Random conductance models with stable-like jumps: Heat kernel estimates and Harnack inequalities

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Abstract

We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of the well known two-sided Gaussian heat kernel estimates by M.T. Barlow for nearest neighbor (short range) random walks on the supercritical percolation cluster. Unlike the cases of nearest neighbor conductance models, we cannot use parabolic Harnack inequalities since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theorem for the models.

Introduction

In analysis, it is one of the central problems to study stability of asymptotic properties for solutions of parabolic equations under perturbations. One typical example is to study stability of Gaussian estimates for fundamental solutions of heat equations (i.e. heat kernels). In 1967, Aronson [8] proved that the heat kernel of any uniform elliptic divergence operator L=i,j=1dxi(aij(x)xj) on Rd enjoys the following estimates:c1td/2exp(c2d(x,y)2/t)p(t,x,y)c3td/2exp(c4d(x,y)2/t) for all t>0 and x,yRd, where d(,) is the Euclidean distance and c1,,c4 are positive constants. The divergence operator can be viewed as a perturbation of the Laplace operator so this result can be viewed as a stability result of the Gaussian heat kernel estimates. More generally, consider the Laplace-Beltrami operator on a Riemannian manifold with the Riemannian metric d(,) and the Riemannian measure μ. Later in the last century, Grigor'yan [40] and Saloff-Coste [53] independently proved that in this setting (1.1), where td/2 is replaced by μ(B(x,t1/2))1, is equivalent to the volume doubling condition (VD) and Poincaré inequalities (PI), both of which are stable under perturbations. It is also equivalent to parabolic Harnack inequalities and the important consequence is that it implies Hölder regularity of the solutions of heat equations. The results were then extended to the setting of general metric measure spaces and graphs. Such detailed heat kernel estimates are heavily related to the control of harmonic functions and the theory behind is sometimes called the De Giorgi-Nash-Moser theory. In this century, there has been extensive study of expanding the stability theory to nearest neighbor random conductance models (symmetric random walks on random media) as we discuss below. Note that in general one cannot expect VD nor PI uniformly on random media, so it is highly non-trivial to utilize the stability theory. For non-local operators (jump processes), such stability theory and the De Giorgi-Nash-Moser theory on metric measure spaces and graphs have been developed quite recently; see for instance [13], [25], [26], [28], [29], [30], [42], [43], [50], [51] and references therein. The aim of this paper is to expand the stability theory to random media with long-range jumps.

Let us explain some more details about nearest neighbor random conductance models. A typical example is a bond percolation on Zd, d2; namely, on each nearest neighbor bond x,yZd with |xy|=1, we put a random conductance wx,y in such a way that {wx,y(ω)=wy,x(ω):x,yZd,|xy|=1} are i.i.d. Bernoulli so that P(wx,y(ω)=1)=p and P(wx,y(ω)=0)=1p for some p[0,1]. It is known that there exists a constant pc(Zd)(0,1) such that almost surely there exists a unique infinite cluster C(ω) (i.e. a connected component of bonds with conductance 1) when p>pc(Zd) and no infinite cluster when p<pc(Zd). Suppose p>pc(Zd) and consider a continuous time simple random walk (Xtω)t0 on the infinite cluster. Let pω(t,x,y) be the heat kernel (also called the transition density function) of Xω, i.e.,pω(t,x,y):=Px(Xtω=y)μy, where μy is a number of bonds whose one end is y. In the cerebrated paper [9], Barlow proved the following detailed heat kernel estimates those are almost sure w.r.t. the randomness of the environment; namely the following quenched estimates:

There exist random variables {Rx(ω)}xZd with Rx(ω)[1,) for all xC(ω) P-a.s. ω such that for all x,yC(ω) with t|xy|Rx(ω), pω(t,x,y) satisfies (1.1).

Note that because of the degenerate structure of the (random) environment, we cannot expect (1.1) to hold for all t1. Barlow's results assert that such a Gaussian estimate holds as a long time estimate despite of the degenerate structure.

When the conductances are bounded from above and below (the uniformly elliptic case) and the global volume doubling condition holds, it is well known that the associated heat kernel for the nearest neighbor conductance models obeys two-sided Gaussian estimates for all t1 (see e.g. [35]). However, this is no longer true when the conductances are non-uniformly elliptic, such as the supercritical bond percolation discussed above.

