Magnetic energies and Feynman-Kac-It\^o formulas for symmetric Markov processes

TL;DR

This paper develops bilinear forms related to symmetric Markov processes, generalizing electromagnetic energy forms, and establishes conditions under which these forms coincide and describes the associated Hamiltonian.

Contribution

It introduces a new bilinear form via semigroup approximation and proves its equivalence to a closed form derived from a Feynman-Kac-Itô formula for Feller processes.

Findings

The two bilinear forms agree on a core for Feller processes.

Explicit form of the Hamiltonian associated with the forms.

Conditions under which the forms coincide and are closed.

Abstract

Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an electromagnetic field. We obtain one bilinear form by semigroup approximation and another, closed one, by using a Feynman-Kac-It\^o formula. If the given process is Feller, its energy measures have densities and its jump measure has a kernel, then the two forms agree on a core and the second is a closed extension of the first. In this case we provide the explicit form of the associated Hamiltonian.