On the domain of fractional Laplacians and related generators of Feller processes

TL;DR

This paper investigates the domain of fractional Laplacians and related generators of Feller processes, providing conditions on symbols and analyzing small-time asymptotics of process moments, with applications to various Lévy-type processes.

Contribution

It establishes new conditions for domain inclusion of fractional Laplacians and analyzes small-time process behavior for non-smooth functions.

Findings

Conditions on symbols ensure certain Hölder spaces are in the domain.

Pointwise limits of generalized moments exist under Hölder conditions.

Results apply to stable-like, relativistic stable-like, and Lévy-driven SDE processes.

Abstract

In this paper we study the domain of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of L\'evy- and L\'evy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) H\"{o}lder and H\"{o}lder-Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a L\'evy or L\'evy-type process $(X_t)_{t \geq 0}$, \begin{equation*} \lim_{t \to 0} \frac 1t\left(\mathbb{E}^x f(X_t)-f(x)\right), \quad x\in\mathbb{R}^d, \end{equation*} for functions $f$ which are not necessarily bounded or differentiable. The pointwise limit exists for fixed $x \in \mathbb{R}^d$ if $f$ satisfies a H\"{o}lder condition at $x$. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of $(X_t)_{t \geq 0}$ contains certain H\"{o}lder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of L\'evy-driven SDEs and L\'evy processes.