Generalizing Dirichlet-to-Neumann operators

TL;DR

This paper explores the generalization of Dirichlet-to-Neumann operators within the framework of Dirichlet forms, establishing their probabilistic counterparts and analyzing their behavior under perturbations.

Contribution

It introduces a novel connection between Dirichlet-to-Neumann operators and trace Dirichlet forms, extending understanding to perturbed forms and associated Markov processes.

Findings

Dirichlet-to-Neumann operators linked to trace Dirichlet forms for time-changed processes

Perturbed Dirichlet forms yield quasi-regular positivity preserving coercive forms

Existence of Markov processes via Doob's h-transformations for these operators

Abstract

The aim of this paper is to study the Dirichlet-to-Neumann operators in the context of Dirichlet forms and especially to figure out their probabilistic counterparts. Regarding irreducible Dirichlet forms, we will show that the Dirichlet-to-Neumann operators for them are associated with the trace Dirichlet forms corresponding to the time changed processes on the boundary. Furthermore, the Dirichlet-to-Neumann operators for perturbations of Dirichlet forms will be also explored. It turns out that for typical cases such a Dirichlet-to-Neumann operator corresponds to a quasi-regular positivity preserving (symmetric) coercive form, so that there exists a family of Markov processes associated with it via Doob's $h$-transformations.