) e∈Ed

(M) − [j−e Fˆ (A)]2

 1 √ (M) (M) + Ac (e)[ j−e Fˆ (A) + 1 − j−e Fˆ (A ∪ {e})]2 2 #  1 √ (M) ˆ (M) ˆ 2 + A (e)[ j−e F (A\{e}) + 1 − j−e F (A)] 2 and (4.2) follows.



(4.6)

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4.2. A finite dimensional scheme for approximating the corrector Suppose that  > 0 and integers M, N 1 are fixed. We consider the following finite dimensional system of linear equations ⎡ − + B (M) B (M) ⎤ ⎡ wˆ (N) ⎤ 0 0 ··· 0 ⎡ ˆ⎤ 1,1 1,2 ∗,1 f∗ (M) (M) ⎢ B (M) ⎥ ⎢ wˆ (N) ⎥ B 0 · · · 0 B 2,1 2,2 2,3 ⎢ ⎥ ⎢ ∗,2 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ (N) ⎥ ⎢ ⎥ (M) (M) (M) ⎢ ⎥ ⎢ wˆ ∗,3 ⎥ ⎢ 0 ⎥ − + B3,3 B3,4 ··· 0 0 B3,2 ⎢ ⎥ ⎢ (N) ⎥ = ⎢ ⎥ . (4.7) (M) (M) ⎢ ⎥ ⎢0⎥ ⎥ ⎢ wˆ 0 0 B4,3 − + B4,4 · · · 0 ⎢ ⎥ ⎢ ∗,4 ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎣ .. ⎦ .. .. .. .. .. .. ⎣ ⎦⎣ . ⎦ . . . . . . . 0 (M) (N) 0 0 0 0 · · · − + BN,N wˆ ∗,N Our main result of this section can be stated as follows. Theorem 4.2. Suppose that d 3,  ∈ (0, 1) and M, N 1 are some positive integers. Then, system (4.7) has a unique (N) solution wˆ ∗ ∈ B(M). In addition, for an arbitrary # ∈ (0, 1) and an integer k 1 there exists C# > 0 independent of , M, N and such that   N    1 −2M 1−# 1 2(d)−# (N) 2  . (4.8) [je uˆ ∗ (A) − j(M) w ˆ (A)] C + e + # ∗ e Nk 4 d n=1 A∈Fn (M) e∈E

Here (d) := (d − 2)/(d + 8). −1 ((d)−

Remark 4.3. Let ∗ ∈ (0, (d)) be arbitrary. Choose then # ∈ (0, 2∗ ) and let (M) := M −2(2(d)−# ) and define  M,N := 2[¯ +  (1 − )j1 wˆ ∗(N) (−e1 )]. We obtain from (4.8) that for any k 1 there exists C˜ # > 0 such that   1 −(d)+∗ ˜ |M,N − |  C# M ∀N, M 1.  + k N

∗)

(4.9)

(4.10) (N)

Proof of the theorem. The proof of the existence and uniqueness of the solution wˆ ∗ ∈ B(M) is standard and relies on the nondegeneracy of the matrix appearing on the left-hand side of (4.7), see also (4.2). The proof of (4.8) shall be divided into three steps. In the first one we show that u∗ , the solution of the cell problem (2.4), can be approximated in the H+ -norm by the solutions of the resolvent equation ( − L)u∗, = −f∗ ,

(4.11) (M)

as  → 0+. In the next step we prove that for each # ∈ (0, 1) one can find constants C1 , C2 > 0 such that uˆ ∗, -the Lˆ 2 projection of the function uˆ ∗, : F → R onto B(M)-satisfies the following estimate: uˆ ∗, − uˆ ∗, Lˆ 2 C1 e−C2 M (M)

1−#

1  1−# +  (1 − )e−M 

∀M 1,  ∈ (0, 1).

(4.12)

Finally, we show that for any integer k 1 one can find a constant C3 > 0 such that for all  ∈ (0, 1) and M, N 1 we have N 





n=1 A∈Fn (M) e∈Ed

(M)

2 [j(M) ˆ ∗, (A) − j(M) ˆ (N) ∗ (A)]  e u e w

C3 . Nk

(4.13)

As we have already mentioned our first result deals with estimating the convergence rate of solutions to (4.11) towards the solution of the cell problem (3.1). It can be stated as follows.

