EINSTEIN RELATION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS
T.KOMOROWSKI AND S. OLLA
Abstract. We consider a tracer particle performing a continuous time nearest neighbor random walk on Zd in dimension d ≥ 3 with random jump rates. The kind of a walk con-sidered 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.
1. 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 = [Dp,q], where for each t > 0
(1.1) Dp,q:= t−1E[Xp(t)Xq(t)], p, q = 1, . . . , d.
The mobility σ is defined in the following way. Suppose that the moving particle is electri-cally charged and an exterior uniform electric field E = El is applied in a given direction l, represented by a unit vector in Rd. 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
(1.2) lim
E→0+
v(E) E =: σ
Date: March 23, 2004.
2000 Mathematics Subject Classification. Primary 60F17, 35B27; Secondary 60G44.
Key words and phrases. Passive Tracer, Random Walk in Random Environment, Einstein Relation. The research of T.K. was supported by KBN grant nr 2PO3A 031 23.
defines the mobility of the particle. The Einstein relation, established in [3], says that σ = βDl, where β−1 = kBT , T being the temperature of the environment fluid, and kB 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 challeng-ing 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 con-vergence 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 sta-tionary 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 Zd. The dynamics can be described then as follows. The particle located at given time t at site x waits for an exponential random time of unit intensity and performs a jump from site x to a neighboring site x + e with the probability
c(x, x + e) := P γ({x, x + e})
|e0|=1γ({x, x + e0})
where γ({x, x + e}) = γ({x + e, x}), x, e ∈ Zd, |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−1X(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−1/2Xη(t) converge weakly to a centered normal distribution N (0, D), with D
the effective diffusivity matrix.
For a given direction l ∈ Sd−1and 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
(1.3) c(α)(x, x + e) := eαl·ec(x, x + e).
The degenerate case of this model describing biased random walks on the supercritical per-colation 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 the mean velocity v(α) = limt→+∞t−1X(α)(t).
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.
2. Preliminaries and the statement of the main result
2.1. The description of the model. We denote by Bd the set of bonds on Zd, i.e. the set consisting of all unordered pairs {x, x + e}, where x, e ∈ Zd and |e| = 1. Let Ω be the compact metric space {0, 1}Bd
. By B(Ω) we denote the Borel σ–algebra of Ω. In fact, for any metric space X we denote by B(X) the σ–algebra of Borel sets. In addition, if A is any σ–algebra of subsets of X we denote by Bb(A) the set of all bounded and A–measurable real
valued functions.
For any subset A ⊂ Bdlet Ω
A:= {0, 1}A. When η ∈ Ω we denote by ηAthe restriction of η
to set A. Let 0 < γ−< γ+< +∞ be fixed and γ : {0, 1} → {γ−, γ+} be given by γ(0) = γ−,
γ(1) = γ+. Let C(Ω) denotes the space of all real valued continuous functions on Ω. By
C0(Ω) we denote the space of all local functions F : Ω → R, i.e. those for which there exists
a finite set A ⊂ Bd and a function G : ΩA→ R such that F (η) = G(ηA).
Let ρ ∈ [0, 1] and Bρbe the Bernoulli probability measure on {0, 1} given by Bρ[{1}] = ρ,
Bρ[{0}] = 1 − ρ. Denote by ¯γ := (1 − ρ)γ− + ργ+. Let µρ := BN B d ρ be the product measure given on B(Ω). By µA ρ := B N A
ρ we denote the respective product measure induced
on (ΩA, B(ΩA)).
Suppose that R > 0 is a certain integer. Denote by ΛR the set of those bonds b = {v, w}
that satisfy |v|, |w| < R. Let us fix s < t and l ∈ Sd−1:= [m ∈ Rd: |m| = 1]. Let Vst be the
σ-algebra generated by bonds b having non-empty intersection with the slab [x ∈ Zd : s ≤ l · x ≤ t] and that do not intersect the half-lattice H := [x ∈ Zd: x · l > t]. For a fixed s ∈ R we let V+
s :=
W
t:s<tVstand for a fixed t ∈ R we let Vt−:=
W
s:s<tVst.
