Harnack inequalities on weighted graphs and some applications to the random conductance model
We establish elliptic and parabolic Harnack inequalities on graphs with unbounded weights. As an application we prove a local limit theorem for a continuous time random walk in an environment of ergodic random conductances taking values in satisfying some moment conditions.
Key words and phrases:Harnack inequality, Moser iteration, Random conductance model, local limit theorem, ergodic
2000 Mathematics Subject Classification:31B05, 39A12, 60J35, 60K37, 82C41
- 1 Introduction
- 2 Sobolev and Poincaré inequalities
- 3 Elliptic Harnack inequality
- 4 Parabolic Harnack inequality
- 5 Local limit theorem
- 6 Improving the lower moment condition
- A Technical estimates
Consider the operator in divergence form,
acting on functions on , where the symmetric positive definite matrix is bounded, measurable and uniformly elliptic, i.e. for some
Then, a celebrated result of Moser  states that the elliptic Harnack inequality (EHI) holds, that is there exists a constant such that for any positive function which is harmonic on the ball , i.e. for all , we have
A few years later, Moser also proved that the parabolic Harnack inequality (PHI) holds (see ), that is there exists a constant such that for any positive caloric function , i.e. any weak solution of the heat equation on , we have
where and .
Both EHI and PHI have had a big influence on PDE theory and differential geometry, in particular they play a prominent role in elliptic regularity theory. One application of Harnack inequalities – observed by Nash in  – is to show Hölder regularity for solutions of elliptic or parabolic equations. For instance, the PHI implies the Hölder continuity of the fundamental solution of the parabolic equation
Furthermore, the PHI is also one of the key tools in Aronson’s proof  of Gaussian type bounds for .
The remarkable fact about these results is that they only rely on the uniform ellipticity and not on the regularity of the matrix . We refer to the monograph  for more details on this topic.
1.1. The model
In this paper we will be dealing with a discretised version of the operator . We consider an infinite, connected, locally finite graph with vertex set and edge set . We will write if and for if the edges and have a common vertex. A path of length between and in is a sequence with the property that , and . Let be the natural graph distance on , i.e. is the minimal length of a path between and . We denote by the closed ball with center and radius , i.e. .
The graph is endowed with the counting measure, i.e. the measure of is simply the number of elements in . For any non-empty, finite and , we introduce space-averaged -norms on functions by the usual formula
For a given set , we define the relative internal boundary of by
and we simply write instead of . Throughout the paper we will make the following assumption on .
For some the graph satisfies the following conditions:
volume regularity of order , that is there exists such that
relative isoperimetric inequality of order , that is there exists such that for all and ,
The Euclidean lattice, , satisfies the Assumption 1.1.
Assume that the graph, , is endowed with positive weights, i.e. we consider a family . With an abuse of notation we also denote the conductance matrix by , that is for we set if and otherwise. We also refer to as the conductance of the corresponding edge and we call a weighted graph. Let us further define measures and on by
For any fixed we consider a reversible continuous time Markov chain, , on with generator acting on bounded functions as
We denote by the law of the process starting at the vertex . The corresponding expectation will be denoted by . Setting , this random walk waits at an exponential time with mean and chooses its next position with probability . Since the law of the waiting times does not depend on the location, is also called the constant speed random walk (CSRW). Furthermore, is a reversible Markov chain with symmetrising measure given by . In  we mainly consider the variable speed random walk (VSRW) , which waits at an exponential time with mean , with generator
1.2. Harnack inequalities on graphs
In the case of uniformly bounded conductances, i.e. there exist such that
both the EHI and the PHI have been derived in the setting of an infinite, connected, locally finite graph by Delmotte in [20, 21] with constant and depending on and only. It is obvious that for a given ball the constants and can be replaced by
Our main result shows that the uniform upper and lower bound can be replaced by bounds.
Theorem 1.3 (Elliptic Harnack inequality).
For any and let . Suppose that is harmonic on , i.e. on . Then, for any with
there exists such that
The constant is more explicitly given by
for some positive and .
