Maximum principles for nonlocal parabolic Waldenfels operators

TL;DR

This paper establishes weak and strong maximum principles for parabolic equations involving nonlocal Waldenfels operators, which are important in stochastic processes with jumps and boundary conditions.

Contribution

It provides the first rigorous proof of maximum principles for parabolic equations with nonlocal Waldenfels operators, expanding the theoretical framework for such stochastic models.

Findings

Proved weak maximum principle for nonlocal Waldenfels operators.

Proved strong maximum principle for nonlocal Waldenfels operators.

Applied results to stochastic differential equations with $\alpha$-stable Lévy processes.

Abstract

As a class of L\'evy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under L\'evy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for `parabolic' equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with $\alpha$-stable L\'evy processes are presented to illustrate the maximum principles.