Heat kernel estimates for random walks with degenerate weights

We establish Gaussian-type upper bounds on the heat kernel for a continuous-time random walk on a graph with unbounded weights under an ergodicity assumption. For the proof we use Davies' perturbation method, where we show a maximal inequality for the perturbed heat kernel via Moser iteration.


INTRODUCTION
A well known theorem by Delmotte [12] states that Gaussian bounds on the heat kernel hold for constant-speed random walks on locally finite graphs, provided the jump rates are uniformly elliptic, that is they are uniformly bounded and bounded away from zero.In a recent work [13], Folz showed Gaussian upper bounds for the heat kernel of continuous-time, elliptic random walks with arbitrary speed measure under the assumption that on-diagonal upper bounds for the heat kernel at two points are given and the speed measure is uniformly bounded from below.In this paper we relax the uniform ellipticity condition and show a Gaussian-type upper bound for constant-speed random walks with unbounded jump rates satisfying a certain ergodicity condition.
1.1.Setting and Result.Let G = (V, E) be an infinite, connected, locally finite graph with vertex set V and (nonoriented) edge set E. We will write x ∼ y if {x, y} ∈ E. The graph G is endowed with the counting measure, i.e. the measure of A ⊂ V is simply the number |A| of elements in A. Further, we denote by B(x, r) the closed ball with center x and radius r with respect to the natural graph distance d, that is B(x, r) := {y ∈ V | d(x, y) ≤ r}.
For a given set B ⊂ V , we define the relative internal boundary of A ⊂ B by and we simply write ∂A instead of ∂ V A. Throughout the paper we will make the following assumption on G.
Assumption 1.1.The graph G satisfies the following conditions: (i) volume regularity of order d for large balls, that is there exists d ≥ 2 and C reg ∈ (0, ∞) such that for all x ∈ V there exists N 1 (x) < ∞ with (ii) local Sobolev inequality (S 1 d ′ ) for large balls, that is there exists d ′ ≥ d and C S 1 ∈ (0, ∞) such that for all x ∈ V the following holds.There exists N 2 (x) < ∞ such that for all n ≥ N 2 (x), for all u : V → R with supp u ⊂ B(x, n).
Remark 1.3.It was recently shown in [15], that the infinite cluster of a supercritical Bernoulli percolation satisfies the Assumption 1.1 for some d ′ > d.
Remark 1.4.The following relative isoperimetric inequality for large balls holds.For all n ≥ N 2 (x), Consider a family of positive weights ω = {ω(e) ∈ (0, ∞) : e ∈ E}.With an abuse of notation we also denote the conductance matrix by ω, that is for x, y ∈ V we set ω(x, y) = ω(y, x) = ω({x, y}) if {x, y} ∈ E and ω(x, y) = 0 otherwise.Let us further define measures µ ω and ν ω on V by For any fixed ω we consider a continuous time Markov chain, Y = {Y t : t ≥ 0}, on V with generator L ω ≡ L ω Y acting on bounded functions f : V → R as Let us stress the fact that the Markov chain, Y , is reversible with respect to the measure µ ω .Setting p ω (x, y) := ω(x, y)/µ ω (x), this stochastic process waits at x an exponential time with mean 1 and chooses its next position y with probability p ω (x, y).Since the law of the waiting times does not depend on the location, Y is also called the constant speed random walk (CSRW).We denote by P ω x the law of the process starting at the vertex x ∈ V and by q ω (t, x, y) for x, y ∈ V and t ≥ 0 the transition density (or the heat kernel associated with L ω ) with respect to the measure µ ω , i.e.
For any non-empty, finite A ⊂ V and p ∈ [1, ∞), we introduce space-averaged ℓ p -norms on functions f : A → R by the usual formula For our main results we need to make the following assumption on the ergodicity of the conductances.
Assumption 1.5.There exist p, q ∈ (1, ∞] with such that, setting for any x ∈ V , μp (x) := lim sup n→∞ µ ω p,B(x,n) and νq (x) := lim sup we have that In particular, for every x ∈ V there exists N (x) ≡ N (x, ω) such that Our aim is to continue the program initiated in [3,2].In [3] we showed a quenched invariance principle for the random walk X on the integer lattice Z d under ergodic, degenerate random conductances satisfying a moment condition and thus Assumption 1.5 (cf.Remark 1.7 below).In [2] we established elliptic and parabolic Harnack inequalities for the operator L ω , from which a local limit theorem for the heat kernel could be deduced.In this paper we prove a Gaussian-type upper bound on the heat kernel.Theorem 1.6.Suppose that Assumption 1.5 holds.Then, there exist constants c i = c i (d, p, q, μp , νq ) such that for any given t and x with √ t ≥ N (x) ∨ 2(N 1 (x) ∨ N 2 (x)) and all y ∈ V the following hold.
and write E for the expectation with respect to P. The space shift by z ∈ Z d is the map τ z : Ω → Ω defined by (τ z ω)(x, y) := ω(x + z, y + z) for all {x, y} ∈ E d .Now assume that P satisfies the following conditions: (i) P is ergodic with respect to translations of Z d , i.e.P • τ −1 x = P for all x ∈ Z d and P for any e ∈ E d .
Then, the spatial ergodic theorem gives that for P-a.e. ω, In particular, Assumption 1.5 is fulfilled in this case and therefore for P-a.e. ω the upper estimates on q ω t (x, y) in Theorem 1.6 hold.It has been been shown in [2,Theorem 5.4] that the moment condition in (1.6) is optimal for a local limit theorem to hold.In particular, this moment condition is also necessary for both upper and lower Gaussian near-diagonal bounds to be satisfied.