For a given η ∈ Ω, l ∈ Sd−1 and α ∈ R we consider a continuous time nearest neighbor
random walk on Zd, starting at 0, with the generator
L(α)η f (x) := X
|e|=1
c(α)(x, e; η)∂ef (x), f ∈ C0(Zd),
where C0(Zd) is the space of compactly supported functions of Zd, ∂ef (x) := f (x + e) − f(x)
and
c(α)(x, e; η) := eαl·eγ(x, e; η) Z(x, η) ,
with γ(x, e; η) := γ(η(x, x + e)), Z(x, η) := P|e|=1γ(x, e; η). When α = 0 the generator of the walk can be rewritten (regardless of the direction l) in the following form
(2.1) Lηf (x) := −Z−1(x, η) d
X
i=1
∂i∗[γi(x; η)∂if (x)] , f ∈ C0(Zd).
Here ∂if := ∂eif , γi(x; η) := γ(x, ei; η), where e1, . . . , edis the canonical basis in Z
d. Let D :=
D([0, +∞); Zd) be equipped with the standard σ–algebra M and the filtration (Mt). The
transition of probabilities of the walk, its path measures on D and the respective expectations shall be denoted correspondingly by p(α)η (t, x, y), Px,η(α), E(α)x,η, x, y ∈ Zd. We shall always assume
that the random walk is defined over the canonical path space Tx,η(α) := (D, M, Px,η(α)) by the
formula (X(t))t≥0, for X ∈ D. The subscript x shall be suppressed when the walk starts at 0.
2.2. The environment process. For any y ∈ Zd we define a shift operator Ty : Bd → Bd
via Ty{x, x + e} := {x + y, x + y + e}. With the help of Tx, we define the shift operator on
Ω, which we also denote Tx, via Tx(η)(b) := η(Tx(b)), b ∈ Bd. For any function F : Ω → R
we let DxF := F ◦ Tx− F and DpF := DepF , p = 1, . . . , d.
Let L(α): C(Ω) → C(Ω) be a linear bounded operator given by
(2.2) L(α)F (η) := X
|e|=1
c(α)(e; η)DeF (η), F ∈ C(Ω),
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
(2.3) ζη(t; X) := TX(t)(η)
defined over Tα
η , where (X(t)) is the random walk process defined in the previous section.
The corresponding semigroup is given by formula
(2.4) PαtF (η) := X
x∈Zd
p(α)η (t, 0, x)F (Txη), F ∈ C(Ω).
For any Borel probability measure ν on Ω we denote by P(α)ν the path measure on the space
DΩ := D([0, +∞); Ω) corresponding to the process starting with the initial distribution ν. In
case when ν = δη 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 factor ¯Z := R Zdµρ = 2d¯γ. Let P(α) denote the
path measure of the environment process starting with the equilibrium measure.
When α = 0 a standard calculation shows that measure ¯µ0 is invariant, reversible and
ergodic under the semigroup defined by (2.4). The semigroup extends then to a C0–continuous
semigroup of self-adjoint operators over L2(¯µ
0). The generator and the Dirichlet form of the
semigroup are given respectively by
(2.6) LF (η) := − 1 Z(η) d X p=1 Dp∗(γp(η)DpF (η)) , F ∈ C(Ω), E(F, F ) := Z F (η)(−LF )(η)d¯µ0(η) = d X p=1 Z cp(η)(DeF (η))2d¯µ0(η), F ∈ C(Ω). Here γp(η) := γp(0; η), cp(η) := c(0, ep; η), p = 1, . . . , d.
When α 6= 0 the measure ¯µ0(dη) is not anymore invariant. We will consider the
(non-stationary) process ζ(t; η, X) := ζη(t; X) with the initial condition η distributed by µρ, i.e.
the process defined over (D × Ω, M ⊗ B(Ω), Q(α)), where Q(α) is the semi-product measure
Q(α)(dX, dη) = Pη(α)(dX) ⊗ µρ(dη).
2.3. Central limit theorem for trajectory fluctuations – homogenization. Here we recall certain well known facts concerning the motion of an unperturbed tracer in the equi-librium 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]. Theorem 2.1. (i) lim t→∞ X(t) t = 0, Q − a.s.,
(ii) The laws of the r.v. t−1/2X(t), converge weakly, as t → +∞, to a mean zero Gaussian
random vector with co-variance matrix D = [Dp,q].
