Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances

We establish heat kernel upper bounds for a continuous-time random walk under unbounded conductances satisfying an integrability assumption, where we correct and extend recent results by the authors to a general class of speed measures. The resulting heat kernel estimates are governed by the intrinsic metric induced by the speed measure. We also provide a comparison result of this metric with the usual graph distance, which is optimal in the context of the random conductance model with ergodic conductances.


INTRODUCTION
Let G = (V, E) be an infinite, connected, locally finite graph with vertex set V and (non-oriented) edge set E. We will write x ∼ y if {x, y} ∈ E. Consider a family of positive weights ω = {ω(e) ∈ (0, ∞) : e ∈ E} ∈ Ω, where Ω = R E + is the set of all possible configurations.We also refer to ω(e) as the conductance of the edge e.With an abuse of notation, 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 µ ω (x) := y∼x ω(x, y) and ν ω (x) := y∼x 1 ω(x, y) .
Given a speed measure θ : V → (0, ∞) we consider a continuous time continuous time Markov chain, X = {X t : t ≥ 0}, on V with generator L ω θ acting on bounded functions f : V → R as Then the Markov chain, X, is reversible with respect to the speed measure θ, and regardless of the particular choice of θ the jump probabilities of X are given by p ω (x, y) := ω(x, y)/µ ω (x), x, y ∈ V , and the various random walks corresponding to different speed measures will be time-changes of each other.The maybe most natural choice for the speed measure is θ ≡ θ ω = µ ω , for which we obtain the constant speed random walk (CSRW) that spends i.i.d.Exp(1)-distributed waiting times at all visited vertices.Another frequently arising choice for θ is the counting measure, i.e. θ(x) = 1 for all x ∈ V , under which the random walk waits at x an exponential time with mean 1/µ ω (x).Since the law of the waiting times does depend on the location, X is also called the variable speed random walk (VSRW).
For any choice of θ we denote by P ω x the law of the process X starting at the vertex x ∈ V .For x, y ∈ V and t ≥ 0 let p ω θ (t, x, y) be the transition densities of X with respect to the reversible measure (or the heat kernel associated with L ω θ ), i.e.
As our first main result we establish upper bounds on the heat kernel under a certain integrability condition on the conductances, see Theorem 2.5 below.The resulting bounds are of Gaussian type apart from an additional factor which may vanish for specific choices of the speed measure or the conductances (see Remark 2.6 below).
It is well known that Gaussian bounds hold, for instance, for the CSRW on locally finite graphs in the uniformly elliptic case, that is c −1 ≤ ω(e) ≤ c for all e ∈ E for some c ≥ 1, see [12].More recently, Folz showed in [15] upper Gaussian estimates for elliptic random walk for general speed measures that need to be bounded away from zero, provided on-diagonal upper bounds at two vertices are given.In [3] we weakened the strict ellipticity condition and showed heat kernel upper bounds for the CSRW and VSRW under a similar integrability condition as in Theorem 2.5, while in the present paper we extend this result to general speed measures.Notice that some integrability assumption on the conductances is necessary for Gaussian bounds to hold.In fact, it is well known that due to a trapping phenomenon under random i.i.d.conductances with sufficiently heavy tails at the zero the subdiffusive heat kernel decay may occur, see [6,7] and cf.[8].For the proof of Theorem 2.5 we use the same strategy as in [3] which is based on a combination of Davies' perturbation method (cf.e.g.[10,11,9]) with a Moser iteration following an idea in [20].We refer to [3, Section 1.2] for a more detailed outline of the method.Naturally, the heat kernel upper bounds in Theorem 2.5 are governed by the distance function d ω θ on V × V defined by where Γ xy is the set of all nearest-neighbor paths γ = (z 0 , . . ., z lγ ) connecting x and y (cf.[11,5,15,17,3]).Note that d ω θ is a metric which is adapted to the transition rates and the speed measure of the random walk.Further, for the CSRW, i.e. θ ≡ θ ω = µ ω , the metric d ω θ coincides with the usual graph distance d, and for a VSRW d ω θ becomes the so-called chemical distance.In general, d ω θ can be identified with the intrinsic metric generated by the Dirichlet form associated with L ω θ and X, see Proposition 2.3 below.Further, notice that d ω θ (x, y) ≤ d(x, y) for all x, y ∈ V .In fact, the distance d ω θ can become much smaller than the graph distance, see [3, Lemma 1.12] for an example in the context of a VSRW under random conductances.As our second main result stated in Theorem 3.3 below, for any x, y ∈ V sufficiently far apart, we provide under a suitable integrability condition on ω a lower bound on d ω θ (x, y) in terms of a certain power of d(x, y).This lower bound turns out to be optimal within our general framework up to an arbitrarily small correction in the exponent.
The rest of the paper is organised as follows.In Section 2 we show the heat kernel upper bounds.The lower bound on the chemical distance in terms of the graph distance is proven in Section 3 and in Section 4 we discuss its optimality by providing an example in the context of the random conductance model on Z d .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.