As already mentioned in the beginning, for random walks on weighted graphs Gaussian type estimates on the heat kernel have been proven by Delmotte [12] in the case, where the conductances are uniformly elliptic, i.e. c −1 ≤ ω(e) ≤ c, e ∈ E, for some c ≥ 1.Later this has been improved for the VSRW under conductances, which are only uniformly bounded away from zero, see [6, Theorems 2.19 and 3.3].However, the distance function that appears in these estimates for the VSRW is the so called chemical distance which is adapted to the transition rates of the random walk [14,13].In general, the chemical metric might be quite different compared to the graph metric.In [6] it is shown that in the case of i.i.d.conductance the chemical distance d chem (x, y) can be compared with the graph distance d(x, y) provided that d(x, y) ≥ N ω (x) where {N ω (x) : x ∈ V } is a family of random variables with streched-exponential tails.In order to provide some control on the size of {N ω (x) : x ∈ V } in our context we would need some information on the speed of convergence in the ergodic theorem, which is not available in this general framework unless we make additional mixing assumptions.
Moreover, if the conductances are i.i.d.but have fat tails at zero, due to a trapping phenomenon the heat kernel decay may be sub-diffusive and Gaussian estimates do not hold in general -see [7,8].
1.2.The method.It is well known that Gaussian lower and upper bounds on the heat kernel are equivalent to a parabolic Harnack inequality in many situations, for instance in the case of uniformly elliptic conductances, see [12].Recently, this equivalence has also been established on locally irregular graphs in [5].In our context such a parabolic Harnack inequality has been recently proven in [2].Unfortunately, due to special structure of the Harnack constant, in particular its dependence on µ ω p,B(x,n) and ν ω q,B(x,n) , we cannot directly deduce off-diagonal Gaussian bounds from it.More precisely, in order to get effective Gaussian offdiagonal bounds using the established chaining argument (see e.g.[4]), one needs to apply the Harnack inequality on a number of balls with radius n having a distance of order n 2 .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 [1]).
Nevertheless, in order in to prove the Gaussian upper bound in Theorem 1.6 another technique is quite performing, which is known as Davies' method in the literature (see e.g.[10,11,9]).In contrast to the chaining argument mentioned above the main advantage of Davies' technique is that we only need to apply the ergodic theorem (or Assumption 1.5, respectively) on balls with one fixed center point x 0 .
We now briefly sketch the idea of Davies' method.Instead of studying the original semigroup {P t : t ≥ 0} which is generating the the random walk Y , that is Davies suggests to consider the semigroup {P ψ t : t ≥ 0} given by with generator for a suitable class of test functions ψ.Clearly, this semigroup has a kernel which is given by e ψ(x) q ω (t, x, y)e −ψ(y) and satisfies the heat equation ∂ t v − L ψ v = 0. Note that P ψ t is symmetric with respect to the measure e −2ψ µ ω .In the classical setting of symmetric Markov semigroups whose generator is a second order elliptic operator, the Nash inequality and equivalently Gaussian ondiagonal estimates do hold.Then, Davies used the classical Leibniz rule to derive a bound on the kernel of {P ψ t : t ≥ 0}, which can be rewritten as where Γ denotes the carré du champ operator.Finally, by varying over ψ Gaussian upper bounds can be obtained.For further details we refer to [9].The method has also been used to obtain the Gaussian upper bounds in [12].
In our setting, where the conductances are unbounded from below, we do not have a Nash inequality available.Therefore, we follow an approach used by Zhikov in [16], where some upper bounds for the solution kernel of certain degenerate Cauchy problems on R d are obtained.More precisely, we use Moser's iteration technique to show a maximal inequality for solution of ∂ t v − L ψ v = 0 and combine it with Davies' method.Similarly to [9], where Davies' method has been carried out for processes generated by non-local Dirichlet forms, one difficulty is the absence of a Leibniz rule in the discrete setting of a graph.In [2] we already established a Moser iteration scheme and a maximal inequality for solutions of the original heat equation ∂ t u − L ω u = 0, so we adapt here the arguments in [2] to deal with the perturbed semigroup {P ψ t : t ≥ 0}.In the rest of the paper we prove Theorem 1.6, while the appendix contains a collection of some elementary estimates needed in the proofs.Throughout the paper we write c to denote a positive constant which may change on each appearance.Constants denoted C i will be the same through each argument.where for each non-oriented edge e ∈ E we specify out of its two endpoints one as its initial vertex e + and the other one as its terminal vertex e − .Nothing of what will follow depend on the particular choice.Since for all f ∈ ℓ 2 (V ) and

