A semigroup approach to nonlinear L\'evy processes

TL;DR

This paper explores the connection between nonlinear Le9vy processes, nonlinear semigroups, and fully nonlinear PDEs, establishing conditions for their existence and illustrating with examples.

Contribution

It introduces a framework linking nonlinear Le9vy processes with nonlinear Markovian convolution semigroups and provides conditions for their existence via infinitesimal generators.

Findings

Established a one-to-one relation between nonlinear Le9vy processes and nonlinear semigroups.

Provided a condition on generators ensuring the existence of nonlinear Le9vy processes.

Demonstrated the theory with multiple examples.

Abstract

We study the relation between L\'evy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear L\'evy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators $(A_\lambda)_{\lambda\in\Lambda}$ of linear L\'evy processes which guarantees the existence of a nonlinear L\'evy processes such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE $\partial_t u=\sup_{\lambda\in \Lambda} A_\lambda u$. The results are illustrated with several examples.