Intrinsic scaling properties for nonlocal operators

TL;DR

This paper investigates the regularity and growth properties of generators of Markov processes with nonlocal operators, focusing on intrinsic scaling to understand the boundary between integro-differential and integral operators.

Contribution

It introduces a novel approach using intrinsic scaling properties to analyze regularity for generators with arbitrary differentiability less than 2, bridging integro-differential and integral operators.

Findings

Established growth lemmas for nonlocal operators.

Analyzed regularity up to the phase boundary between operator types.

Provided a framework for studying operators with variable differentiability.

Abstract

We study growth lemmas and questions of regularity for generators of Markov processes. The generators are allowed to have an arbitrary order of differentiability less than 2. In general, this order is represented by a function and not by a number. The approach enables a careful study of regularity issues up to the phase boundary between integro-differential (positive order of differentiability) and integral operators (nonnegative order of differentiability). The proof is based on intrinsic scaling properties of the underlying operators and stochastic processes.