Vol. 8, No. 1, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Hölder continuity and bounds for fundamental solutions to nondivergence form parabolic equations

Seiichiro Kusuoka

Vol. 8 (2015), No. 1, 1–32
Abstract

We consider nondegenerate second-order parabolic partial differential equations in nondivergence form with bounded measurable coefficients (not necessary continuous). Under certain assumptions weaker than the Hölder continuity of the coefficients, we obtain Gaussian bounds and Hölder continuity of the fundamental solution with respect to the initial point. Our proofs employ pinned diffusion processes for the probabilistic representation of fundamental solutions and the coupling method.

Keywords
parabolic partial differential equation, diffusion, fundamental solution, Hölder continuity, Gaussian estimate, stochastic differential equation, coupling method
Mathematical Subject Classification 2010
Primary: 35B65, 35K10, 60H30
Secondary: 60H10, 60J60
Milestones
Received: 16 October 2013
Revised: 6 November 2014
Accepted: 21 December 2014
Published: 15 April 2015
Authors
Seiichiro Kusuoka
Graduate School of Science
Tohoku University
6-3 Aramaki Aza-Aoba
Aoba-ku
Sendai 980-8578
Japan