M. Cudna, T. Komorowski / Journal of Computational and Applied Mathematics 213 (2008) 186 – 204

197

Proposition 4.4. Suppose that d 3. Then, for any # > 0 there exists a constant C > 0 such that for all  ∈ (0, 1) we have uˆ ∗ − uˆ ∗, + C(d)−# .

(4.14)

This is an analog of the result due to Yurinskii [20] proven for random diffusions. We essentially adapt that proof to the discrete setting. For the sake of clarity of the main line of our argument we postpone the presentation of the proof of the proposition till Section 5. (M) Before stating our next result we introduce some notation. Let u∗, be the L2 projection of u∗, onto the sub (M) space L2 (M ) spanned by A , A ∈ F(M). Obviously, u∗, = A∈F(M) uˆ ∗, (A)A . Estimate (4.12) is then a direct consequence of the following. Proposition 4.5. For each # ∈ (0, 1) there exists constant C > 0 such that for all  ∈ (0, 1) and M 1 we have (M)

u∗, − u∗, L2 () 

C −M 1−# . e 

(4.15)

Proof. Using a standard probabilistic representation of the solution of the resolvent equation, see, e.g., [11, p. 16], we can write  +∞ u∗, () = − e−t Ef ∗ (TX() ) dt. (4.16) t

0

Dividing the integration on the right-hand side into two parts from 0 to M 1−# and from M 1−# to +∞ and denoting the respective two terms by g1 and g2 we can write  +∞  1  1−# g2 L2 ()   (1 − ) e−t dt   (1 − )e−M . (4.17) 1−  # M The other term equals $  M 1−# −t g1 () = − e E f∗ (TX() ), 0

 −

$

M 1−#

e 0

−t

t

E f∗ (TX() ), t

% sup t∈[0,M 1−# ]

sup t∈[0,M 1−# ]

() |Xt |∞ M

dt %

() |Xt |∞ > M

dt.

(4.18)

Denote the respective integrals that appear on the right-hand side of the above equality by g1,1 and g1,2 . Of course g1,1 ∈ L2 (M ) and therefore the corresponding function gˆ 1,1 ∈ B(M). On the other hand, $ %  () 1−# P sup |Xt |∞ > M . g1,2 L2 ()   (1 − )M t∈[0,M 1−# ]

Estimate (4.15) is a consequence of the following bound $ % #   #  1 M () P sup − + M 1−# |Xt |∞ > M d exp − M 1−# log 2 2e+ t∈[0,M 1−# ]

(4.19)

for M # > 2+ , see, e.g., [14, Proposition 2.1, p. 320]. The above estimate proves (4.15) when 21 log(M # (2e+ )−1 ) > 1. We can justify (4.15) for other M’s by adjusting suitably the constant C.  From estimate (4.15) we conclude that there exists a constant C > 0 such that  C 1−# [uˆ ∗, (A)]2  2 e−2M  A∈ / F(M)

(4.20)

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M. Cudna, T. Komorowski / Journal of Computational and Applied Mathematics 213 (2008) 186 – 204

for all  ∈ (0, 1) and M 1. In addition, since je is a bounded operator on Lˆ 2 (whose norm equals 2) from the above estimate we obtain also that   C 1−# (M) je uˆ ∗, − je uˆ ∗, 2Lˆ 2 8d [uˆ ∗, (A)]2  2 e−2M . (4.21)  d A∈ / F(M) e∈E

Suppose now that M 2. Denote by F(M, j) he family of those sets A ∈ F(M), for which A ∩ jM  = ∅. Observe that  (M) (M) uˆ ∗, (A⊕M e) − uˆ ∗, (A + e) if A ∈ F(M), (M) (M) j(M) u ˆ (A) − j u ˆ (A) = (4.22) e ∗, e (M) ∗, if otherwise. −uˆ ∗, (A + e) The right-hand side of the above equality does not vanish if and only if A ∈ F(M, j) ∪ F(M + 1, j) hence using again estimate (4.20) we obtain  C 1−# (M) (M) . (4.23) j(M) ˆ ∗, − je uˆ ∗, 2Lˆ 2  2 e−2M e u  d e∈E

