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A remark on subharmonicity for symmetric Dirichlet forms

Kazuhiro Kuwae, Rong Lei, Ludovico Marini

TL;DR

The article extends stochastic characterizations of subharmonicity to general symmetric regular Dirichlet forms by removing the Local Boundedness assumption, enabling a unified treatment of $\mathscr{E}$-subharmonic functions for $\alpha\ge0$. It develops an energy-based definition, proves equivalence with stochastic subharmonicity notions, and shows invariance under time-change, culminating in a strong maximum principle under irreducibility. The results are illustrated through Brownian motion and relativistic $\alpha$-stable processes, highlighting applicability to non-diffusion settings and non-locally bounded functions. This broadens the potential-theoretic toolkit for Dirichlet forms and supports future analyses of irregular Markov processes.

Abstract

We remove the local boundedness for $\mathscr{E}_α$-subharmonicity in the framework of (not necessarily strongly local) regular symmetric Dirichlet form $(\mathscr{E},D(\mathscr{E}))$ with $α\geq0$ and establish the stochastic characterization for $\mathscr{E}$-subharmonic functions without assuming the local boundedness.

A remark on subharmonicity for symmetric Dirichlet forms

TL;DR

The article extends stochastic characterizations of subharmonicity to general symmetric regular Dirichlet forms by removing the Local Boundedness assumption, enabling a unified treatment of -subharmonic functions for . It develops an energy-based definition, proves equivalence with stochastic subharmonicity notions, and shows invariance under time-change, culminating in a strong maximum principle under irreducibility. The results are illustrated through Brownian motion and relativistic -stable processes, highlighting applicability to non-diffusion settings and non-locally bounded functions. This broadens the potential-theoretic toolkit for Dirichlet forms and supports future analyses of irregular Markov processes.

Abstract

We remove the local boundedness for -subharmonicity in the framework of (not necessarily strongly local) regular symmetric Dirichlet form with and establish the stochastic characterization for -subharmonic functions without assuming the local boundedness.

Paper Structure

This paper contains 6 sections, 34 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Framework
  3. $\mathscr{E}$-subharmonicity
  4. Main Results
  5. Strong Maximum Principle
  6. Examples

Key Result

Lemma 3.1

Let $T:\mathbb{R}^N\to\mathbb{R}$ be a generalized normal contraction. For $u_i\in D(\mathscr{E})_{D,{\rm loc}}^{\dag}$ (resp. $u_i\in D(\mathscr{E})_{D,{\rm loc}}^{\diamond}$, $u_i\in D(\mathscr{E})_{D,{\rm loc}}$) ($i=1,2,\cdots, N$), then $T(u_1,\cdots,u_N)\in D(\mathscr{E})_{D,{\rm loc}}^{\dag}$

Theorems & Definitions (73)

  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • proof : Proof of Lemma \ref{['lem:weldefinedness']}.
  • Corollary 3.4
  • proof : Proof.
  • Proposition 3.5
  • proof : Proof.
  • Corollary 3.6
  • proof : Proof.
  • ...and 63 more