Right Markov processes and systems of semilinear equations with measure data

TL;DR

This paper establishes the existence of solutions to systems of semilinear equations involving measure data, using probabilistic methods and properties of right Markov processes with order compact resolvents.

Contribution

It introduces a new compactness property relating Markov processes to convergence, applicable to both local and nonlocal operators, expanding the scope of solvable systems.

Findings

Existence of solutions for systems with measure data and generalized sign conditions.

Characterization of the compactness property via Meyer's property and resolvent compactness.

Applicability to operators beyond variational forms, including nonlocal operators.

Abstract

In the paper we prove the existence of probabilistic solutions to systems of the form $-Au=F(x,u)+\mu$, where $F$ satisfies a generalized sign condition and $\mu$ is a smooth measure. As for $A$ we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on $L^1$. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer's property (L) of Markov processes and in terms of order compactness of the associated resolvent.