From (4.21) and (4.23) we conclude therefore that  C 1−# (M) j(M) ˆ ∗, − je uˆ ∗, 2Lˆ 2  2 e−2M . e u  d

(4.24)

e∈E

Let vˆ∗ ∈ B(M) be the unique solution of the equation ˆ (M) )vˆ∗ = −fˆ∗ . ( − L

(4.25)

A simple calculation shows that ˆ (M) )uˆ ˆ = −fˆ∗ + R, ( − L ∗, (M)

(4.26)

ˆ ˆ where the remainder Rˆ ∈ B(M) is given by R(A)=0 for any A ∈ F(M)\F(M, j) and R(A)= Rˆ 1 (A)+ Rˆ 2 (A)+ Rˆ 3 (A) for A ∈ F(M, j) and   Rˆ 1 (A) :=  (1 − ) Ac (e)[uˆ ∗, (A ∪ {e} − e) − uˆ ∗, (A ∪ {e}M e)], Rˆ 2 (A) :=



e∈Ed

{− + [ + (1 − 2)A (e)]}[uˆ ∗, (A − e) − uˆ ∗, (AM e)],

e∈Ed

  Rˆ 3 (A) :=  (1 − ) A (e)[uˆ ∗, (A\{e} − e) − uˆ ∗, (A\{e}M e)]. e∈E

(4.27)

d

Thanks to estimate (4.20) we conclude that ˆ ˆ 2  C e−M 1−# R L 

∀ ∈ (0, 1), M 1.

(4.28)

Subtracting (4.25) from (4.26) we get ˆ (M) )(uˆ ˆ − vˆ∗ ) = R. ( − L ∗, (M)

(M)

Performing the scalar multiplication of both sides of the above equality by uˆ ∗, − vˆ∗ we obtain, thanks to (4.2) (M)

(M)

(M)

(M)

uˆ ∗, − vˆ∗ 2Lˆ 2 uˆ ∗, − vˆ∗ 2Lˆ 2 + − uˆ ∗, − vˆ∗ 2+,M RLˆ 2 uˆ ∗, − vˆ∗ Lˆ 2 . Using (4.28) we obtain that (M)

uˆ ∗, − vˆ∗ Lˆ 2 

C 2



e−M

1−#

(4.29)

M. Cudna, T. Komorowski / Journal of Computational and Applied Mathematics 213 (2008) 186 – 204

for all  ∈ (0, 1) and M 1. Summarizing from (4.14), (4.15), (4.24) and (4.29) we can conclude that     1 −2M 1−# (d) 2  . C + e v ˆ (A)] [je uˆ ∗ (A) − j(M) # ∗ e 4 d A∈F(M)

199

(4.30)

e∈E

Finally, to replace vˆ∗ by wˆ ∗ , as in the statement of the theorem, we can repeat word by word the proof of Theorem 3.1 and obtain the following estimate: vˆ∗ − wˆ ∗ +,M 

Cˆ , Nk

where Cˆ is given by (3.7).

(4.31) 

5. Proof of Proposition 4.4 Let us introduce some notation. For any integer M 1 denote by M the set of all vertices x ∈ Zd for which |x|∞ M. The boundary of M is given by / M and ∃y ∈ M s.t. |y − x|∞ = 1]. jM := [x ∈ Zd : x ∈ For any bounded function f : Zd → R we define  1 f (x). f M := (2M + 1)d x∈ M

d

Let |∇x f (x)|2 := p=1 [jp f (x)]2 . For any F :  → R we shall write F˜ (x; ) := F (Tx ). Note also that taking the scalar product of both sides of (4.11) with u∗, and using (2.5) we obtain easily that sup (u∗, 2L2 () + u∗, 2+ ) < + ∞.

(5.1)

∈(0,1)

It is also well known, see, e.g., [5, formula (2.58), p. 806], that lim u∗, − u∗ + = 0.

(5.2)

→0+

To show the proposition in question it suffices to prove that u∗, − u∗ + C(d) ,

 ∈ (0, 1).

