Abstract
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 in [3] 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.
Highlights
Let G = (V, E) be an infinite, connected, locally finite graph with vertex set V and edge set E
The resulting heat kernel estimates are governed by the intrinsic metric induced by the speed measure
We 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
Summary
Y ∈ V and t ≥ 0 let pωθ (t, x, y) be the transition densities of X with respect to the reversible measure (or the heat kernel associated with Lωθ ), i.e. As one of our main results we establish upper bounds on the heat kernel under a certain integrability condition on the conductances, see Theorem 3.2 below. In [3] we weakened the uniform ellipticity condition and showed heat kernel upper bounds for the CSRW and VSRW under a similar integrability condition as in Theorem 3.2, while in the present paper we extend this result to general speed measures. Constants denoted Ci will be the same through each argument
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