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We prove that weak invariance is equivalent to strong invariance in the framework of quasi-regular local positivity preserving forms with lower bounds. As an application, we give an ergodic decomposition of Markov processes associated with quasi-regular local semi-Dirichlet forms with lower bounds. As consequences, criteria of transience and recurrence for Markov processes associated with quasi-regular semi-Dirichlet forms with lower bounds are presented.
Forum Mathematicum – de Gruyter
Published: Nov 1, 2011
Keywords: Semi-Dirichlet form; Dirichlet form; weakly invariant set; strongly invariant set; irreducibility; strict irreducibility; absolute continuity condition; dissipative part; conservative part; transience; recurrence; ergodic decomposition; Feller diffusion process; strong Feller diffusion process; doubly Feller diffusion process
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