Quenched invariance principle for random walks with time-dependent ergodic degenerate weights
We study a continuous-time random walk, , on in an environment of dynamic random conductances taking values in . We assume that the law of the conductances is ergodic with respect to space-time shifts. We prove a quenched invariance principle for the Markov process under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser’s iteration scheme.
Key words and phrases:time dependent dynamics, random walk, Moser iteration
2010 Mathematics Subject Classification:60K37; 60F17; 82C41
- 1 Introduction
- 2 Harmonic embedding and the corrector
- 3 Sublinearity of the corrector
- 4 Quenched invariance principle
- 5 Mean value inequality for time-inhomogeneous Poisson equation
- A Technical estimates
Random walks in random environment is a topic of major interest in probability theory. A specific model for such a random walks that has been intensively studied during the last decade is the Random Conductance Model (RCM). The question whether a quenched invariance principle or quenched functional central limit theorem (QFCLT) holds is of particular interest. In the case of an environment generated by static i.i.d. random variables this question has been object of very active research (see [2, 13] and references therein). Recently, in  a QFCLT has been proven for random walks under general ergodic conductances satisfying a certain moment condition.
Quenched invariance principles have also been shown for various models for random walks evolving in dynamic random environments (see [1, 8, 15, 19, 26, 35, 34]). Here analytic, probabilistic and ergodic techniques were invoked, but assumptions on the ellipticity and the mixing behaviour of the environment remained a pivotal requirement. For instance, the QFCLT for the time-dynamic RCM in  required strict ellipticity, i.e. the conductances are almost surely uniformly bounded and bounded away from zero, as well as polynomial mixing, i.e. the polynomial decay of the correlations of the conductances in space and time. In this paper we significantly relax these assumptions and show a QFLCT for the dynamic RCM with degenerate space-time ergodic conductances that only need to satisfy a moment condition. In contrast to the earlier results mentioned above the environment is not assumed to be strictly elliptic or mixing or Markovian in time and we also do not require any regularity with respect to the time parameter.
1.1. The setting
Consider the -dimensional Euclidean lattice, , for , whose edge set, , is given by the set of all non-oriented nearest neighbor bonds, that is . For any we denote by the cardinality of the set . Further, we denote by the closed ball with center and radius with respect to the natural graph distance , and we write . We also write , , for closed balls in with respect to the -norm with center at the origin and radius . The canonical basis vectors in will be denoted by .
The graph is endowed with time-dependent positive weights, that is, we consider a family . We refer to as the conductance on an edge at time . To simplify notation, for and we set if and otherwise. A space-time shift by is a map defined by
The set together with the operation defines the group of space-time shifts.
Finally, let be equipped with a -algebra, , and a probability measure, , so that becomes a probability space. We also write to denote the expectation with respect to .
Assume that satisfies the following conditions:
and for all and .
is ergodic and stationary with respect to space-time shifts, that is for all , , and for any such that for all , .
For every the mapping is jointly measurable with respect to the -algebra .
(i) Note that Assumption 1.1(i) implies that for all and almost all .
(ii) The static model where the conductances are constant in time and ergodic with respect to space shifts is included as a special case.
We denote by the space of -valued càdlàg functions on . We will study the dynamic nearest-neighbour random conductance model. For a given and for and , let be the probability measure on , under which the coordinate process is the continuous-time Markov chain on starting in at time with time-dependent generator (in the sense) acting on bounded functions as
That is, is the time-inhomogeneous random walk, whose time-dependent jump rates are given by the conductances. Note that the counting measure, independent of , is an invariant measure for . Further, the total jump rate out of any site is not normalised, in particular the sojourn time at site depends on . Therefore, the random walk is sometimes called the variable speed random walk (VSRW).
1.2. Main Results
We are interested in the -almost sure or quenched long time behaviour of this process. Our main objective is to establish a quenched functional central limit theorem for the process in the sense of the following definition.
Set , . We say that the Quenched Functional CLT (QFCLT) or quenched invariance principle holds for if for -a.e. under , converges in law to a Brownian motion on with covariance matrix . That is, for every and every bounded continuous function on the Skorohod space , setting and with being a Brownian motion started at , we have that -a.s.
As our main result we establish a QFCLT for under some additional moment conditions on the conductances. In order to formulate this moment condition we first define measures and on by
In addition, for arbitrary numbers and any non-empty compact interval and any finite let us introduce a space-time averaged -norm on functions by
Note that by Jensen’s inequality if and .
There exist satisfying
such that -a.s.