(5.3)

It will become apparent during the course of the proof that estimate (5.3) holds for more general measures  than the product Bernoulli measure on Bd . Namely one can take  := [−, +]Bd for some 0 < − < + , an arbitrary Borel d probability measure  on [− , + ] and  := B . Estimate (5.3) will result from the series of auxiliary steps formulated as lemmas. Lemma 5.1. Suppose that  ∈ (0, 1), M 1 and L ∈ F are fixed. Assume also that configurations ,  ∈  are such that (b) =  (b) for all b ∈ / L. Then, ⎧ ⎫1/2 ⎨ ⎬

 1/2 2 |u˜ ∗, (·; ) − u˜ ∗, (·;  )|2 M  (d + |∇ u ˜ (x; )| ) . (5.4) x ∗,  ⎭ [2− (2M + 1)d ]1/2 ⎩ x∈L˜

Here L˜ is the set of all vertices belonging to bonds from L. (M  )

Proof. Suppose now that M  M is an integer and u˜ ∗, (x; ) is the unique solution of the boundary value problem

(M  ) ( − L )u˜ ∗, (x; ) = −f˜∗ (x; ), x ∈ M  , (5.5) (M  ) u˜ ∗, (x; ) = 0 x ∈ jM  .

200

M. Cudna, T. Komorowski / Journal of Computational and Applied Mathematics 213 (2008) 186 – 204 (M  )

(M  )

A simple calculation shows that v(x) := u˜ ∗, (x; ) − u˜ ∗, (x;  ) satisfies 

( − L )v(x) = f˜∗ (x) + v(x) = 0,

d

(M  ) ∗ ˜ ∗, (x; )), p=1 jp ( p (x)jp u

x ∈ M  , x ∈ jM  ,

(5.6)

where p (x) = p (x; ) − p (x;  ) and f˜∗ (x) := −j∗1 p (x). Multiplying both sides of (5.6) and summing up over all x ∈ M  we get  x∈M 

(v 2 (x) + − |∇v(x)|2 )   

d 

(M  )

(1 + |jp u˜ ∗, (x; )|)|jp v(x)|

x∈L˜ p=1 2 

( ) 2−

(M  )

(d + |∇ u˜ ∗, (x; )|2 ) +

x∈L˜

−  |∇v(x)|2 . 2 x∈M 

From the above estimate we conclude in particular that v 2 M 



( )2 2− (2M + 1)

d

(M  )

(d + |∇ u˜ ∗, (x; )|2 ).

x∈L˜

The conclusion of the lemma can be obtained upon taking the limit, as M  → +∞, in light of the following estimate: for any ∈ (0, 1) there exist constants C1 , C2 > 0 such that (M  )

|u˜ ∗, (x; ) − u˜ ∗, (x; )| C1 u∗, L∞ () exp{−C2 (M  ) } ∀x ∈ M ,  ∈ ,  ∈ (0, 1), provided that M  2M, see [9, (4.32), p. 1296].

(5.7)

 ()

()

We define a (random) set L(t, x; , ) ⊂ Bd as the union of all bonds {Xs − (x; ), Xs (x; )} for all s t such that

()

()

Xs − (x; )  = Xs (x; ). For any 0 t  + ∞ and x ∈ Zd we let  U∗, (t, x; , ) := −

0

t

e−s f∗ (TX(n) (x;) ) ds. s

(5.8)

In the case x = 0 the parameter x shall be dropped from our notation. Lemma 5.2. There exists a constant C > 0 such that   1/2   1/2   2  EU ∗, (t, x)u˜ ∗, M d C t + 1 EU ∗, (t) d (1 + u∗, + )   M d

(5.9)

for all t 0, x ∈ Zd ,  ∈ (0, 1), M 1. Proof. To simplify the notation we assume that x = 0, the proof in the general case is identical. The expression under the absolute value on the left-hand side of (5.9) can be rewritten in the form  L∈F

=

E[U∗, (t; , ), L(t; , ) = L]u˜ ∗, (·; )M (d)

  L∈F

E[U∗, (t; , ), L(t; , ) = L]u˜ ∗, (·; )M (d)(d ),

(5.10)

