Einstein relation for random walks in random environments

. We consider a tracer particle performing a continuous time nearest neighbor random walk on Z d in dimension d ≥ 3 with random jump rates. The kind of a walk considered here models the motion of an electrically charged particle under a constant external electric ﬁeld. We prove the existence of the mobility coeﬃcient, and that it equals to the diﬀusivity coeﬃcient of the particle.


Introduction
Consider a particle moving in a random medium, which can be constituted by the molecules of a fluid in thermal equilibrium, or by atoms in a fixed periodic or random lattice. The trajectory X(t) of this particle, in a large space-time scale, can be regarded as a centered Brownian motion whose mean square displacement is proportional to time. The diffusivity of a Brownian particle is defined as a matrix D = [D p,q ], where for each t > 0 (1.1) D p,q := t −1 E[X p (t)X q (t)], p, q = 1, . . . , d.
The mobility σ is defined in the following way. Suppose that the moving particle is electrically charged and an exterior uniform electric field E = El is applied in a given direction l, represented by a unit vector in R d . In the corresponding stationary state, the particle will pick up a mean velocity v(E) corresponding to the magnitude E of the field. The limit vector defines the mobility of the particle. The Einstein relation, established in [3], says that σ = βDl, where β −1 = k B T , T being the temperature of the environment fluid, and k B is the Boltzmann constant. A heuristic derivation of this relation can be found in section 8.8 of [18].
A rigorous derivation of Einstein relation for a physically realistic model is a challenging problem and there are only few mathematical results on the subject available, see e.g. [9,10,8]. For a purely mechanical system the question seems to be out of reach of current mathematical methods. Therefore it is natural to look first at those models where the convergence to Brownian motion (of the rescaled path) is known, like e.g. for certain tracer dynamics in stochastic environments (cf. references at the end of Chapter 8 in [18]). The main difficulties one finds when trying to prove the Einstein relation, are: first to establish the existence of a stationary state and later to show good properties of relaxation to this stationary state of the dynamics. These facts could be rigorously established using perturbative argument when the environment is time depended and has the spectral gap property, see [8].
In the present paper we consider a particle motion in a fixed random environment, that is modeled by a continuous time nearest neighbor random walk in Z d . The dynamics can be described then as follows. The particle located at given time t at site x waits for an where γ({x, x + e}) = γ({x + e, x}), x, e ∈ Z d , |e| = 1 are independent identically distributed random variables with values in the interval [γ − , γ + ] ⊂ R + . This type of walk is sometimes called a random walk among random conductances, see [17]. In this paper we consider only the case where the i.i.d. random variable γ({x, x + e}) take only two possible values γ − and γ + , with 0 < γ − < γ + < ∞. A degenerate version of this model has been discussed in the physics literature in the context of random walks on an infinite percolation cluster. In that case γ({x, x + e}) can take only two values 0, or 1.
It is quite standard, using individual ergodic theorem, to show that t −1 X(t) converge to 0, as t → +∞ almost surely (with respect to the realization of the environment and the random jumps of the walk). Using the argument of Kipnis and Varadhan, see [6], one can show that the laws of t −1/2 X η (t) converge weakly to a centered normal distribution N (0, D), with D the effective diffusivity matrix.
For a given direction l ∈ S d−1 and α ∈ R we can consider the perturbed trajectory process X (α) (t) t≥0 that correspond to the motion under an external forcing field. The jump rates are now given by The degenerate case of this model describing biased random walks on the supercritical percolation cluster can be found in the theoretical physics literature, see [4]. We also add here that the results concerning the law of large numbers for biased random walks, with jumps rates as in (1.3), in the degenerate case have been recently obtained in [19,1].