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 ep is given by the formula
(2.7) Xp(t) = X(t) · ep=
Z t 0
where up(η) = −Z−1(η)D∗pγp(η), and {Mt(p)}dp=1 are martingales with joint quadratic varia-tion given by (2.8) DM(p), M(q)E t= 2δp,q Z t 0 dp(ζ(s)) ds,
where dp(η) := Z−1(η)[γp(η) + γp(T−epη)]. By the time reversibility of ¯µ0 (cf. [2])
(2.9) EM(p) t M (q) t = E Z t 0 up(ζ(s)) ds Z t 0 uq(ζ(s)) ds + E (Xp(t)Xq(t))
On the other hand by (2.8), we have
(2.10) EM(p) t M (q) t = tδp,q Z dp d¯µ0 = t dδp,q. So the asymptotic variance is given by
(2.11) Dp,q= t dδp,q− limt→∞ 1 tE Z t 0 up(ζ(s)) ds Z t 0 uq(ζ(s)) ds
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
R
F d¯µ0 = 0 in the
norm kF k+:= 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
kΨk2−= (Ψ, (−L)−1Ψ)L2(¯µ0) = sup
F ∈H0 +
[2(Ψ, F )L2(¯µ0)− kF k2+].
Here (−L)−1 is understood as the inverse operator to the extension L : H+→ H−.
Let λ > 0 and χ(p)λ := (λ − L)−1up ∈ C(Ω), p = 1, . . . , d. The following proposition gathers
some useful properties of up and χ(p)λ .
Proposition 2.2. For each p = 1, . . . , d we have. i) sup λ∈(0,1]λkχ (p) λ k∞< +∞. ii) lim λ→0+λkχ (p) λ k−= 0.
iii) χ(p)λ and Dqχ(p)λ converge in H+ and L2(¯µ0), respectively, as λ → 0+. Denote by χ(p)q ,
Φ(p)q their respective limits.
iv) The functional (up, ·)L2(¯µ0) has a continuous extension from H0+ to H+, which we denote
by the same symbol (i.e. up∈ H−). We have
The limiting variance D = [Dp,q] of Theorem 2.1 equals (2.13) Dp,q= 2 1 2dδp,q+ Φ(q)p ,γp Z L2(¯µ0) .
Proof. The results follow from the standard homogenization theory. Part i) follows immedi-ately from the fact that up ∈ 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 ∈ Sd−1 and assume that α 6= 0. Below
we formulate a result proven in [7] that asserts the existence of a steady state ¯µα for the
environment process corresponding to the perturbed trajectory. This measure is equivalent to ¯µ0 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 (θt)t≥0 be the
semi-dynamical system defined by the temporal shifts on DΩ, i.e. θtξ(·) := ξ(· + t), ξ ∈ DΩ.
For any a ∈ R we denote by Oa+the smallest sub-σ–algebra of B(DΩ) generated by mappings
ξ → F (ξ(t)), ξ ∈ DΩ, where F is Va+–measurable and t ≥ 0. Note that each θt is O+a to
O+a–measurable, i.e. θ−1t (A) ∈ Oa+, when A ∈ Oa+.
Theorem 2.3. There exists a Borel probability measure ¯µα on Ω satisfying the following
conditions 1) it is invariant (2.14) Z PαtF d¯µα= Z F d¯µα, ∀ t ≥ 0, F ∈ C(Ω).
2) for an arbitrary N ≥ 0, ¯µα is equivalent with µ¯0, when restricted to V−N+ , i.e.
(2.15) µ¯0(A) = 0 iff ¯µα(A) = 0 for all A ∈ V−N+ .
3) it is ergodic, i.e. if F ∈ C(Ω) is such that Pt
αF = F for all t ≥ 0 we have F =const
¯ µα-a.s.
4) the law of large numbers holds, i.e. for any N ≥ 0 and F ∈ Bb(O+−N) we have
(2.16) lim T ↑+∞ 1 T T Z 0 F (θtξ)dt = Z F dPµ¯α, for P (α)− a.s. ξ ∈ D Ω.
5) ¯µα is unique, i.e. any other Borel measure on Ω satisfying conditions 1) − 4) listed
above coincides with µ¯α.
By substituting for F the components of a random vector u(α):= (u(α)1 , . . . , u(α)d ), where (2.17) u(α)p (η) = Z−1(η)heαlpγ
p(η) − e−αlpγp(T−epη)
i
, η ∈ Ω
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]).
Corollary 2.4. For each α ∈ R we have
(2.18) v(α) := lim t↑+∞ X(t) t = Z u(α)d¯µα, P(α)− a.s.
Remark 2.5. It follows from the proof of Theorem 3.1 of [7] that the component of the mean
velocity v(α) in the direction l is non-zero.
2.5. Einstein relation between the mobility and self diffusivity coefficient. Our result concerning the Einstein relation can be stated as follows.
Theorem 2.6. The function α → v(α) defined by (2.18) is differentiable at 0 and
(2.19) v0(0) = Dl.
