Decomposition Formulae for Dirichlet Forms and Their Corollaries

Abstract

We provide decompositions of symmetric Dirichlet forms into recurrent and transient parts as well as into conservative and dissipative parts, in the framework of $$\sigma $$ -finite measure 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 $${\mathcal {E}}^{(t)}$$ and $${\mathcal {E}}^{(\beta )}$$ . Finally, we show the equivalence between conservativeness (resp. dissipativity) of a Dirichlet form and the conservativeness (reps. dissipativity) of $${\mathcal {E}}^{(t)}$$ and $${\mathcal {E}}^{(\beta )}$$ . The elaborated results are enlightened by some examples.