Barlow's results have many applications. For example, they were applied crucially in the proofs of the quenched invariance principle and the quenched local central limit theorem for the model. The results have been extended to nearest neighbor ergodic conductance models in [4], [54]. Large time heat kernel estimates (and parabolic Harnack inequalities) have played [1], [2], [3], [4], [5], [6], [10], [11], [17], [21], [22], [47], [48], [49], [52], [54], [55] and [20], [46] for surveys on these topics.

In this paper, we consider quenched heat kernel estimates for random conductance models that allow big jumps. In particular, we establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic and possibly degenerate stable-like jumps on graphs. Despite of the fundamental importance of the problem, so far there are only a few results for conductance models with long range jumps. As far as we are aware, this is the first work on detailed heat kernel estimates for possibly degenerate random walks with long range jumps. We now explain our framework and a result.

Suppose that G=(V,EV) is a locally finite connected infinite graph, where V and EV denote the collection of vertices and edges respectively. For xyV, we write ρ(x,y) for the graph distance, i.e., ρ(x,y) is the smallest positive length of a path (that is, a sequence x0=x, x1, ⋯, xl=y such that (xi,xi+1)EV for all 0il1) joining x and y. We set ρ(x,x)=0 for all xV. Let B(x,r):={yV:ρ(y,x)r} denote the ball with center xV and radius r1. Let μ be a measure on V such that the following assumption holds.

Assumption (d-Vol). There are constants R01, κ>0, cμ1, θ(0,1) and d>0 (all four are independent of R0) such that the following hold:0<μxcμ,for allxV,infxB(0,R)μxRκ,for allRR0,cμ1rdμ(B(x,r))cμrd,for allRR0,xB(0,6R)andRθ/2r2R. Under (1.4), (G,μ) only satisfies the d-set condition for large scale. For p1, let Lp(V;μ)={fRV:xV|f(x)|pμx<}, and fp be the Lp(V;μ) norm of f with respect to μ. Let L(V;μ) be the space of bounded measurable functions on V, and f be the L(V;μ) norm of f.

Suppose that {wx,y:x,yV} is a sequence such that wx,y0 and wx,y=wy,x for all xy, andyV:yxwx,yμyρ(x,y)d+α<,xV, where α(0,2). For simplicity, we set wx,x=0 for all xV. We can define a regular Dirichlet form (D,F) as follows (see the first statement in [27, Theorem 3.2])D(f,f)=12x,yV(f(x)f(y))2wx,yρ(x,y)d+αμxμy,F={fL2(V;μ):D(f,f)<}. It is easy to verify that the infinitesimal generator L associated with (D,F) is given byLf(x)=zV(f(z)f(x))wx,zμzρ(x,z)d+α. Let X:=(Xt)t0 be the symmetric Hunt process associated with (D,F). When μ is a counting measure on G (resp. μx is chosen so that zVwx,zμzρ(x,z)d+α=1 for all xV), the associated process X is called the variable speed random walk (resp. the constant speed random walk) in the literature.

In the accompanied paper [24], we discussed the quenched invariance principle for conductance models with stable-like jumps. As a continuation of [24], we consider heat kernel estimates for the conductance models. Here the main difficulty is neither conductances are uniformly elliptic (possibly degenerate) nor the global d-set condition is supposed to be satisfied. To illustrate our contribution, we state the following result for random conductance models on L:=Z+d1×Zd2 with d1,d2Z+:={0,1,2,} such that d1+d21 (i.e., V=L and the coefficients wx,y given in (1.5) are random variables).

Theorem 1.1

(Heat kernel estimates for Variable speed random walks) Let α(0,2), V=L with d>42α and E={(x,y):x,yLandxy} be the collection of all unorder pairs on L. Define {wx,y(ω):(x,y)E} as a sequence of independent random variables on some probability space (Ω,FΩ,P) such that for any xy, wx,y=wy,x0 andsupx,yL:xyP(wx,y=0)<1/2,supx,yL:xy(E[wx,y2p]+E[wx,y2q1{wx,y>0}])< for somep>max{(d+1+θ0)/(dθ0),(d+1)/(2θ0(2α))},q>(d+1+θ0)/(dθ0), where θ0:=α/(2d+α). Let (Xtω)t0 be the symmetric Hunt process corresponding to the Dirichlet form (D,F) above with random variables {wx,y(ω):(x,y)E} and μ being the counting measure on L. Denote by pω(t,x,y) the heat kernel of the process (Xtω)t0. Then, P-a.s. ωΩ, for any xL, there is a constant Rx(ω)1 such that for all t>0 and yL with t(|xy|Rx(ω))θα,C1(td/αt|xy|d+α)pω(t,x,y)C2(td/αt|xy|d+α), where θ(0,1) and C1,C2>0 are constants independent of Rx(ω), t, x and y.