HEAT KERNEL UPPER BOUNDS
2.1.Preliminaries.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}.Throughout the paper we will make the following assumption on G. (i) Uniformly bounded vertex degree, that is there exists (ii) Volume regularity of order d for large balls, that is there exist d ≥ 2 and C reg ∈ (0, ∞) such that for all x ∈ V there exists N 1 (x) < ∞ with (iii) 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 Remark 2.2.The Euclidean lattice, (Z d , E d ), satisfies the Assumption 2.1 with For f : V → R we define the operator ∇ by where for each non-oriented edge e ∈ E we specify one of its two endpoints as its initial vertex e + and the other one as its terminal vertex e − .Further, the corresponding adjoint operator ∇ * F : V → R acting on functions F : Notice that in the discrete setting the product rule reads where av(f )(e) := 1 2 (f (e + ) + f (e − )).On the weighted Hilbert space ℓ 2 (V, θ) the Dirichlet form associated with L ω θ is given by where dΓ ω (f, g) := ω∇f ∇g and As a first step, we identify the metric d θ as the intrinsic metric of the Dirichlet form E ω on ℓ 2 (V, θ).Proposition 2.3.For every x, y ∈ V , Proof.We follow the argument in [17,Proposition 10.4].For any x, y ∈ V set Then, for any function ψ : V → R with the properties that ∇ψ ∞ ≤ 1 and dΓ ω (ψ, ψ)(e) ≤ θ(e + ) ∧ θ(e − ) for all e ∈ E we obtain .
Let γ ∈ Γ x,y be a nearest neighbour path connecting x and y.By summing over all consecutive vertices in γ, we get that ψ(y) − ψ(x) = lγ −1 i=0 ψ(z i+1 ) − ψ(z i ).Thus, ∆ ω θ (x, y) ≤ d ω θ (x, y).In order to obtain ∆ ω θ (x, y) ≥ d ω θ (x, y), set ψ(z) := d ω θ (x, z) for all z ∈ V .Then, for any edge e ∈ E an application of the triangle inequality and the definition of Likewise, it follows that, for any e ∈ E, and any non-empty, finite B ⊂ V , we define space-averaged weighted ℓ p -norms on functions f : B → R by If φ ≡ 1, we simply write f p,B := f p,B,φ .
2.2.Result.Our main objective in this section is to prove Gaussian-like upper bound on the heat kernel p θ in term of the intrinsic distance d θ .For that purpose, we impose the following assumption on the integrability of the conductances.
there exists Theorem 2.5.Suppose that ω ∈ Ω satisfies Assumption 2.4.Then, there exist constants c i = c i (d, p, q, C int ) and γ = γ(d, p, q, C int ) such that for any given t and x with ) and all y ∈ V the following hold.
Remark 2.6.(i) If the distance d ω θ and the graph distance d are comparable, as for instance in the case of CSRW, the estimates in Theorem 2.5 turn into Gaussian upper bounds since then the additional term (1 + d(x, y)/ √ t) γ can be absorbed by the exponential term into a constant.
(ii) In the case of CSRW or VSRW Theorem 2.5 has been established in [3].However, the term (1 + d(x, y)/ √ t) γ is erroneously missing in the result for the VSRW in [3,Theorem 1.10].
(iii) The on-diagonal decay t −d/2 corresponds to 1/ B(x, √ t) .In general we expect a stronger decay to hold resulting from the volume of a ball with radius √ t w.r.t. the distance d ω θ under the speed measure θ.In the remainder of this section we explain how the proof of [3,Theorem 1.6] needs to be adjusted in order to prove Theorem 2.5, that is to obtain Gaussian-like upper bounds on the the heat kernel for a larger class of speed measures θ.We also take the opportunity to streamline the arguments in [3] and to correct some technical mistakes leading to the error mentioned in Remark 2.6.

