Extension Technique for Functions of Diffusion Operators: a stochastic approach

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.

33 pages, corrected two mistakes: added condition (S4) to be satisfied by Krein strings, and added part (b) to Remark 2.13 clarifying the missing contraction property of certain semigroups