Explosion by Killing and Maximum Principle in Symmetric Markov Processes

TL;DR

This paper characterizes stochastic completeness and explosion phenomena in symmetric Markov processes, relates them to maximum principles for Schrödinger operators, and establishes conditions for Liouville properties and refined maximum principles.

Contribution

It introduces the concept of explosion by killing (EK), links it to stochastic completeness, and connects these to maximum principles and eigenvalue conditions for Schrödinger-type operators.

Findings

(EK) is equivalent to stochastic completeness and the process not exploding.

If (EK) holds, then () > 1 implies Liouville property for solutions.

Refined maximum principle holds if and only if the principal eigenvalue exceeds 1.

Abstract

Keller and Lenz \cite{KL} define a concept of {\it stochastic completeness at infinity} (SCI) for a regular symmetric Dirichlet form $(\cE,\cF)$. We show that (SCI) can be characterized probabilistically by using the predictable part $\zeta^p$ of the life time $\zeta$ of the symmetric Markov process $X=({\bf P}_x,X_t)$ generated by $(\cE,\cF)$, that is, (SCI) is equivalent to $\bfP_x(\zeta=\zeta^p<\infty)=0$. We define a concept, {\it explosion by killing} (EK), by $\bfP_x(\zeta=\zeta^i<\infty)=1$. Here $\zeta^i$ is the totally inaccessible part of $\zeta$. We see that (EK) is equivalent to (SCI) and $\bfP_x(\zeta=\infty)=1$. Let $X^{\rm res}$ be the {\it resurrected process} generated by the {\it resurrected form}, a regular Dirichlet form constructed by removing the killing part from $(\cE, \cF)$. Extending work of Masamune and Schmidt (\cite{MS}), we show that (EK) is also equivalent to the ordinary conservation property of time changed process of $X^{\rm res}$ by $A^k_t$, where the $A^k_t$ is the positive continuous additive functional in the Revuz correspondence to the killing measure $k$ in the Beurling-Deny formula (Theorem \ref{ma-sh}). We consider the maximum principle for Schr\"odinger-type operator $\cL^\mu=\cL-\mu$. Here $\cL$ is the self-adjoint operator associated with $(\cE,\cF)$ %with non-local part and $\mu$ is a Green-tight Kato measure. Let $\lambda(\mu)$ be the principal eigenvalue of the trace of $(\cE,\cF)$ relative to $\mu$. We prove that if (EK) holds, then $\lambda(\mu)>1$ implies a Liouville property that every bounded solution to $\cL^\mu u=0$ is zero quasi-everywhere and that the {\it refined maximum principle} in the sense of Berestycki-Nirenberg-Varadhan \cite{BNV} holds for $\cL^\mu$ if and only if $\lambda(\mu)>1$ (Theorem \ref{RMP}).