Note that (1.9) is the typical heat kernel estimates for stable-like jumps, and it corresponds to the Gaussian estimates (1.1) for the nearest neighbor cases. We can also obtain large scale Hölder regularity of associated parabolic functions as in Theorem 2.10 below.

Remark 1.2

We give some remarks on the statement of Theorem 1.1.

  • (i)

    The integrability condition in (1.7) with p,q given by (1.8) is not optimal, and we do not even know what could be the optimal integrability condition. In particular, by tracking the proof of Theorem 1.1 (see [24, Proposition 5.8] for more details), the negative moment condition supx,yLE[wx,y2q1{wx,y>0}]< in (1.7) is only required to guarantee the local Poincaré inequality which will be used to establish on diagonal upper bound for the Dirichlet heat kernel in Proposition 2.1.

  • (ii)

    The probability 1/2 in (1.7) is also not optimal. In fact, the critical probability pc allowing the conductances to be degenerate heavily depends on large scale properties of the long-range percolation cluster associated with the random conductance model in our paper, which are different from these of the nearest neighbor percolation cluster (see e.g. [1]). We do not know the exact value of pc, and even whether pc=0 or pc>0.

  • (iii)

    It is clear that wx,y can be unbounded and/or degenerate under (1.7). As mentioned above, unlike the uniformly elliptic case (which corresponds to p=q= in (1.7)), we cannot expect two-sided heat kernel estimates (1.9) to be held for all t1 when p,q<.

  • (iv)

    According to the proof of Theorem 1.1 (also see that of [24, Proposition 5.8] for more details), the random variables {Rx(ω)}xL enjoy the polynomial tails, i.e., there are constants c0,η>0 such that for all xL and n1, P(Rx(ω)n)c0nη. For the i.i.d. nearest neighbor percolation cluster on Zd, the associated random variables (Rx(ω))xZd enjoy the exponential tails (see [9, (0.5) and Lemma 2.24]), which is due to the structure of simple random walks on the percolation cluster. However, for the random conductance model in our paper, the polynomial tails of {Rx(ω)}xL are just deduced from the integrability condition (1.7); see Remark 4.2.

Let us explain some related work. As we mentioned above, there are only a few results for conductance models with stable-like (long range) jumps. When the conductances are uniformly elliptic and the global volume doubling condition holds, heat kernel estimates like (1.9) have been discussed, for instance in [13], [16], [50], [51]. In these aforementioned papers, a lot of arguments are heavily based on uniformly elliptic conductances, and two-sided pointwise bounds of conductances are also necessary and frequently used. The corresponding results for non-local Dirichlet forms on general metric measure spaces have been also obtained, as mentioned in the first paragraph of this section. Crawford and Sly [31] proved on-diagonal heat kernel upper bounds for random walks on the infinite cluster of supercritical long range percolation, see [32] for the scaling limit of random walks on long range percolation clusters. Due to the singularity of long range percolation cluster, off-diagonal heat kernel estimates are still unknown. To the best of our knowledge, heat kernel estimates for conductance models with non-uniformly elliptic stable-like jumps under non-uniformly volume doubling condition are not available till now. In this paper we will address this problem completely.

We summarize some difficulties of our problem as follows.

  • (i)

    As for nearest neighbor non-uniformly elliptic conductance models, a usual (and powerful) idea is to establish first elliptic and parabolic Harnack inequalities, and then deduce heat kernel bounds. For example, see [2], [3], [4] for the recent study on ergodic environments of nearest neighbor random conductance models under some integrability conditions. However, in the present setting elliptic Harnack inequalities do not hold in general even for large balls, hence parabolic Harnack inequalities do not hold either, when conductances are not uniformly elliptic. This is totally different from uniformly elliptic stable-like jumps, see e.g. [13], [16], [50], [51], or uniformly elliptic stable-like jumps with variable orders on Rd, see e.g. [15]. We refer readers to Proposition 4.8 and Example 4.9 below for details.

  • (ii)

    For nearest neighbor conductance models, off-diagonal upper bounds of the heat kernel can be deduced from on diagonal upper bounds using the maximum principle initiated by Grigor'yan on manifolds [41] and developed in [39] on graphs, see e.g. the proofs of [22, Proposition 1.2] or [10, Proposition 3.3]. Because of the effect of long range jumps, such approach does not seem to be applicable in our model.