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
(3.1) Z F d¯µα− Z F d¯µ0 ≤ c1α.
The constantc1 > 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
(3.2) u(α)p (η) = `p
d + up(η) + αF (η) + O(α
where (3.3) F (η) := `p Z(η) 2¯γ −Z(η) d + ˜γp(η) + ˜γp(T−epη)
and ˜γp(η) := γp(η) − ¯γ. Since F is local and RF d¯µ0= 0, by Proposition 3.1 R F d¯µα→ 0 as
α → 0. It remains therefore to prove that
(3.4) lim α→0 1 α Z up(η) d¯µα= 2 X q Φ(q)p ,γp Z L2(¯µ 0) `q
Formally one can see this by the following argument. By stationarity of ¯µα we have
R
L(α)χpd¯µα= 0. With this and (2.12) we obtain
1 α Z up(η) d¯µα= 1 α Z L(α)− Lχp(η) d¯µα = d X q=1 `q Z Z−1 γqDqχp+ γq◦ T−eqDqχp◦ T−eq d¯µα+ O(α) (3.5)
and (3.4) would follow by Proposition 3.1 if Dqχp were a bounded local function.
Unfortu-nately we only know that Dqχp is in L2(¯µ0). 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 L2(µ
ρ). We adopt the notation of [15]. Recall
ξb(η) := pη(b) − ρ
ρ(1 − ρ), b ∈ B
d
, η ∈ Ω.
Suppose that Z ⊆ Bd. Denote by Fn(Z) the family of all subsets of Z of cardinality n. Let
also F(Z) := S
n≥1Fn
(Z). We shall omit writing the set Z if it equals Bd. For A ∈ F we let
ξA(η) := Q b∈A ξb(η), if A 6= φ, 1, if A = φ. The functions ξA, A ⊆ Ed form an orthonormal basis of L2(µρ) and
L2(µρ) = +∞ M n=0 Hn, where Hn:= span{ξA: A ∈ Fn}.
3.2. The Glauber dynamics. Let us fix an integer R > 0 and consider a Markovian dy-namics on ΩΛR given by the generator
(3.6) GΛRF (η) := X b∈ΛR 1 ρ η(b) 1 1 − ρ 1−η(b) [F (ηb) − F (η)], ∀ F ∈ Bb(ΛR). Here ηb(b0) = η(b0), if b0 6= b, 1 − η(b), if b0 = b.
The corresponding Dirichlet form is given by
(3.7) EΛg R(F, F ) := X b∈ΛR Z [F (ηb) − F (η)]2µΛR ρ (dη), ∀ F ∈ Bb(ΛR).
It is well known that this form satisfies the spectral gap estimate
(3.8) EΛg R(F, F ) ≥ kF k 2 L2(µΛR ρ ) , ∀ F ∈ L20(µΛR ρ ).
If F ∈ L20(µΛρR) then there is a unique G ∈ L20(µΛρR) (which is obviously also bounded)
satisfying
(3.9) F = −GΛRG.
We also note, after a direct calculation, that, if
(3.10) F =X A ˆ FAξA then (3.11) Z [F (ηb) − F (η)]2µρ(dη) = 1 (1 − ρ)ρ X b∈A ˆ FA2. Let (3.12) Esh(F, F ) := 1 2d X |e|=1 Z (DeF )2dµρ, F ∈ C(Ω)
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 c2 > 0,
depending on R, such that
(3.13) EΛg R(F, F ) ≤ c2E sh(F, F ), ∀ F ∈ B b(ΛR). Proof. By (3.11), (3.14) EΛg R(F, F ) = 1 (1 − ρ)ρ X b∈ΛR X A3b ˆ FA2.
Define τe(A) := [τe(b) : b ∈ A] and suppose that F , given by (3.10), belongs to Hn for some
n. We have then Esh(F, F ) = 1 2d X |A|=n X |e|=1 ( ˆFτe(A)− ˆFA) 2, F ∈ Hn.
This is a Dirichlet form of the process ξ(t) := τXt(A), t ≥ 0, where (Xt)t≥0 is a symmetric,
simple, random walk on Zd starting at 0. The state space of this process is F
n– the family of
sets of cardinality n. The transition of probability from set A to a set B (both of cardinality n) in time t for this process equals
P (t, A, B) = X
y∈Zd
δ(B, τy(A))p(t, 0, y),
where p(t, x, y) is the transition of probability of the symmetric simple random walk. In fact, only one term of this sum could possibly be non-zero, corresponding to the eventual value of y = r(A, B) such that B = τy(A).