GAUSSIAN UPPER
, ∇ * can be seen as the adjoint of ∇.We also define the products f • F and F • f between a function, f , defined on the vertex set and a function, F , defined on the edge set in the following way Then, the discrete analog of the product rule can be written as where av(f )(e) := 1 2 (f (e + ) + f (e − )).On the weighted Hilbert space ℓ 2 (V, µ ω ) the Dirichlet form or energy associated to L ω Y is given by where dΓ ω (f, g) := ω∇f ∇g.For a given function η : B ⊂ V → R, we denote by E ω η 2 (u) the Dirichlet form where ω(e) is replaced by av(η 2 ) ω(e) for e ∈ E.

2.2.
Maximal inequality for the perturbed Cauchy problem.Our next aim is to derive a maximal inequality for the function v.For that purpose we will adapt the arguments given in [2, Section 4] and setup a Moser iteration scheme.For any non-empty, finite B ⊂ V and p ∈ [1, ∞), we introduce a space-averaged norm on functions V : B → R by For a given φ > 0 with φ, φ −1 ∈ ℓ ∞ (V ), let v t ≥ 0 be a solution of where A(φ) := max e∈E av(φ)(e) av(φ −1 )(e). Proof.
As an easy consequence we obtain now the analogue to [2, Lemma 4.1].

Corollary 2.3. Under the assumptions of Lemma 2.2 consider a function
(2.12) Proof.By multiplying both sides of (2.7) with ζ(t) and integrating the resulting inequality over [s 1 , s] for any s ∈ I, we get Thus, by neglecting the first term on the left-hand side of (2.13), (2.11) is immediate, whereas (2.12) follows once we neglect the second term on the left-hand side of (2.13).

Corollary 2.6. Under the assumptions of Corollary 2.5 there exists
Proof.Choosing σ ′ = 1/2 and σ = 1 we combine Corollary 2.5 with the a-priori estimate in (2.4) to obtain and the claim follows.
Notice that P −ψ t is the adjoint of P ψ t in ℓ 2 (V, µ ω ).Since h(φ) remains unchanged if we replace ψ by −ψ, (2.19) also holds true for ψ replaced by −ψ.Therefore, we get by duality that for all g ∈ ℓ 1 (V, µ ω ),  (ii) The statement is trivial for a = b so we may assume that 0 ≤ a < b and set z := a/b.Then, (A.2) is equivalent to But this follows from the fact that the left hand side is increasing in z and converges to 1 − 1 α as z → 1. (iv) With x = e y , we find that BOUNDS It is convenient to introduce a potential theoretic setup.First of all, for f : V → R and F : E → R we define the operators ∇f : E → R and ∇ * F : V → R by ∇f (e) := f (e + ) − f (e − ), and ∇ * F (x) := e:e + = x F (e) − e:e − = x F (e), ) where h(φ) := max e∈E φ(e + ) φ(e − ) + φ(e − ) φ(e + ) − 2 .
) .Remark 1.7.Consider the d-dimensional Euclidean lattice Z d with d ≥ 2, and let E d be the set of all non oriented nearest neighbour bonds, i.e.E d := {{x, y} : x, y ∈ Z d , |x − y| = 1}.Then, (Z d , E d ) satisfies the Assumption 1.1 as pointed out in Remark 1.2.Further, let P be a probability measure on the measurable space