Schauder and Sobolev Estimates of Parabolic Equations

TL;DR

This paper demonstrates how probabilistic methods can extend Schauder and Sobolev estimates from one-dimensional to multidimensional heat equations, generalizing previous results by Krylov-Priola.

Contribution

It introduces a probabilistic approach to derive multidimensional Schauder and Sobolev estimates for parabolic equations, expanding existing theoretical frameworks.

Findings

Probabilistic method effectively derives multidimensional estimates.

Generalization of Krylov-Priola results to higher dimensions.

Provides a new perspective on heat equation regularity estimates.

Abstract

In this note, we use the non-homogeneous Poisson stochastic process to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The method is probability. We generalize the result of Krylov-Priola [7].