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

Matthew Folz (University of British Columbia)

Abstract


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.

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Pages: 1693-1722

Publication Date: September 12, 2011

DOI: 10.1214/EJP.v16-926

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