M. Cudna, T. Komorowski / Journal of Computational and Applied Mathematics 213 (2008) 186 – 204

201

 where ˜ (b) equals (b) for b ∈ L and  (b) for b ∈ / L. Since u˜ ∗, (·;  )M (d ) = 0 we conclude from (5.10) that the left-hand side of (5.9) can be estimated by   E[U∗, (t; , ), L(t; , ) = L]|u˜ ∗, (·; ˜ ) − u˜ ∗, (·;  )M |(d)(d ) L∈F (5.4)



C

1/2 1/2 2 [L(t; , ) = L] (E[U∗,  (t; , ), L(t; , ) = L]) P (M d )1/2 L∈F ⎫1/2 ⎧ ⎬ ⎨ (1 + |∇y u˜ ∗, (y; )|2 ) (d). × ⎭ ⎩



(5.11)

y∈L˜

Using Cauchy–Schwartz inequality and subsequently performing the summation over all L we obtain that the right-hand side of (5.11) can be estimated by ⎧ 1/2 ⎨



C (M d )1/2

EU 2∗, (t) d

⎩

⎡ E⎣



⎫1/2 ⎬

⎤ (1 + |∇y u˜ ∗, (y)|2 ), L(t) = L⎦ d

⎭

y∈L˜

.

(5.12) ()

()

Recall now, see [8, Sections 2 and 3 of Appendix 1], that the random walk in question can be written as Xt = Xn () () () for Tn t < Tn+1 , where Xn is a discrete time, nearest neighbor, random walk on a lattice Zd starting at 0, with  the transition of probabilities p(y, y + e; ) := e (y; )/Z(y; ), e ∈ Ed , y ∈ Zd and Z(y; ) := e e (y; ). The () () random variables Tn on the other hand have independent increments, when conditioned upon the sequence Xn , () () () the respective conditional law of Tn+1 − Tn is exponential with the intensity Z(Xn ; ). Denote by p(n, x, y; ) the transition of probability of the discrete random walk in n steps. Using the discrete random walk we can rewrite the second factor of (5.12) containing the expectation E as estimated by ⎧ ⎨ ⎩





0  k  n y∈Zd

()

⎫1/2 ⎬

()

(1 + |∇ u˜ ∗, (y)|2 )p(k, 0, y; )P [Tn t < Tn+1 ] d

⎭

.

(5.13)

Invoking Gaussian estimates of the transition of probability, see, e.g., [6, Theorem 5.3, p. 690], we can write  # C2 |y|2∞ C1 exp − p(k, 0, y; ) k+1 (k + 1)d/2 for some constants C1 , C2 > 0 and all k 0. The expression in (5.13) can be therefore estimated by ⎧ +∞  ⎨

1/2 C1 ⎩

k=0 y∈Zd

1 (k + 1)d/2

1/2 = C1 (1 + u∗, 2+ )1/2

C2 |y|21 exp − k+1



+∞  k=0

&

k (k + 1)

⎫1/2 ⎬

() (1 + |∇ u˜ ∗, (y; )|2 )P [Tk t](d) ⎭

&1/2 () P [Tk t](d) d/2

.

(5.14)

 () Here k := y∈Zd exp{−C2 |y|2∞ /(k + 1)} ∼ k d/2 for k?1. Since the intensity parameter of increments of Tn is  bounded by ∗ := 2d+ we can estimate the sum appearing on the right-hand side of (5.14) by C k  0 P [N∗ t k] = C(1 + ∗ t), where N∗ t is a Poisson random variable with intensity ∗ t and C > 0 is a certain constant. Estimate (5.9) then follows.  For a given  > 0 we let t () := −1 log2 . We have the following estimate.

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M. Cudna, T. Komorowski / Journal of Computational and Applied Mathematics 213 (2008) 186 – 204

Lemma 5.3. There exists a constant C > 0 such that for all  ∈ (0, 1)  C EU 2∗, (t ()) d  .  Proof. We show first that  C EU 2∗, (∞) d  

(5.16)

for some C > 0 and all  ∈ (0, 1). A straightforward computation shows that   EU 2∗, (∞) d = 2 v∗, d, where (see (2.2)) v∗, () :=



∞ 0

(5.15)

e−2s E(f∗ u∗, )( s ) ds.