  • (iii)

    As mentioned before, in order to establish heat kernel estimates for uniformly elliptic stable-like jumps, pointwise upper and lower bounds of conductances are crucially used in [13], [16], [50], [51]. In particular, uniform lower bounds of conductances yield Nash/Sobolev inequalities for the associated Dirichlet form, which in turn imply on-diagonal heat kernel upper bounds immediately. Furthermore, based on Nash/Sobolev inequalities, the Davies method was adopted in [13], [16], [50], [51] to derive off-diagonal upper bound estimates for heat kernel. However, in the setting of our paper Nash/Sobolev inequalities do not hold, and so the approaches above are not applicable.

To establish two-sided heat kernel estimates for long range and non-uniformly elliptic conductance models with stable-like jumps, we will apply the localization argument for Dirichlet heat kernel estimates, and then pass through these to global heat kernel estimates via the Dynkin-Hunt formula. For this, we make full use of estimates for the exit time of the process obtained in [24]. Though part of ideas in the proofs are motivated by the study of global heat kernel estimates for the uniformly elliptic case (for instance, see [13], [14]), a lot of non-trivial modifications and new ideas are required. Actually, in this paper we will establish heat kernel estimates under a quite general framework beyond Theorem 1.1, see Theorem 2.8 and Theorem 2.12 below. In particular, only the d-set condition with large scale (d-Vol) and locally summable conditions on conductances (see Assumptions (HK1)(HK3) below) are assumed. These conditions can be regarded as a generalization of “good ball” conditions for nearest neighbor conductance models in [10] into long range conductance models. As an application of our estimates, we can also justify the local limit theorem for our model, see Theorem 4.5 below.

The organization of this paper is as follows. The next section is devoted to heat kernel estimates for large time. This part is split into three subsections. We first consider on-diagonal upper bounds, later study off-diagonal upper bounds, and then lower bound estimates. In Section 3, we present some estimates for Green functions, and also give a consequence of elliptic Harnack inequalities. In the last section, we apply our previous results to random conductances with stable-like jumps.

Section snippets

Heat kernel estimates: large time

To obtain heat kernel estimates for large time, we need the following three assumptions on {wx,y:x,yV}. Throughout this section, we always assume that α(0,2). We fix 0V, and define Bzw(x,r):={yB(x,r):wy,z>0} for all x,zV and r>0. Set Bw(x,r):=Bxw(x,r) for simplicity.

Assumption (HK1). Suppose that there exist R01, θ(0,1), c0>1/2 and C1>0 (all three are independent of R0) such that

  • (i)

    For every R>R0 and Rθ/2r2R,supxB(0,6R)yV:ρ(x,y)rwx,yμyρ(x,y)d+α2C1r2α,μ(Bzw(x,r))c0μ(B(x,r)),x,zB(0

Green function estimates and elliptic Harnack inequalities

In this section, we give some estimates of Green functions and then give a consequence of elliptic Harnack inequalities. The results in this section is used in Subsection 4.2.2. Throughout this section, let α(0,2) and assume that (G,μ) satisfies the global d-set condition; that is,cμ1μxcμ,cμ1rdμ(B(x,r))cμrd,r1,xV. We also suppose in this section that wx,y>0 for all x,yV with xy.

Application: random conductance model

We will apply results in the previous two sections to study heat kernel estimates and elliptic Harnack inequalities for random conductance models on L:=Z+d1×Zd2 (with d1,d2Z+ such that d1+d21) with stable-like jumps. Let E={(x,y):x,yL and xy} denote the collection of all unorder pairs on L.

Let V=L, and {wx,y(ω):(x,y)E} be a sequence of independent (but not necessarily identically distributed) random variables on some probability space (Ω,FΩ,P) such that wx,y(ω)=wy,x(ω)0 for any xy. Let μ

Acknowledgements

The authors are very grateful to the referee for helpful suggestions and comments. The research of Xin Chen is supported by the National Natural Science Foundation of China (Nos. 11501361 and 11871338). The research of Takashi Kumagai is supported by JSPS KAKENHI Grant Number JP17H01093 and by the Alexander von Humboldt Foundation. The research of Jian Wang is supported by the National Natural Science Foundation of China (No. 11831014), the Program for Probability and Statistics: Theory and

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