The parametrix method for parabolic SPDEs

TL;DR

This paper extends the parametrix method to analyze linear stochastic PDEs with time-measurable coefficients, establishing existence, regularity, and estimates for their fundamental solutions, and applying these results via a change of variables.

Contribution

It introduces a novel extension of the parametrix method to SPDEs with minimal regularity assumptions on coefficients, providing new analytical tools.

Findings

Proved existence and regularity of fundamental solutions for the considered SPDEs.

Derived upper and lower bounds for the fundamental solutions.

Applied the method to transform SPDEs into PDEs with random coefficients.

Abstract

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in H\"older classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Ito-Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.