Coming back to the situation of nondegenerate rates considered here it has been shown in [7] that the environment process corresponding to the particle motion, see Section 2.2 for its precise definition, possesses an invariant measure whose properties guarantee the existence of In the main theorem of this paper, see Theorem 2.6, we prove the existence of the mobility coefficient (1.2) for the particle and establish the Einstein relation between the mobility and self-diffusivity if the dimension d ≥ 3. In order to prove the Einstein relation, we adopt an appropriate modification of the method of Loulakis (cf. [10]), that was used in the proof of a version of the Einstein relation for the symmetric simple exclusion model. We should stress however some important differences between the results in [10] and those obtained in the present article. First, since in [10] there is no proof of the existence of a steady state, the definition of mobility used there (see Theorem 1, p. 351 in [10]) is weaker than the one established in this paper. Secondly, the environment of the exclusion process is time dependent, while in our case it is static. This fact causes some additional difficulty and forces us to adopt an assumption that γ(b), for any bond b = (x, x + e), can take only two different values. The only place in the proof where we use this fact comes in establishing inequality (3.16), which we can only prove using a duality argument. This, essentially, is the only reason why we have to use the aforementioned assumption. We add here that we are not aware of any other static model (i.e. when the environment is time independent), besides some periodic ones (see [14]), where the validity of the Einstein relation has been established. Suppose that R > 0 is a certain integer. Denote by Λ R the set of those bonds b = {v, w} that satisfy |v|, |w| < R. Let us fix s < t and l ∈ S d−1 := [m ∈ R d : |m| = 1]. Let V t s be the σ-algebra generated by bonds b having non-empty intersection with the slab [x ∈ Z d : s ≤ l · x ≤ t] and that do not intersect the half-lattice H := [x ∈ Z d : x · l > t]. For a fixed s ∈ R we let V + s := t:s<t V t s and for a fixed t ∈ R we let V − t := s:s<t V t s . For a given η ∈ Ω, l ∈ S d−1 and α ∈ R we consider a continuous time nearest neighbor random walk on Z d , starting at 0, with the generator and c (α) (x, e; η) := e αl·e γ(x, e; η) Z(x, η) , with γ(x, e; η) := γ(η(x, x + e)), Z(x, η) := |e|=1 γ(x, e; η). When α = 0 the generator of the walk can be rewritten (regardless of the direction l) in the following form Here x,η , x, y ∈ Z d . We shall always assume that the random walk is defined over the canonical path space T (α) The subscript x shall be suppressed when the walk starts at 0.

2.2.
The environment process. For any y ∈ Z d we define a shift operator T y : Let L (α) : C(Ω) → C(Ω) be a linear bounded operator given by with c (α) (e; η) := c (α) (0, e; η). It is the generator of the Ω-valued, Markov process, that we shall call the environment process, defined by where (X(t)) is the random walk process defined in the previous section. The corresponding semigroup is given by formula For any Borel probability measure ν on Ω we denote by P (α) ν the path measure on the space ; Ω) corresponding to the process starting with the initial distribution ν. In case when ν = δ η the corresponding measure shall be denoted by by P (α) η . We define the equilibrium measure with Z(η) := Z(0, η) and the normalizing factorZ := Zdµ ρ = 2dγ. Let P (α) denote the path measure of the environment process starting with the equilibrium measure.
When α = 0 the measureμ 0 (dη) is not anymore invariant. We will consider the (non-

Central limit theorem for trajectory fluctuations -homogenization.
Here we recall certain well known facts concerning the motion of an unperturbed tracer in the equilibrium environment, i.e. when α = 0. In that case we shall omit the index α in the notation.
The following theorem follows directly from the argument contained in [6], see also Chapters 2 and 3 of [12].
, converge weakly, as t → +∞, to a mean zero Gaussian Let us describe in more details the limiting co-variance matrix appearing in part (ii) of the above theorem. The position of the tracer at time t in the direction e p is given by the formula t } d p=1 are martingales with joint quadratic variation given by On the other hand by (2.8), we have So the asymptotic variance is given by In order to compute the second term on the right hand side of (2.11) we introduce H + , the completion of the subspace H 0 + of C(Ω) consisting of those F for which F dμ 0 = 0 in the norm F + := E(F, F ) 1/2 . The dual of H + will be denoted by H − . The operator L extends to a unitary isomorphism mapping H + onto H − . The norm of Ψ ∈ H − can be characterized via the following variational principle Here q their respective limits. iv) The functional (u p , ·) L 2 (μ 0 ) has a continuous extension from H 0 + to H + , which we denote by the same symbol (i.e. u p ∈ H − ). We have The limiting variance D = [D p,q ] of Theorem 2.1 equals Proof. The results follow from the standard homogenization theory. Part i) follows immediately from the fact that u p ∈ C(Ω) and (2.4). The other assertions can be proven using a suitable adaptation of the argument of (e.g.) [13].
2.4. The existence of a stationary state and the law of large numbers for the perturbed tracer particle. We fix a direction l ∈ S d−1 and assume that α = 0. Below we formulate a result proven in [7] that asserts the existence of a steady stateμ α for the environment process corresponding to the perturbed trajectory. This measure is equivalent toμ 0 when restricted to the σ-algebra that can be associated with the "forward bonds" in the direction pointed by the drift l, i.e. V + −N for any N ≥ 1. Also, we assert a version of the strong law of large numbers holding w.r.t. Q (α) .
