For Brox's diffusion in a Brownian environment, the heat kernel satisfies two-sided Gaussian-type bounds at small times and the annealed on-diagonal kernel decays like (log t)^{-2}, up to log-log factors.
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Quenched and annealed heat kernel estimates for Brox's diffusion
For Brox's diffusion in a Brownian environment, the heat kernel satisfies two-sided Gaussian-type bounds at small times and the annealed on-diagonal kernel decays like (log t)^{-2}, up to log-log factors.