Elliptic equations for invariant measures on finite and infinite dimensional manifolds

By Vladimir Bogachev, Michael Röckner, Feng-Yu Wang. 2001

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Abstract We obtain sufficient conditions in terms of Lyapunov functions for the existence of invariant measures for diffusions on finite-dimensional manifolds and prove some regularity results for such measures. These results are extended to countable products of finite-dimensional manifolds. We introduce and study a new concept of weak elliptic equations for measures on infinite-dimensional manifolds. Then we apply our results to Gibbs distributions in the case where the single spin spaces are Riemannian manifolds. In particular, we obtain some a priori estimates for such Gibbs distributions and prove a general existence result applicable to a wide class of models. We also apply our techniques to prove absolute continuity of invariant measures on the infinite dimensional torus, improving a recent result of A.F. Ramirez. Furthermore, we obtain a new result concerning the question whether invariant measures are Gibbsian.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4b5d4677ae486e6655b5b02738a7359a1194171f

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Citations in the corpus

Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds (2005, methodology): “Moreover, we refer the reader to Cruzeiro and Malliavin [19] and Boga Röckner and Wang [14] and the reference therein for the study of existence, uniq and regularity of invariant measures for diffusion operators (which are not neces to be symmetric) on finite or infinite-dimensional manifolds under ”
Elliptic and parabolic equations for measures (2009, methodology): “According to this corollary, drifts of symmetrizable diffusions in R are logarithmic gradients of measures (an analogous fact is valid also for manifolds, see [48]).”
On Parabolic Equations for Measures (2008, result): “In this respect, our approach is closer to that of [4], [5], [6] and [9] in the elliptic case.”
Uniqueness of solutions to weak parabolic equations for measures (2007, result): “For the so far most general existence results we refer to [3] in the finite dimensional case and to [4] (see also [13]) in the infinite dimensional case.”
Supercontractivity and ultracontractivity for (nonsymmetric) diffusion semigroups on manifolds (2003, background): “6 in [9] Pt possesses a unique invariant probability measure μ.”