To make the statement of the result precise we need some notation. Let (θ t ) t≥0 be the semi-dynamical system defined by the temporal shifts on D Ω , i.e. θ t ξ(·) := ξ(· + t), ξ ∈ D Ω .
For any a ∈ R we denote by O + a the smallest sub-σ-algebra of B(D Ω ) generated by mappings Theorem 2.3. There exists a Borel probability measureμ α on Ω satisfying the following conditions 1) it is invariant 2) for an arbitrary N ≥ 0,μ α is equivalent withμ 0 , when restricted to V + −N , i.e.
By substituting for F the components of a random vector u (α) := (u We can immediately conclude from part 4) of the previous theorem the following annealed version of the strong law of large numbers (it has been proved for the walks in discrete time in [16]). 3. The proof of Theorem 2.6 In the proof of (2.19) we will make wide use of the following proposition Proposition 3.1. For any local function F we have The constant c 1 > 0 may depend on F but it does not depend on α.
We postpone the proof of this proposition until Section 3.4. Instead, in the following we outline the role of this result in the demonstration of (2.19). By (2.17) we can rewrite Since F is local and F dμ 0 = 0, by Proposition 3.1 F dμ α → 0 as α → 0. It remains therefore to prove that Formally one can see this by the following argument. By stationarity ofμ α we have L (α) χ p dμ α = 0. With this and (2.12) we obtain and (3.4) would follow by Proposition 3.1 if D q χ p were a bounded local function. Unfortunately we only know that D q χ p is in L 2 (μ 0 ). So in order to make this last argument rigorous we need some local approximation of this function. In order to prove all this, we need to exploit the structure of the distribution of the random environment, in particular its duality properties.
3.1. The duality structure of L 2 (µ ρ ). We adopt the notation of [15]. Recall Suppose that Z ⊆ B d . Denote by F n (Z) the family of all subsets of Z of cardinality n. Let also F(Z) := n≥1 F n (Z). We shall omit writing the set Z if it equals B d . For A ∈ F we let The functions ξ A , A ⊆ E d form an orthonormal basis of L 2 (µ ρ ) and where H n := span{ξ A : A ∈ F n }.

The Glauber dynamics. Let us fix an integer R > 0 and consider a Markovian dy-
namics on Ω Λ R given by the generator Here The corresponding Dirichlet form is given by It is well known that this form satisfies the spectral gap estimate We also note, after a direct calculation, that, if be the Dirichlet form of the environment process corresponding to the symmetric simple random walk on the random lattice. A crucial estimate of the Glauber form by the Dirichlet form (3.12) is provided by the following lemma.
Lemma 3.2. Suppose that d ≥ 3. Then, for any integer R > 0 there exists a constant c 2 > 0, depending on R, such that Proof. By (3.11), Define τ e (A) := [τ e (b) : b ∈ A] and suppose that F , given by (3.10), belongs to H n for some n. We have then This is a Dirichlet form of the process ξ(t) According to Lemma 3.1 p. 984 of [15] we have the following bound stemming from transience of the process (ξ(t)) t≥0 and summing over A b and b ∈ Λ R we obtain (3.13).
3.3. The eigenvalue estimate. Lemma 3.3. Suppose that Ψ is local, supported in Λ R for a certain R > 0 and such that Ψdµ ρ = 0. Then there exists a constant c 4 > 0 depending on Ψ, ρ ∈ (0, 1) such that The expression under the absolute value on the left hand side of (3.16) equals where for any fixed B ∈ F(Λ c R ) We can write then that the utmost right hand side of (3.17) equals The absolute value of the expression on the right hand side of (3.18) can be estimated by Using the result of Lemma 3.2 we can further estimate (3.19) by Using Cauchy-Schwartz inequality we can bound this expression by For any bounded function F on Ω denote by, the supremum of the L 2 (μ 0 )-spectrum of L + F . Proof. We have On the other hand since ZF dµ ρ = 0 and F ∈ C 0 (Ω) there exists Λ R such that ZF ∈ B b (Λ R ) for a sufficiently large positive integer R and using (3.16) we can estimate the right hand side of (3.22) by for some constant c 5 > 0 depending only on F and γ − . Let F ∈ C 0 such that F dμ 0 = 0. Applying the entropy inequality, see e.g. [5] p. 347, we Here h T,η (α) is the relative entropy of P α η w.r.t. P η on interval [0, T ], i.e. for Ψ T (α) := dP α η /dP η [0,T ] , h T,η (α) := Ψ T (α) log Ψ T (α)dP η . A straightforward calculation using Proposition 2.6, p. 320 of [5] shows that h T,η (α) ≤ c 8 α 2 T for some deterministic c 8 > 0 independent of T . Using (3.23), (2.16) and Jensen's inequality we obtain that Applying the Feynman-Kac formula, we conclude that In addition, H ∈ H − and H − < ε.
