Poisson stochastic process and basic Schauder and Sobolev estimates in the theory of parabolic equations

TL;DR

This paper demonstrates how Poisson stochastic processes can be used to derive Schauder and Sobolev estimates for multidimensional parabolic equations from one-dimensional cases, offering a novel probabilistic approach.

Contribution

It introduces a new probabilistic method using Poisson processes to extend estimates from 1D to multidimensional parabolic equations with time-dependent coefficients.

Findings

Derivation of multidimensional estimates from 1D heat equation estimates

Use of Poisson stochastic process as a key tool

Potential for a new approach in parabolic PDE analysis

Abstract

We show among other things how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the {\em same\/} constants as in the case of the one-dimensional heat equation. The method is based on using the Poisson stochastic process. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results.