Let g the Green function of the simple symmetric random walk. Then the Green function of ξ(t) is given by G(A, B) = g(0, r(A, B)) if A is a parallel translation of B, G(A, B) = 0 otherwise.
According to Lemma 3.1 p. 984 of [15] we have the following bound stemming from transience of the process (ξ(t))t≥0
(3.15) FˆA2 ≤ c3 sup B∈Fn
G(A, B) Esh(F, F ) ≤ c3g(0, 0)Esh(F, F )
3.3. The eigenvalue estimate.
Lemma 3.3. Suppose that Ψ is local, supported in ΛR for a certain R > 0 and such that
R
Ψdµρ= 0. Then there exists a constant c4> 0 depending on Ψ, ρ ∈ (0, 1) such that
(3.16) Z Ψφ2dµρ ≤ c4kφkL2(µ ρ)E sh(φ, φ)1/2, ∀ φ ∈ L2(µρ).
Proof. Let φ = PAφˆAξA. The expression under the absolute value on the left hand side of
(3.16) equals (3.17) X A,A0 ˆ φAφˆA0 Z ΨξAξA0dµρ= X A,A0 ˆ φAφˆA0 Z ΨξA∩ΛRξA0∩ΛRdµρ Z ξA∩Λc RξA0∩ΛcRdµρ = X B∈F (Λc R) X B1,B2∈F (ΛR) ˆ φB∪B1φˆB∪B2 Z ΨξB1ξB2dµρ= X B∈F (Λc R) Z Ψφ2Bdµρ,
where for any fixed B ∈ F(Λc R) φB := X B1∈F (ΛR) ˆ φB∪B1ξB1.
Suppose that G ∈ Bb(ΛR) is such that GΛRG = Ψ. We can write then that the utmost right
hand side of (3.17) equals
(3.18) X B∈F (Λc R) Z GGΛR(φ 2 B)dµρ= 1 ρ(1 − ρ) X B∈F (Λc R) X b∈ΛR Z G(η)[φ2B(ηb) − φ2B(η)]dµρ.
The absolute value of the expression on the right hand side of (3.18) can be estimated by
(3.19) 2kGk∞ ρ(1 − ρ) X B∈F (Λc R) X b∈ΛR kφBkL2(µρ) Z [φB(ηb) − φB(η)]2dµρ 1/2 .
Using the result of Lemma 3.2 we can further estimate (3.19) by 2c2kGk∞ ρ(1 − ρ) X B∈F (Λc R) kφBkL2(µ ρ)E sh(φ B, φB)1/2.
Using Cauchy-Schwartz inequality we can bound this expression by 2c2kGk∞ ρ(1 − ρ) X B∈F (Λc R) kφBk2L2(µρ) 1/2 X B∈F (Λc R) Esh(φB, φB) 1/2 ,
Note however that X B∈F (Λc R) kφBk2L2(µρ)= X B∈F (Λc R) X B1∈F (ΛR) ˆ φ2B∪B1 ≤ kφk2L2(µρ).
and X B∈F (Λc R) Esh(φB, φB) = 1 2d X B∈F (Λc R) X B1∈F (ΛR) X |e|=1 ( ˆφτe(B∪B1)− ˆφB∪B1) 2 = Esh(φ, φ). For any bounded function F on Ω denote by,
(3.20) λ0(L + F ) := sup
kφkL2( ¯µ0)≤1((L + F )φ, φ)L
2(¯µ0)
the supremum of the L2(¯µ0)–spectrum of L + F .
Lemma 3.4. Suppose that F ∈ C0 such that RF d¯µ0 = 0, and d ≥ 3. Then, there exists a
constant c5> 0, depending only on F and γ−, such that
(3.21) λ0(L + αF ) ≤ c5α2. Proof. We have λ0(L + αF ) = sup kφkL2( ¯µ0)≤1 " α Z F φ2d¯µ0− 1 2 d X i=1 Z ci(η)(Diφ)2d¯µ0 # sup kφkL2( ¯µ0)≤1 " α Z ZF φ2dµρ− 1 2 d X i=1 Z γi(η)(Diφ)2dµρ # (3.22)
On the other hand sinceRZF dµρ= 0 and F ∈ C0(Ω) there exists ΛRsuch that ZF ∈ Bb(ΛR)
for a sufficiently large positive integer R and using (3.16) we can estimate the right hand side of (3.22) by sup φ h c6αEsh(φ, φ)1/2− γ−Esh(φ, φ) i ≤ c5α2
for some constant c5> 0 depending only on F and γ−.