(5.17)

(5.18)

It satisfies the equation (2 − L)v∗, = f∗ u∗, . Hence    (2.5) 1 D1 u∗, d + u∗, + . 2 v∗, d = f∗ u∗, d = Estimate (5.16) can be concluded from (5.1). Bound (5.15) is an easy consequence of (5.16) since, as  it can2be easily seen by elementary calculation, for any positive N > 0 one can find a constant C > 0 such that |E[U∗,  (∞) − 2 (t ())]| d CN for all  ∈ (0, 1). U∗, 



Lemma 5.4. There exists a constant C > 0 such that for all  ∈ (0, 1), M 1 we have % $   t () 1/2 1 −log2  2 . + 1/2 e  u˜ ∗, (·; )M (d)C Md 

(5.19)

Proof. The left-hand side of (5.19) equals    EU ∗, (t (), x)u˜ ∗, (·)M d (2M + 1)d x∈ M    + E[U∗, (∞, x) − U∗, (t (), x)]u˜ ∗, (·)M d. (2M + 1)d x∈ M

The second term can be bounded from above by  +∞ (5.1) C 2 e−s ds  1/2 e−log  . Cu∗, L2 ()  t () Estimate (5.19) is then a straightforward consequence of the above bound and of (5.9) together with (5.15).



Suppose now that 0 <  < . We have u∗, − Lu∗, = −f∗

(5.20)

 u∗, − Lu∗, = −f∗ .

(5.21)

and

Subtracting (5.21) from (5.20) and taking the inner product of the difference against u∗, − u∗, we obtain  u∗, − u∗, 2L2 () + − u∗, − u∗, 2+ ( − )(u∗, , u∗, − u∗, )L2 () .

(5.22)

M. Cudna, T. Komorowski / Journal of Computational and Applied Mathematics 213 (2008) 186 – 204

203

Denote by u¯ ∗, := u˜ ∗, M . The scalar product on the right-hand side of the above inequality equals   u˜ ∗, (u˜ ∗, − u˜ ∗, )M d = u¯ ∗, (u¯ ∗, − u¯ ∗, ) d  + (u˜ ∗, − u¯ ∗, )[u˜ ∗, − u˜ ∗, − (u¯ ∗, − u¯ ∗, )]M d.

(5.23)

The first term on the right-hand side of (5.23) can be estimated by u¯ ∗, L2 () (u¯ ∗, L2 () + u¯ ∗, L2 () ). Using (5.1) and (5.19) we can further estimate this expression by C 

( )1/2

Q(,  M),

(5.24)

where Q(,  , M) := q(, M)[q(, M) + q( , M)] and   1 t () 1/4 2 + 1/4 e−(1/2) log  . q(, M) := d M  To estimate the second term on the right-hand side of (5.23) we use the Poincare inequality on M ⊂ Zd and obtain that the term in question is less than, or equal to  (5.1) 1/2 1/2 CM 2 |∇ u˜ ∗, |2 M |∇ u˜ ∗, − ∇ u˜ ∗, |2 M d  CM 2 u∗, − u∗, + . (5.25) We have shown therefore that  1/2  Q(,  , M) + CM 2 u∗, − u∗, + , u∗, − u∗, 2+ C   where C > 0 is a certain constant. Thus, $ %  1/4   2 1/2 u∗, − u∗, + C M  + Q (,  , M) . 

(5.26) 1+

Choose an arbitrary  > 0 and define recursively a sequence 0 :=  and k+1 := k for k 1. Let also Mk := −5/(d+8) [k ]. Thanks to (5.26) we conclude that one can choose a constant C > 0 such that u∗,k − u∗,k+1 + Ck ∗

∀k 1,

(5.27)

where ∗ := (d) − /2 and we assume that ∗ > 0. Obviously, k = (1+) and the conclusion of Proposition 4.4 follows from (5.2) and the estimate k

u∗, − u∗, + C

  C(1+)∗ k ∗  C(1+)∗ ∗ (1+) 1− k 1

∀ ∈ (0, 1).



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