ii) We have Moreover, there exists a constant c 9 > 0, depending only on c * ,Z, such that one can choose Proof of i). The proof relies on the following lemma.
Lemma 3.6. Let us fix λ > 0 and p ∈ {1, . . . , d}. Then, for any ε > 0 there exists F ∈ C 0 (Ω) such that Proof. Let us fix R > 0 and let χ (p) λ,R be the unique solution of the Dirichlet boundary value problem where L η is given by (2.1). Then δχ λ,R (x; η) satisfies the Dirichlet boundary value problem A standard bound on Green's function of the penalized Dirichlet boundary value problem, see Appendix A, yields that λ,R (0; η) ∈ C 0 (Ω). Returning to the proof of Theorem 3.5 we choose G ∈ C 0 , such that We have therefore LG. The conclusion of part i) of the theorem follows from (3.33) and part i) of Proposition 2.2, provided that λ is chosen in such a way that λ χ (p) λ − < ε/2. Proof of ii). Note that (3.27) follows easily from i) since, according to (3.26) and (2.6) we Denote by L 2 div (µ ρ ) the space of those square integrable, centered, divergenceless random vectors L = (L 1 , . . . , L d ), i.e. the fields that satisfy d q=1 L q D q φ dµ ρ = 0, for all φ ∈ C(Ω).
Let K (0) q := γ q D q G + γ p δ p,q and K (0) = (K pot is a potential field and K (0) div is divergenceless. Denoting c * := sup c p we can write Since C div (Ω), the space of all divergenceless local vector fields, is Then, the field K := K (0) − F satisfies the conclusions of part ii) of the theorem.
3.6. The Proof of (3.4). Let ε > 0 be chosen arbitrarily. As a rule all constants appearing in the following shall not depend on ε and α. Suppose that G, H are as in the statement of Theorem 3.5. We can write then that Denoting the first and second terms on the right hand side of (3.34) by I(α), II(α) respectively we can write that Sinceμ α is a steady state we conclude that the last term on the right hand side of (3.35) vanishes. Using (2.2) we conclude that Note that Γ q ∈ C 0 so by proposition 3.1, the first term on the utmost right hand side of (3.36) is of order of magnitude O(α 2 ). We also have for some constant c 10 > 0. The utmost right hand side of (3.24) is less than c 10 ε by virtue of part i) of Theorem 3.5. We have proved therefore that To estimate II(α) we choose A ∈ (ε −1 , 2ε −1 ). As a rule all the following constants shall not depend on A, α and ε. Repeating the argument leading up to (3.25) we conclude that Using once more variational principle to calculate λ 0 (L + αAH) we get By virtue of the representation (3.27) we can rewrite the expression on the right hand side of (3.39) in the following form for some constants c 11 , c 12 , c 13 > 0. Here J 2 := q K 2 q , J 2 := q (K q • T −eq ) 2 . We deal with the two terms appearing on the utmost right hand side of (3.40) in the same fashion so we only show how to estimate the first one. The term in question can be estimated by c 12 c 13 α 2 A 2 sup φ 2 J 2 dµ ρ : φ L 2 (µρ) ≤ 1, φ + ≤ c 13 αA J ∞ .
Note that where J := J 2 − J 2 L 2 (µρ) . Since φ L 2 (µρ) ≤ 1 we can estimate the first term on the right hand side of (3.41), with the help of (3.28) by c 2 9 H 2 − < c 2 9 ε 2 . To estimate the second term on the right hand side of (3.41) we use once more Lemma 3.3 and obtain that it is bounded by c 4 c 13 αA J ∞ . Summarizing, we have just shown that Appendix A. Proof of (3.32).
Let δ ∈ (0, 1) be fixed. Let τ R denote the exit time from the box Λ R . We have Note that According to Girsanov theorem, see [5], Proposition 2.6, p. 320, we can rewrite the second term on the right hand side of (A.1) in the form Appendix B. Density argument.