Distribution flows associated with positivity preserving coercive forms

TL;DR

This paper constructs distribution flows linked to positivity preserving coercive forms, enabling stochastic analysis by establishing connections with Markov processes, optional measures, and Revuz correspondence.

Contribution

It introduces a method to associate distribution flows with coercive forms, facilitating stochastic analysis and measure construction in this context.

Findings

Distribution flows behave like strong Markov processes.

Established Revuz correspondence for additive functionals.

Constructed optional measures for coercive forms.

Abstract

For a given quasi-regular positivity preserving coercive form, we construct a family of ($\sigma$-finite) distribution flows associated with the semigroup of the form. The canonical cadlag process equipped with the distribution flows behaves like a strong Markov process. Moreover, employing distribution flows we can construct optional measures and establish Revuz correspondence between additive functionals and smooth measures. The results obtained in this paper will enable us to perform a kind of stochastic analysis related to positivity preserving coercive forms.