(i) Assume that for any with ,
where denotes the -algebra of sets invariant under time-shifts and the -algebra of sets invariant under space-shifts. Then, a sufficient moment condition for (1.5) to hold is
Indeed, for all the function is time-invariant and is space-invariant which yields
by the ergodic theorem and similarly for . In particular, notice that if the measure is space-ergodic we always have
so that we can choose and to be infinite.
(ii) Clearly the example in (i) can be made more general by considering conductances which are a mixture of products where is time-invariant and is space invariant. For example let
with being time-invariant and space-invariant. In this case for (1.5) to hold one needs to assume that
For the static RCM a QFCLT is proven in  for stationary ergodic conductances satisfying and for such that . Since in the static case we can choose , the moment condition for the static model can be recovered in (1.4).
In the setting of general ergodic environments it is natural to expect that some moment conditions are needed in view of the results in , where Barlow, Burdzy and Timar give an example for a static RCM on for which the QFCLT fails but a weak moment condition is fulfilled.
One motivation to study the dynamic RCM is to consider random walks in an environment generated by some interacting particle systems like zero-range or exclusion processes (cf. [17, 33]). Recently, some on-diagonal upper bounds for the transition kernel of a degenerate time-dependent conductances model are obtained in , where the conductances are uniformly bounded from above but they are allowed to be zero at a a given time satisfying a lower moment condition. In  it is shown that for uniformly elliptic dynamic RCM in discrete time – in contrast to the time-static case – two-sided Gaussian heat kernel estimates are not stable under perturbations. In a time dynamic balanced environment a QFCLT under moment conditions has been recently shown in .
An annealed FCLT has been obtained for strictly elliptic conductances in , for non-elliptic conductances generated by an exclusion process in  and for a similar one-dimensional model allowing some local drift in  and recently for environments generated by random walks in . In [12, 32] random walks on the backbone of an oriented percolation cluster are considered, which are interpreted as the ancestral lines in a population model.
Finally, let us remark that there is a link between the time dynamic RCM and Ginzburg-Landau interface models as such random walks appear in the so-called Helffer-Sjöstrand representation of the space-time covariance in these models (cf. [16, 1]). However, in this context the annealed FCLT is relevant.
1.3. The method
We follow the most common approach to prove a QFLCT for the RCM and introduce the so-called harmonic coordinates, that is we construct a corrector such that
is a space-time harmonic function. In other words,
This can be rephrased by saying that is a solution of the time-inhomogeneous Poisson equation
where denotes the identity mapping on . Recall that one property of the static RCM – being one its main differences to other models for random walks in random media – is the reversibility of the random walk w.r.t. its speed measure. In our setting, the generator of the space-time process is asymmetric and the construction of the corrector as carried out for instance in [2, 13] fails, since it is based on a simple projection argument using the symmetry of the generator and an integration by parts. In  it was possible to construct the corrector by techniques close to the original method by Kipnis and Varadhan, since in the case of strictly elliptic conductances the asymmetric part can be controlled and a sector condition holds. In our degenerate situation, the construction of the corrector is indeed one of the most challenging parts to prove the QFCLT. Following the approach in , we first solve a regularised corrector equation by an application of the Lax-Milgram lemma and then we obtain the harmonic coordinates by taking limits in a suitable distribution space. The resulting corrector function consists of two parts, one part being time-homogeneous and invariant w.r.t. space shifts in the sense that for every fixed it satisfies -a.s. the cocycle property (see Definition 2.2 below) and a second part which is only depending on the time variable and which therefore does not appear in the corrector for the time-static model.
Given the harmonic coordinates as a solution of (1.6) the process
is a martingale under for -a.e. , and a QFCLT for the martingale part can be easily shown by standard arguments. We thus get a QFCLT for once we verify that -almost surely the corrector is sublinear:
This control on the corrector implies that for any and -a.e ,
Once the corrector is constructed, the remaining difficulty in the proof of the QFCLT is to prove (1.8). In a first step we show that the rescaled corrector converges in the space-time averaged -norm to zero (see Proposition 3.3 below). This is based on some input from ergodic theory, see Section 3 for more details. In a second step we establish a maximal inequality for the corrector as a solution of (1.7) using Moser iteration, that is we show that the maximum of the rescaled corrector in (1.8) can be controlled by its -norm (see Proposition 3.2 below). In the case of static conductances Moser iteration has already been implemented in order to show the QFCLT in , but also to obtain a local limit theorem and elliptic and parabolic Harnack inequalities in  as well as upper Gaussian estimates on the heat kernel in . In the present time-inhomogeneous setting involving a time-dependent operator a space-time version of the Sobolev inequality in  is needed and the actual iteration procedure has to be carried out in both the space and the time parameter of the space-time averaged norm (cf. ).
