# Decomposition formulae for Dirichlet forms and their corollaries

## TL;DR

This paper develops decomposition formulae for Dirichlet forms into recurrent, transient, conservative, and dissipative parts within Hausdorff spaces, and studies their invariants under convergence.

## Contribution

It introduces new decomposition formulae for Dirichlet forms and analyzes invariance properties under Mosco convergence, providing a comprehensive framework for understanding their structure.

## Key findings

- Decomposition of Dirichlet forms into recurrent, transient, conservative, and dissipative parts.
- Mosco convergence preserves invariant sets of Dirichlet forms.
- Equivalence of conservativeness and dissipativity between Dirichlet forms and their approximations.

## Abstract

We provide decompositions of Dirichlet forms into recurrent and transient parts as well as into conservative and dissipative parts, in the framework of Hausdorff state spaces. Combining both formulae we write every Dirichlet form as the sum of a recurrent, dissipative and transient conservative Dirichlet forms. Besides, we prove that Mosco convergence preserves invariant sets and that a Dirichlet form shares the same invariants sets with its approximating Dirichlet forms E(t) and E(?). Finally we show the equivalence between conservativeness (resp. dissipativity) of a Dirichlet form and the conservativeness (reps. dissipativity) of E(t) and E(?). The elaborated results are enlightened by some examples.