A note on a stochastic approach to Caffarelli-Silvestre Theorem

TL;DR

This paper explores the Caffarelli-Silvestre extension via stochastic analysis, providing an explicit kernel formulation for the associated Dirichlet problem, thus offering a new perspective on this mathematical theorem.

Contribution

It introduces a stochastic approach to explicitly formulate the kernel for the Caffarelli-Silvestre extension, connecting stochastic analysis with fractional Laplacian theory.

Findings

Explicit kernel formulation for the Caffarelli-Silvestre extension

Application of stochastic analysis to Dirichlet problems

New insights into the structure of the extension function

Abstract

In this note we analyze the Caffarelli-Silvestre extension function using tools from the theory of stochastic analysis applied to Dirichlet problems. We use a stochastic approach to give the explicit formulation of the kernel associated to the Dirichlet problem which defines the Cafferelli-Silvestre extension function.