For abbreviation we introduce .
Theorem 1.4 (Parabolic Harnack inequality).
For any , and let . Suppose that is caloric on , i.e. on . Then, for any with
there exists such that
where and . The constant is more explicitly given by
for some positive and .
In the context of elliptic PDEs on discussed above, the classical results on EHI and regularity of harmonic function have been extended to the situation, where the uniform ellipticity assumption on the coefficient (see (1.1) ) is weakened to a certain integrability condition very similar to ours, see [23, 27, 30, 34, 35]. In particular, the same -condition is obtained. This condition also appears in , where an upper bound on the solution of a degenerate parabolic equation is derived by using Moser iteration and Davies’ method (cf. ).
Given a speed measure , one can also consider the process, on that is defined by a time change of the VSRW , i.e. for , where denotes the right continuous inverse of the functional
Its generator is given by
The maybe most natural choice for the speed measure is , for which we get again the CSRW . Since we have the same harmonic functions w.r.t. for any choice of , it is obvious that the EHI in Theorem 1.3 also holds for . On the other hand, for the PHI one can show the following statement along the lines of the proof of Theorem 1.4 (see [16, Appendix A] and [4, Section 3] for a few more details). Let be caloric w.r.t. on , then for any with
there exists with such that
In particular, for the VSRW we have and choosing the condition on and reads again in this case.
Notice that the Harnack constants and in Theorem 1.3 and Theorem 1.4 do depend on the ball under consideration, namely on its center point as well as on its radius . As a consequence one cannot deduce directly Hölder continuity estimates from these results. This requires ergodicity w.r.t. translations as an additional assumption (see Proposition 3.8 and Proposition 4.8 below).
In the applications to the random conductance model discussed below the conductances will be stationary ergodic random variables, so under some moment conditions Assumption 1.6 will be satisfied by the ergodic theorem.
It is well known that a PHI is equivalent to Gaussian lower and upper bounds on the heat kernel in many situations, for instance in the case of uniformly bounded conductances on a locally finite graph, see . Indeed, in a forthcoming paper  we will prove upper Gaussian estimates on the heat kernel under Assumption 1.6 following the approach in , i.e. we combine Moser’s iteration technique with Davies’ method. We remark that in our non-elliptic setting we cannot deduce off-diagonal Gaussian bounds directly from the PHI due to the dependence of the Harnack constant on the underlying ball. More precisely, in order to get effective Gaussian off-diagonal bounds, one needs to apply the PHI on a number of balls with radius having a distance of order . In general, the ergodic theorem does not give the required uniform control on the convergence of space-averages of stationary random variables over such balls (see ). However, we still obtain some near-diagonal estimates (see Proposition 4.7 below).
1.3. The Method
Since the pioneering works [28, 29] Moser’s iteration technique is by far the best-established tool in order to prove Harnack inequalities. Moser’s iteration is based on two main ideas: the Sobolev-type inequality which allows us to control the norm with in terms of the Dirichlet form, and a control on the Dirichlet form of any harmonic (or caloric) function . In the uniformly elliptic case this is rather standard. In our case where the conductances are unbounded from above and not bounded away from zero we need to work with a dimension dependent weighted Sobolev inequality, established in  by using Hölder’s inequality. That is, the coefficient is replaced by
For the Moser iteration we need , of course, which is equivalent to appearing in statements of Theorems 1.3 and 1.4. As a result we obtain a maximum inequality for , that is an estimate for the -norm of on a ball in terms of the -norm of on a slightly bigger ball for some . Since the same holds for , to conclude the Harnack inequality we are left to link the -norm and the -norm of . In the setting of uniformly elliptic conductances (see ) this can be done by using the John-Nirenberg inequality, that is the exponential integrability of BMO functions. In this paper we establish this link by an abstract lemma of Bombieri and Giusti  (see Lemma 2.5 below). In order to apply this lemma, beside the maximum inequalities mentioned above, we need to establish a weighted Poincaré inequality, which is classically obtained by the Whitney covering technique (see e.g. [21, 6, 33]). However, in this paper we can avoid the Whitney covering by using results from a recent work by Dyda and Kassmann .
