Shot-down stable processes

TL;DR

This paper introduces the shot-down process for the fractional Laplacian, exploring its properties, transition density, and boundary behaviors, revealing new insights into nonlocal boundary conditions and their effects on process comparability.

Contribution

It constructs the shot-down process for the fractional Laplacian and analyzes its properties, highlighting differences between convex and nonconvex domains.

Findings

Green function comparable to fractional Laplacian with Dirichlet conditions for smooth bounded sets

Transition density differs from Dirichlet heat kernel in nonconvex domains

Provides new boundary condition perspective for nonlocal differential equations

Abstract

The shot-down process is a strong Markov process which is annihilated, or shot down, when jumping over or to the complement of a given open subset of a vector space. Due to specific features of the shot-down time, such processes suggest new type of boundary conditions for nonlocal differential equations. In this work we construct the shot-down process for the fractional Laplacian in Euclidean space. For smooth bounded sets $D$, we study its transition density and characterize Dirichlet form. We show that the corresponding Green function is comparable to that of the fractional Laplacian with Dirichlet conditions on $D$. However, for nonconvex $D$, the transition density of the shot-down stable process is incomparable with the Dirichlet heat kernel of the fractional Laplacian for $D$.