$L^1$ semigroup generation for Fokker-Planck operators associated with general L\'evy driven SDEs

TL;DR

This paper establishes $L^1$-semigroup generation for a broad class of non-local operators linked to Le9vy-driven SDEs, advancing understanding of their analytical properties and regularity.

Contribution

It introduces a new $L^1$-generation result for complex non-local operators, including those with state-dependent coefficients and singular, heavy-tailed Le9vy measures.

Findings

Proves $L^1$-semigroup generation for general Le9vy operators.

Derives a new elliptic regularity result for these operators.

Handles operators with state-dependent, linearly growing coefficients and singular measures.

Abstract

We prove a new generation result in $L^1$ for a large class of non-local operators with non-degenerate local terms. This class contains the operators appearing in Fokker-Planck or Kolmogorov forward equations associated with L\'evy driven SDEs, i.e. the adjoint operators of the infinitesimal generators of these SDEs. As a byproduct, we also obtain a new elliptic regularity result of independent interest. The main novelty in this paper is that we can consider very general L\'evy operators, including state-space depending coefficients with linear growth and general L\'evy measures which can be singular and have fat tails.