Analysis & PDE

Hölder continuity and bounds for fundamental solutions to nondivergence form parabolic equations

Seiichiro Kusuoka

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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.

Article information

Anal. PDE, Volume 8, Number 1 (2015), 1-32.

Received: 16 October 2013
Revised: 6 November 2014
Accepted: 21 December 2014
First available in Project Euclid: 28 November 2017

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Zentralblatt MATH identifier

Primary: 35B65: Smoothness and regularity of solutions 35K10: Second-order parabolic equations 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

parabolic partial differential equation diffusion fundamental solution Hölder continuity Gaussian estimate stochastic differential equation coupling method


Kusuoka, Seiichiro. Hölder continuity and bounds for fundamental solutions to nondivergence form parabolic equations. Anal. PDE 8 (2015), no. 1, 1--32. doi:10.2140/apde.2015.8.1.

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