1.4. Local Limit Theorem for the Random Conductance Model
Our main motivation for deriving the above Harnack inequalities is the quenched local CLT for the random conductance model. Consider the -dimensional Euclidean lattice, , for . The vertex set, , of this graph equals and the edge set, , is given by the set of all non oriented nearest neighbour bonds, i.e. .
Let be a measurable space. Let the graph be endowed with a configuration of conductances. We will henceforth denote by a probability measure on , and we write to denote the expectation with respect to . A space shift by is a map ,
The set together with the operation defines the group of space shifts. We will study the nearest-neighbor random conductance model. For any fixed realization it is a reversible continuous time Markov chain, , on with generator defined as in (1.4). We denote by for and the transition density (or heat kernel associated with ) of the CSRW with respect to the reversible measure , i.e.
As a consequence of (1.9) we have .
Assume that satisfies the following conditions:
for all and .
is ergodic with respect to translations of , i.e. for all and for any such that for all .
There exist satisfying such that
for any .
We are interested in the -a.s. or quenched long range behaviour, in particular in obtaining a quenched functional limit theorem (QFCLT) or invariance principle for the process starting in and a quenched local limit theorem for . We first recall that the following QFCLT has been recently obtained as the main result in .
Theorem 1.10 (Qfclt).
See Theorem 1.3 and Remark 1.5 in . ∎
We refer to  for a quenched invariance principle for diffusions under an analogue moment condition. In this paper our main concern is to establish a local limit theorem for . It roughly describes how the transition probabilities of the random walk can be rescaled in order to get the Gaussian transition density of the Brownian motion, which appears as the limit process in Theorem 1.10. Write
for the Gaussian heat kernel with covariance matrix . For write .
Theorem 1.11 (Quenched Local Limit Theorem).
Similarly to Remark 1.5 it is possible to state a local limit theorem for the process , obtained as a time-change of the VSRW in terms of a speed measure with generator defined in (1.7). Indeed, if for every , if and if Assumptions 1.8 and 1.9 hold, we have a QFCLT also for the process with some non-degenerate covariance matrix , where denotes the covariance matrix in the QFCLT for the VSRW (see Remark 1.5 in ). If in addition there exists such that , and satisfy (1.8) and , writing for the heat kernel associated with and , we have
Analogous results have been obtained in other settings, for instance for random walks on supercritical percolation clusters, see . In , the arguments of  have been used in order to establish a general criterion for a local limit theorem to hold, which is applicable in a number of different situations like graph trees converging to a continuum random tree or local homogenisation for nested fractals.
For the random conductance model a local limit theorem has been proven in the case, where the conductances are i.i.d. random variables and uniformly elliptic (see Theorem 5.7 in ). Later this has been improved for the VSRW under i.i.d. conductances, which are only uniformly bounded away from zero (Theorem 5.14 in ). However, if the conductances are i.i.d. but have fat tails at zero, the QFCLT still holds (see ), but due to a trapping phenomenon the heat kernel decay is sub-diffusive, so the transition density does not have enough regularity for a local limit theorem – see [9, 10].
Hence, it is clear that some moment conditions are needed. It turns out that the moment conditions in Assumption 1.9 are optimal in the sense that for any and satisfying there exists an environment of ergodic random conductances with and , under which the local limit theorem fails (see Theorem 5.4 below). In the case when the conductances are bounded from above, i.e. , the optimality of the condition is already indicated by the examples in [12, 24].
However, in Section 6 below we will show that for some examples of conductances our moment condition can be improved if it is replaced by a moment condition on the heat kernel. While for the example in , where the conductances are given by for a family of random variables bounded from above, no improvement can be achieved by this method (see Proposition 6.2), we obtain that is sufficient if for instance (see Proposition 6.5). Moreover, in the case of i.i.d. conductances uniformly bounded from above this procedure gives the following result (see Proposition 6.3).
