Gaussian Upper Bounds for Heat Kernels of Continuous Time Simple Random Walks

By M. Folz. 2011

79 citations recorded by the source corpus.

We consider continuous time simple random walks with arbitrary speed measure $\theta$ on infinite weighted graphs. Write $p_t(x,y)$ for the heat kernel of this process. Given on-diagonal upper bounds for the heat kernel at two points $x_1,x_2$, we obtain a Gaussian upper bound for $p_t(x_1,x_2)$ . The distance function which appears in this estimate is not in general the graph metric, but a new metric which is adapted to the random walk. Long-range non-Gaussian bounds in this new metric are also established. Applications to heat kernel bounds for various models of random walks in random environments are discussed.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/af81ceeab1b7e3982acc374dd74c15c58e1f2ffd

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In Applied Mathematics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Volume growth and stochastic completeness of graphs (2012, methodology): “We call a metric ρ adapted to the VSRW X on (Γ , π ) if there exists C ρ ≥ 0 such that for all x ∈ G , This condition has appeared previously in [10], [12], and [17], and it has apparently close connections to the condition in the deﬁnition of E X .”
A note on self-adjoint extensions of the Laplacian on weighted graphs (2012, background): “Independently, Folz [8] came up with similar ideas in the context of heat kernel estimates on locally finite graphs, see also [9, 10].”
On stochastic completeness of jump processes (2012, background): “Folz [ 10 ] also came up with similar ideas independently in the context of heat kernel estimates.”
Cheeger inequalities for unbounded graph Laplacians (2012, background): “[14, 15, 16, 23, 25, 30, 31, 32, 42], where certain variations of the concept also go under the name of adapted metrics.”