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

We consider continuous time simple random walks with arbitrary speed measure θ on infinite weighted graphs. Write pₜ(x,y) for the heat kernel of this process. Given on-diagonal upper bounds for the heat kernel at two points x₁,x₂, we obtain a Gaussian upper bound for pₜ(x₁,x₂). 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.