3.4. Proof of proposition 3.1. Recall about our convention of omitting superscript in relevant expressions when α = 0. We have now all the ingredients to prove proposition 3.1. Let F ∈ C0 such that RF d¯µ0 = 0. Applying the entropy inequality, see e.g. [5] p. 347, we
have (3.23) Z α Z T 0 F (ζ(s)) ds Pηα(dX) ≤ log Z exp α Z T 0 F (ζ(s)) ds Pη(dX) + hT,η(α).
Here hT,η(α) is the relative entropy of Pηα w.r.t. Pη on interval [0, T ], i.e. for
ΨT(α) := dPηα/dPη
[0,T ], hT,η(α) :=
R
using Proposition 2.6, p. 320 of [5] shows that hT,η(α) ≤ c8α2T for some deterministic c8 > 0
independent of T . Using (3.23), (2.16) and Jensen’s inequality we obtain that (3.24) α Z F d¯µα = limT →∞ Z Z 1 T Z T 0 αF (ζ(s)) ds Pηα(dX) ¯µ0(dη) ≤ lim sup T →+∞ 1 T log Z Z exp Z T 0 αF (ζ(s)) ds Pη(dX) ¯µ0(dη) + hT(α) T with hT(α) :=RhT,η(α)¯µ0(dη) ≤ c7α2T.
Applying the Feynman-Kac formula, we conclude that (3.25) α Z F d¯µα ≤ λ0(L + αF ) + c7α2 (3.21) ≤ c8α2 and (3.1) follows.
3.5. Localization. Denote by C0 := [ F ∈ C0(Ω) : R F d¯µ0 = 0 ]. Let also c∗ := inf cp,
c∗:= sup cp. Our first principal observation is the following analogue of Theorem 4.2 of [11].
Theorem 3.5. For any ε > 0 , p = 1, . . . , d, there exist G, H ∈ C0 such that
i) (3.26) up= −LG + H. In addition, H ∈ H− and kHk−< ε. ii) We have (3.27) ZH = d X q=1 D∗qKq, where Kq∈ C0, q = 1, . . . , d.
Moreover, there exists a constant c9 > 0, depending only on c∗, ¯Z, such that one can choose
K = (K1, . . . , Kd), for which
(3.28) kKkL2(µ
ρ)≤ c9kHk−.
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 ∈ C0(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 (3.30) λχ(p)λ,R(x; η) − Lηχ(p)λ,R(x; η) = up(x; η), x ∈ ΛR χ(p)λ,R(x; η) = 0, x ∈ ∂ΛR,
where Lη is given by (2.1). Then δχ(p)λ (x; η) := χ(p)λ (x; η) − χ(p)λ,R(x; η) satisfies the Dirichlet
boundary value problem
(3.31) λδχ(p)λ (x; η) − Lηδχ(p)λ (x; η) = 0, x ∈ ΛR δχ(p)λ (x; η) = χ(p)λ (x; η), x ∈ ∂ΛR.
A standard bound on Green’s function of the penalized Dirichlet boundary value problem, see Appendix A, yields that
(3.32) |δχ(p)λ (x; η)| ≤ c5kχ(p)λ k∞exp{−c4Rδ}, ∀ x ∈ ΛR/2,
where a parameter δ ∈ (0, 1) while deterministic constants c4, c5 > 0 depend only on δ, λ, d,
c∗, c∗. The proof of the lemma follows if we choose F (η) := χ(p)λ,R(0; η) ∈ C0(Ω).
Returning to the proof of Theorem 3.5 we choose G ∈ C0, such that
kG − χ(p)λ k∞+ kDG − Dχ (p) λ k∞< ε/2. Note that (3.33) kLχ(p)λ − LGk2−= 1 2 X |e|=1 Z c(e; η)|DeG(η) − Deχ(p)λ (η)|2µ¯0(dη) ≤ ε2/4. We have therefore up= −LG + up+ Lχ(p)λ − Lχ(p)λ + LG = −LG + λχ(p)λ − Lχ(p)λ + LG.
Set H := λχ(p)λ + Lχ(p)λ − 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 λkχ(p)λ k−< ε/2.