Under the assumptions of Theorem 1.13 a PHI and a local limit theorem have recently been proven in [13, Theorem 1.9] on one hand for the VSRW if again and on the other hand for the CSRW if , which improves the result in Theorem 1.13. We refer to Remark 1.10 in  for a discussion on the optimality of these conditions for both CSRW and VSRW.
We shall prove Theorem 1.11 by using the approach in  and . The two main ingredients of the proof are the QFCLT in Theorem 1.10 and a Hölder-continuity estimate on the heat kernel, which at the end enables us to replace the weak convergence given by the QFCLT by the pointwise convergence in Theorem 1.11.
Finally, notice that our result applies to a random conductance model given by
When we can also establish a local limit theorem for the Green kernel. Indeed, as a direct consequence from the PHI we get some on-diagonal bounds on the heat-kernel (see Proposition 4.7 below) from which we obtain the existence of the Green kernel defined by
Theorem 1.14 (Local limit theorem for the Green kernel).
On one hand, by integration Theorem 1.14 can be deduced from Theorem 1.11, the on-diagonal bounds obtained in Proposition 4.7 or in  and suitable long-range bounds on the heat-kernel (see ). Another possibility to prove Theorem 1.14 is to combine the QFCLT in Theorem 1.10 with the Hölder continuity of the Green kernel obtained from the EHI in Theorem 1.3 in a similar way as the local limit theorem follows from the QFCLT and the PHI.
The paper is organised as follows: In Section 2 we provide a collection of Sobolev-type inequalities needed to setup the Moser iteration and to apply the Bombieri-Giusti criterion. Then, Sections 3 and 4 contain the proof of the EHI and PHI in Theorem 1.3 and 1.4, respectively. The local limit theorem for the random conductance model is proven in Section 5. Some examples for conductances allowing an improvement on the moment conditions are discussed in Section 6. Finally, the appendix contains a collection of some elementary estimates needed in the proofs.
Throughout the paper we write to denote a positive constant which may change on each appearance. Constants denoted will be the same through each argument.
2. Sobolev and Poincaré inequalities
2.1. Setup and Preliminaries
For functions , where either or , the -norm will be taken with respect to the counting measure. The corresponding scalar products in and are denoted by and , respectively. Similarly, the -norm and scalar product w.r.t. to a measure will be denoted by and , i.e. . For any non-empty, finite and , we introduce space-averaged norms on functions by
The operators and are defined by and
for and , where for each non-oriented edge we specify out of its two endpoints one as its initial vertex and the other one as its terminal vertex . Nothing of what will follow depend on the particular choice. Note that is the adjoint of , i.e. for all and it holds . We define the products and between a function, , defined on the vertex set and a function, , defined on the edge set in the following way
Then, the discrete analog of the product rule can be written as
In contrast to the continuum setting, a discrete version of the chain rule cannot be established. However, by means of the estimate (A.1), for can be bounded from above by
|On the other hand, the estimate (A.4) implies the following lower bound|
The Dirichlet form or energy associated to is defined by
For a given function , we denote by the Dirichlet form where is replaced by for .
2.2. Local Poincaré and Sobolev inequality
The main objective in this subsection is to establish a version of a local Poincaré inequality.
Suppose that the graph satisfies the condition (ii) in Assumption 1.1. Then, by means of a discrete version of the co-area formula, the classical local -Poincaré inequality on can be easily established, see e.g. [32, Lemma 3.3], which also implies an -Poincaré inequality for any . Moreover, the condition (i) in Assumption 1.1 ensures that balls in have a regular volume growth. Note that, due to [17, Théorème 4.1], the volume regularity of balls and the local -Poincaré inequality on implies that for and any
This inequality is the starting point to prove a local -Poincaré inequality on the weighted graph .
Proposition 2.1 (Local Poincaré inequality).
Suppose and for any and , let . Then, there exists such that for any
and for any such that ,