The paper is organised as follows: In Section 2 we construct the corrector and show some of its properties. Then, in Section 3 we prove the sublinearity of the corrector (1.8) and complete the proof of the QFCLT in Section 4. The maximal inequality for the time-inhomogeneous Poisson equation in (1.7) is proven in a more general context in Section 5.
Throughout the paper, we write to denote a positive constant which may change on each appearance. Constants denoted by will be the same through each argument.
2. Harmonic embedding and the corrector
Throughout this section we suppose that Assumption 1.1 holds.
2.1. Setup and Preliminaries
Let us denote by the set of all neighbours of the origin in . Further, we endow the space with the measure defined by
It is easy to check that is a Hilbert space. For functions we define the horizontal gradient as . We will also write for with . Notice that for any . Further, we define
to be the closure of the set of gradients in and let be its orthogonal complement in , i.e.
Lemma 2.1 (cycle condition).
For any and any sequence in with and for all , then .
Follows directly from the definition of . ∎
For any we define its extension in the following way. For any choose a sequence in in such a way that , and for all and set
As a consequence of Lemma 2.1, does not depend on the choice of paths.
A measurable function , also called random field, satisfies the cocycle property (in space), if for -a.e. ,
We denote by the set of function which satisfies the cocycle property such that
Although coincides with the norm on , we nevertheless introduce this notation to stress the fact that we apply it to functions that satisfies in addition the cocycle property.
Let . Then
and for all .
, if and only if, -a.s. for all .
(i) follows immediately from the cocycle property. (ii) is obvious due to the stationarity of and the fact that -a.s. for any . ∎
Recall that, by Remark 1.3, the group is a SCCG on , therefore it has an infinitesimal generator , whose domain is dense in ,
whenever the limit exists in . Finally, we denote by and the scalar product in and , respectively.
The operator is antisymmetric in , that is
In particular and .
For every the operators and commute, that is
For every the adjoint of the operator is given by ,
For every the function belongs to -a.e. .
For any with compact support, and ,
For any , the function is weakly differentiable -almost surely. In particular
for almost all , -almost surely.
For every and every ,
(i) By the shift-invariance of we have for any
The second statement is trivial.
(ii) This follows directly from the linearity of as
where we also used that and .
(iii) Again by the shift invariance of we have
(iv) For any compact and
Thus, for -a.e. ,
(v) A simple change of variables gives
(vi) It follows by (iv) that and belong to -almost surely. By definition of weak differentiability, it suffices to show that for -a.e. and all
By Fubini’s theorem and the fact that (v) holds for all , (2.10) follows for any fixed -a.s. The null-set where (2.10) does not hold may depend on . We can remove this ambiguity using that is separable.
(vii) By the shift invariance of we have for any
Since Lemma 2.1 implies that for all , the assertion follows. ∎
2.2. Construction of the corrector
In this subsection we construct the corrector. We introduce the position field with . We write for the -th coordinate of . Obviously, satisfies the cocycle property since . Moreover, for every ,
Next, we state the main result of this subsection.
Suppose that Assumption 1.5 holds. Then, there exists a function with for such that the following hold.
For all ,
is the unique extension of a function in .
The function ,
also called harmonic coordinate, is (time-space) harmonic in the sense that is differentiable for almost every and
The harmonic coordinates have the asymptotics
Before we prove Theorem 2.5 we define the corrector and collect some of its properties.
The corrector is defined as
Let be defined as in the previous theorem and set .
with for all .
For -a.e. , and , the corrector can be written as
The rest of this section is devoted to the construction of the harmonic coordinates and the proof of Theorem 2.5 (i) and (ii). Statement (iii) is equivalent to the sublinearity of the corrector and will be proven in Section 3 below.
Let equipped with the norm given by
and a scalar product defined by polarisation. It is easy to see that is a Hilbert space. Also, is not trivial, since for and the function belongs to .
We want to solve the following equation
For all , is a coercive bounded bilinear form, and for all , is a bounded and linear operator on .
The statement is true basically by definition and Lemma 2.4 (i). Indeed,
and by the Cauchy-Schwarz inequality
Similarly, since it follows that is bounded for all . ∎
By an application of Lax-Milgram Lemma it follows that for every there exists such that holds for all . In particular, the equation is satisfied for . We use this information to obtain a first energy bound.
For all and