Proof of ii). Note that (3.27) follows easily from i) since, according to (3.26) and (2.6) we have ZH = − d X q=1 Dq∗(γqDqG) − Dp∗γp. Denote by L2
div(µρ) the space of those square integrable, centered, divergenceless random
vectors L = (L1, . . . , Ld), i.e. the fields that satisfy d
X
q=1
Z
Let Kq(0):= γqDqG + γpδp,q and K(0)= (K1(0), . . . , K (0)
d ). From Hodge decomposition K(0)=
Kpot(0)+ Kdiv(0), where Kpot(0) is a potential field and Kdiv(0) is divergenceless. Denoting c∗ := sup c p we can write kHk−≥ sup kDφk2 L2( ¯µ0)≤2(c∗)−1 (H, φ)L2(¯µ0) = ¯Z−1 sup kDφk2 L2( ¯µ0)≤2(c ∗)−1 Z ZHφdµρ= √ 2(c∗)−1/2Z¯−1kKpot(0)kL2(µρ).
Since Cdiv(Ω), the space of all divergenceless local vector fields, is L2–dense in L2div(µρ), see
Lemma B.1 we can find F = (F1, . . . , Fd) ∈ Cdiv(Ω) such that kF − Kdiv(0)kL2(µρ) < kHk−.
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
(3.34) Z upd¯µα= − Z LGd¯µα+ Z Hd¯µα.
Denoting the first and second terms on the right hand side of (3.34) by I(α), II(α) respectively we can write that
(3.35) I(α) =
Z
(L(α)G − LG)d¯µα−
Z
L(α)Gd¯µα.
Since ¯µα 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
(3.36) I(α) = α d X q=1 `q Z
Z−1(η)[γq(η)DqG(η) + γq(T−eqη)DqG(T−eqη)]¯µα(dη) + o(α)
= α d X q=1 `q Z Γq(η)¯µα(dη) + 2α d X q=1 `q Z Z−1γqDqGd¯µ0+ o(α), where Γq(η) := Z−1(η)[γq(η)DqG(η) + γq(T−eqη)DqG(T−eqη)] − 2 d X q=1 `q Z Z−1γqDqGd¯µ0.
Note that Γq ∈ C0so by proposition 3.1, the first term on the utmost right hand side of (3.36)
is of order of magnitude O(α2). We also have d X q=1 `q Z Z−1γqDqGd¯µ0− d X q=1 `q Z χ(p)Lχ(q)d¯µ0 ≤ d X q=1 |`q| Z Z−1γq(DqG − Dqχ(p))d¯µ0 ≤ d X q=1 kZ−1γqk2L2(¯µ0) 1/2 d X q=1 kDqG − Dqχ(p))k2L2(¯µ0) 1/2 ≤ c10kuq− LGk−
for some constant c10> 0. The utmost right hand side of (3.24) is less than c10ε by virtue of
part i) of Theorem 3.5. We have proved therefore that
(3.37) lim sup α→0 I(α) α − 2 d X q=1 `q Z Z−1γqDqGd¯µ0 ≤ c10ε.
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
αA Z H d¯µα ≤ λ0(L + αAH) + c7α2, hence (3.38) 1 α Z H d¯µα ≤ 1 α2Aλ0(L + αAH) + c7 A ≤ 1 α2Aλ0(L + αAH) + c7ε.
Using once more variational principle to calculate λ0(L + αAH) we get
(3.39) λ0(L + αAH) = sup kφkL2( ¯µ0)≤1 Z αAHφ2d¯µ0−1 2Z¯ −1 X |e|=1 Z γ(e)(Deφ)2dµρ .
By virtue of the representation (3.27) we can rewrite the expression on the right hand side of (3.39) in the following form
(3.40) Z¯−1 sup kφkL2( ¯µ0)≤1 d X q=1 Z αAKqDqφ2dµρ− d X p=1 Z γp(Dpφ)2dµρ ≤ c11 sup kφkL2(µρ)≤1 αA d X q=1 Z (φ + φ ◦ Teq) 2K2 qdµρ 1/2 kφk+− γ−kφk2+
≤ c12αA sup (Z φ2J2dµρ 1/2 kφk+: kφkL2(µρ) ≤ 1, kφk+ ≤ c13αA Z φ2J2dµρ 1/2) +c12αA sup Z φ2Jb2dµρ 1/2 kφk+: kφkL2(µρ)≤ 1, kφk+≤ c13αA Z φ2Jb2dµρ 1/2 for some constants c11, c12, c13> 0. Here J2 :=PqKq2, bJ2:=Pq(Kq◦ 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
c12c13α2A2sup (Z φ2J2dµρ: kφkL2(µρ)≤ 1, kφk+≤ c13αAkJk∞. ) Note that (3.41) Z φ2J2dµρ= kJk2L2(µρ)kφk2L2(µρ)+ Z φ2Jdµe ρ,
where eJ := J2− kJk2L2(µρ). Since kφkL2(µρ) ≤ 1 we can estimate the first term on the right
hand side of (3.41), with the help of (3.28) by c2
9kHk2− < c29ε2. 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 c4c13αAkJk∞. Summarizing, we have just shown that
1
α2Aλ0(L + αAH) ≤ c14A(ε 2
+ αA) ≤ c14(2ε + αA2)
for a certain constant c14> 0. Hence, we conclude that
lim sup α→0 II(α) α ≤ (c8+ 2c14)ε. and (3.4) follows. Appendix A. Proof of (3.32).
