Absolute continuity of Markov chains ergodic measures by Dirichlet forms methods

Guillaume Poly (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We study the absolute continuity of ergodic measures of Markov chains $X_{n+1}=F(X_n,Y_{n+1})$ for the discrete case, and $dX_t=b(X_t)dt+\sigma(X_t).dW_t$ for the continuous case. In the discrete case, we provide with a method enabling to deal with the case where the chains has several invariant measures whereas previous works (c.f. \cite{coquio1992calcul,gravereaux1988calcul}) made assumptions of contractivity, and hence unique ergodicity. Besides, the smoothness assumptions on $F$ are weakened. In the continuous case, we make stronger smoothness assumptions than \cite{bogachev2009elliptic}, but non-degeneracy assumptions are strongly weakened. The proofs are based on Dirichlet forms theory, and ergodic theory arguments.

Mots clés: Malliavin calculus; Dirichlet forms; Ergodic theory; Markov chains

Domaines: Probabilités [math.PR]

Guillaume Poly. Absolute continuity of Markov chains ergodic measures by Dirichlet forms methods. 2012. ⟨hal-00690738⟩