On the domain of fractional Laplacians and related generators of Feller 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 Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder-Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process $(X_t{)}_{t\ge 0}$, \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 Hölder 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 $(X_t{)}_{t\ge 0}$ contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.

Accepted for publication in Journal of Functional Analysis

References in corpus (4)

• Ten equivalent definitions of the fractional Laplace operator
• Existence and estimates of moments for Lévy-type processes
• Solutions of Lévy-driven SDEs with unbounded coefficients as Feller processes
• Heat content for convolution semigroups

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• Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear Lévy(-type) processes
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• Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators
• Wavelet Analysis of the Besov Regularity of Lévy White Noises
• Schauder estimates for Poisson equations associated with non-local Feller generators
• A probabilistic proof of Schoenberg's theorem
• The Doob-McKean identity for stable Lévy processes
• The Positive Maximum Principle on Lie Groups
• On the association and other forms of positive dependence for Feller processes