Maximal inequality for the perturbed Cauchy problem. We consider the following Cauchy problem
for some function f : V → R. Recall that for any given y ∈ Z d , the function (t, x) → p ω θ (t, x, y) solves the heat equation (2.8) with f = ½ {y} /θ(y).For any positive function φ on V such that φ, φ −1 ∈ ℓ ∞ (V ) we define the operator L ω θ,φ acting on bounded functions g : V → R as As a first step we establish the following a-priori estimate.
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 set up a Moser iteration scheme.For any finite interval I ⊂ R, finite, connected B ⊂ V and p, p ′ ∈ (0, ∞), let us introduce a space-time-averaged norm on functions u : R × V → R by  Then, there exists for any t ∈ I.By applying the product rule (2.4), we obtain (2.12) Let us first focus on the term T 1 .Again, an application of the product rule (2.4) together with the fact that (∇φ t )(∇φ −1 t ) ≤ 0 and − av(φ −1 )(∇φ) = av(φ)(∇φ −1 ), yields the following lower bound where we used that by Hölder's inequality, av(v ) for any α 1 , α 2 ≥ 0. Further, by [3, Lemma B.1], we have that for all e ∈ E. Thus, by combining the estimates above and using that av(φ) an application of Young's inequality, that reads |ab| ≤ 1 2 (εa 2 +b 2 /ε), with ε = 1/(2α) results in Let us now address the term T 2 .Observe that ), an application of the Young inequality yields On the other hand, Thus, by applying again Young's inequality with ε = 1/(4α), we get Hence, the estimates above together with the fact that give rise to the following lower bound dΓ ω (η, η) we obtain that there exists C 1 < ∞ such that for any s ∈ (s 1 , s 2 ].Thus, by multiplying both sides of (2.15) with ζ(t) and integrating the resulting inequality over [s 1 , s] for any s ∈ I, the assertion (2.10) follows by an application of the Hölder inequality.

Heat kernel bounds.
Proposition 2.10.Suppose that Assumption 2.4 hold and let x 0 ∈ V be fixed.Then, for any given x ∈ V and t with Proof.Given (2.16) this follows as in the proof of [3,Proposition 2.7].

COMPARISON RESULT FOR THE INTRINSIC METRIC
In this section we show that on a large scale the metric d ω θ can be bounded from below by a certain power of the graph distance d.The required integrability condition will be formulated in terms of an Orlicz-norm which we introduce first.
Theorem 3.3.Let (Φ, Ψ) be a Legendre-Fenchel pair of Young functions such that r −→ r is monotone increasing.Further, assume that Then, there exists c(m Ψ ) > 0 such that the following holds.For every x ∈ V there exists N 3 (ω, x) < ∞ such that for any y ∈ V with d(x, y) ≥ N 3 (ω, x), Proof.In order to simplify notation, set m ω θ (x) := 1 ∨ µ ω (x)/θ(x) for x ∈ V .Since the function t → 1/ √ t is convex, an application of the Jensen inequality yields Moreover, By combining the estimates and using (2.2) as well as the concavity of Ψ −1 we obtain that there exists c(m Ψ ) > 0 and N 3 (ω, x) < ∞ such that for any y ∈ V with d(x, y) ≥ N 3 (ω, x), Since the function r → r/ Ψ −1 (r d−1 ) is assumed to be monotone increasing, the assertion follows.
For (Φ p (r), Ψ p (r)) and Φ Exp (r), Ψ Exp (r) we obtain the following result.Then, there exists c(m p ) > 0 such that the following holds: for every x ∈ V there exists N 3 (ω, x) < ∞ such that for any y ∈ V with d(x, y) ≥ N 3 (ω, x), , where E d denotes the set of all non-oriented nearest neighbour bonds.As pointed out in Remark 2.2, (Z d , E d ) satisfies the Assumption 2.1.Further, let P be a probability measure on the measurable space (Ω, F) = R E d + , B(R + ) ⊗ E d 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[A] ∈ {0, 1} for any A ∈ F such that τ x (A) = A for all x ∈ Z d .(ii) There exist p, q ∈ (1, ∞] satisfying 1/p + 1/q < 2/d such that E ω(e) p < ∞ and E ω(e) −q < ∞ (4.1) for any e ∈ E d .
Then, the spatial ergodic theorem gives that for P-a.e. ω, In particular, by choosing θ ≡ 1 and r = ∞, Assumption 2.4 is fulfilled for P-a.e. ω and the heat kernel estimates in Theorem 2.5 hold.Nevertheless, for general ergodic environments we cannot control the size of the random variable N 3 (x), x ∈ Z d , as this requires some information on the speed of convergence in the ergodic theorem.However, if we additionally assume, for instance, that the environment satisfies a concentration inequality in form of a spectral gap inequality w.r.t. the so-called vertical derivative, then E[N 3 (x) n ] < ∞ provided a stronger moment condition holds (depending on n), see Assumption 1.3 and Lemma 2.10 in [4].
In the context of the random conductance model we can now provide the following example for which the lower bound in Theorem 3.3 or Corollary 3.4, respectively, is attained up to an arbitrarily small correction in the exponent.In particular, (4.2) holds and (ii) follows from the definition of d ω θ .

Assumption 2 . 1 .
The graph G satisfies the following conditions.