Nonlocal operators with Neumann conditions

TL;DR

This paper constructs a Markov process linked to a specific Dirichlet form to address Neumann boundary problems for the fractional Laplacian, providing a probabilistic approach to boundary value problems.

Contribution

It introduces a novel Markov process framework for solving Neumann boundary problems associated with the fractional Laplacian.

Findings

Successfully constructs a Markov process for the Dirichlet form of Servadei and Valdinoci.

Solves the Neumann boundary problem for the fractional Laplacian on the half-line.

Provides a probabilistic interpretation of Neumann boundary conditions for fractional operators.

Abstract

We construct a strong Markov process corresponding to the Dirichlet form of Servadei and Valdinoci and use the process to solve the corresponding Neumann boundary problem for the fractional Laplacian and the half-line.