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.
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The authors are very grateful to the anonymous referees whose comments led to improve the framework of the paper.
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BenAmor, A., Moussa, R. Decomposition Formulae for Dirichlet Forms and Their Corollaries. Mediterr. J. Math. 18, 17 (2021). https://doi.org/10.1007/s00009-020-01658-5
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DOI: https://doi.org/10.1007/s00009-020-01658-5