Extension Technique for Functions of Diffusion Operators: a stochastic approach

TL;DR

This paper extends the understanding of boundary condition mappings for functions of diffusion operators, using stochastic analysis to generalize previous results from Laplace operators to more general diffusion operators.

Contribution

It introduces a stochastic approach to extend boundary condition mappings from Laplace to general diffusion operators associated with stochastic differential equations.

Findings

Generalized boundary condition mapping to diffusion operators

Connected non-local operators to differential operators via stochastic methods

Enhanced analytical tools for elliptic PDEs with diffusion operators

Abstract

It has recently been shown that complete Bernstein functions of the Laplace operator map the Dirichlet boundary condition of a related elliptic PDE to the Neumann boundary condition. The importance of this mapping consists in being able to convert problems involving non-local operators, like fractional Laplacians, into ones that only involve differential operators. We generalise this result to diffusion operators associated with stochastic differential equations, using a method which is entirely based on stochastic analysis.