Let δ ∈ (0, 1) be fixed. Let τR denote the exit time from the box ΛR. We have
δχ(p)λ (x; η) = Ex,η[e−λτRχ(p)λ (X(τR); η)] ≤ kχ(p)λ k∞Ex,η[e−λτR]. Note that (A.1) Ex,η[e−λτR] ≤ Ex,η[e−λR δ , τR≥ Rδ] + Ex,η[ τR< Rδ] ≤ e−λR δ + Ex,η[ τR< Rδ]
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
(A.2) Ex τR< R δ, exp − X 0≤s≤Rδ log[2dpη(X(s−), X(s))]1[X(s−)6=X(s)] . For any x, y ∈ Zd pη(x, y) := 0, |x − y| > 1 c(y − x; η), |x − y| = 1
and Ex denotes the expectation w.r.t. the path measure Px corresponding to the symmetric
simple random walk in Zdwith unit intensity. Let A
m be the event consisting of those paths
which have exactly m jumps in time interval [0, Rδ]. Obviously, since the rate of jumps equals 1
we have Px[Am] = e−R
δ
Rδm/m!. Hence, choosing appropriately constants c15, c16, c17, c18> 0
we can estimate the expression in (A.2) from above by
∞ X m=0 ec15mP x h τR< Rδ, Am i ≤ X m≥c16R ec15mP x h τR< Rδ, Am i ≤ X m≥c16R ec15m−RδR mδ m! ≤ 1 [c16R]! e(ec15−1)Rδ ≤ c17e−c18R.
Summarizing the left hand side of (A.1) can be therefore estimated by c19e−c20R
δ
for some
constants c19, c20> 0 and (3.32) follows.
Appendix B. Density argument.
Lemma B.1. The space Cdiv(Ω) of local, divergence free fields is L2-dense in L2div(µρ).
Proof. Choose an arbitrary ε > 0 and suppose that F = (F1, . . . , Fd) ∈ L2div(µρ). Let ˆF =
( ˆF1, . . . , ˆFd) be the random spectral measure corresponding to F . We have
Fp(Txη) =
Z
Td
ei x·kFˆp(dk; η).
For any λ > 0 set
Hp,q(λ)(η) := Z Td (ei kq − 1) ˆF p(dk; η) − (ei kp− 1) ˆFq(dk; η) Pd r=1|ei kr− 1|2+ λ .
Obviously Hp,q(λ) = −Hq,p(λ) for all p, q = 1, . . . , d. Since PqDq∗Fq ≡ 0 we have Fp(λ) :=X q D∗qHp,q(λ) = Z Td Pd r=1|ei kr − 1|2 Pd r=1|ei kr− 1|2+ λ ˆ Fp(dk).
Let F(λ)= (F1(λ), . . . , Fd(λ)). Choosing λ > 0 sufficiently small we obtain that kF −F(λ)kL2(µρ)<
ε/2. Selecting suitable local eHp,q satisfying eHp,q(λ) = − eHq,p(λ) for all p, q = 1, . . . , d we can
guar-antee that d X p=1 k d X q=1 Dq∗(Hp,q(λ)− eHp,q)k2L2(µρ)< ε2/4.
Note that G := (G1, . . . , Gd), where Gp :=PqD∗qHep,q∈ Cdiv(Ω) and kF − GkL2(µρ)< ε. References
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Institute of Mathematics UMCS
pl. Marii Curie-Sk lodowskiej 1 Lublin 20-031, Poland
komorow@golem.umcs.lublin.pl
http://golem.umcs.lublin.pl/~komorow
Ceremade, UMR CNRS 7534 Universit´e de Paris IX - Dauphine,
Place du Mar´echal De Lattre De Tassigny 75775 Paris Cedex 16 - France.
olla@ceremade.dauphine.fr
