On instability of global path properties of symmetric Dirichlet forms under Mosco-convergence

TL;DR

This paper investigates how global path properties of symmetric Dirichlet forms behave under Mosco convergence, revealing that properties like recurrence and conservativeness are not necessarily preserved during the convergence process.

Contribution

It provides sufficient conditions for Mosco convergence of various symmetric processes and highlights the non-preservation of key path properties through concrete examples.

Findings

Mosco convergence does not guarantee preservation of recurrence/transience.

Sufficient conditions for Mosco convergence are established for diffusions, Lévy, and jump processes.

Examples demonstrate the non-preservation of global path properties.

Abstract

We give sufficient conditions for Mosco convergences for the following three cases: symmetric locally uniformly elliptic diffusions, symmetric L\'evy processes, and symmetric jump processes in terms of the $L^1(\mathbb R;dx)$-local convergence of the (elliptic) coefficients, the characteristic exponents and the jump density functions,respectively. We stress that the global path properties of the corresponding Markov processes such as recurrence/transience, and conservativeness/explosion are not preserved under Mosco convergences and we give several examples where such situations indeed happen.