$L^p$-Green-tight measures of $L^p$-Kato class for symmetric Markov processes

TL;DR

This paper introduces a new class of measures called $L^p$-Green-tight measures for symmetric Markov processes, establishing their properties, equivalences, and applications to specific stochastic processes.

Contribution

It defines the $L^p$-Green-tight measures of $L^p$-Kato class, proves their properties, and shows their equivalence with existing classes under certain conditions.

Findings

Embedding of Dirichlet space into $L^{2p}$ is compact under $L^p$-Green tightness.

Two classes of $L^p$-Green-tight measures are shown to coincide.

Results apply to Brownian motion and relativistic $ ext{α}$-stable processes.

Abstract

In this paper, we introduce the notion of $L^p$-Green-tight measures of $L^p$-Kato class in the framework of symmetric Markov processes. The class of $L^p$-Green-tight measures of $L^p$-Kato class is defined by the $p$-th power of resolvent kernels. We first prove that under the $L^p$-Green tightness of the measure $\mu$, the embedding of extended Dirichlet space into $L^{2p}(E;\mu)$ is compact under the absolute continuity condition for transient Markov processes, which is an extension of recent seminal work by Takeda. Secondly, we prove the coincidence between two classes of $L^p$-Green-tightness, one is originally introduced by Zhao, and another one is invented by Chen. Finally, we prove that our class of $L^p$-Green-tight measures of $L^p$-Kato class coincides with the class of $L^p$-Green tight measures of Kato class in terms of Green kernel under the global heat kernel estimates. We apply our results to $d$-dimensional Brownian motion androtationally symmetric relativistic $\alpha$-stable processes on $\mathbb{R}^d$.