Estimates of heat kernels for non-local regular Dirichlet forms

Grigor’yan, Alexander; Hu, Jiaxin; Lau, Ka-Sing

doi:10.1090/S0002-9947-2014-06034-0

Estimates of heat kernels for non-local regular Dirichlet forms
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by Alexander Grigor’yan, Jiaxin Hu and Ka-Sing Lau PDF

Trans. Amer. Math. Soc. 366 (2014), 6397-6441 Request permission

Abstract:

In this paper we present new heat kernel upper bounds for a certain class of non-local regular Dirichlet forms on metric measure spaces, including fractal spaces. We use a new purely analytic method where one of the main tools is the parabolic maximum principle. We deduce an off-diagonal upper bound of the heat kernel from the on-diagonal one under the volume regularity hypothesis, restriction of the jump kernel and the survival hypothesis. As an application, we obtain two-sided estimates of heat kernels for non-local regular Dirichlet forms with finite effective resistance, including settings with the walk dimension greater than $2$.

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