Stochastic approach to bounds and regularity of fundamental solutions to non-divergence form parabolic equations with irregular coefficients (Regularity, singularity and long time behavior for partial differential equations with conservation law)

Abstract

This is a digest of a series of the author's works about the stochastic approach to Gaussian two-sided bounds and regularity of fundamental solutions to non-divergence form parabolic equations with irregular coefficients. The argument are decomposed into two parts. One is about the modulus of continuity of the solution to the parabolic equation, and the other is Gaussian two-sided bounds of the fundamental solution. Combining them, we obtain the Gaussian two-sided bounds of the fundamental solution, and the (1-ε)-Hölder continuity and other continuities in a spatial component.