Appl Math Optim 41:87–109 (2000)
2000 Springer-Verlag New York Inc.
Existence and Uniqueness of Invariant Measures:
An Approach via Sectorial Forms
and T. S. Zhang
Department of Mechanics and Mathematics,
Moscow State University,
119899 Moscow, Russia
Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld,
D-33501 Bielefeld, Germany
Faculty of Engineering, HSH,
Sk˚aregt 103, 5500 Haugesund, Norway
Abstract. We prove the existence and uniqueness for invariant measures of strong-
ly continuous semigroups on L
(X; µ), where X is a (possibly inﬁnite-dimensional)
space. Our approach is purely analytic based on the theory of sectorial forms. The
generators covered are, e.g., small perturbations (in the sense of sectorial forms)
of operators generating hypercontractive semigroups. An essential ingredient of
the proofs is a new result on compact embeddings of weighted Sobolev spaces
(ρ · dx) on R
(resp. a Riemannian manifold) into L
(ρ dx). Probabilistic
consequences are also brieﬂy discussed.
Key Words. Invariant measures, Sectorial forms, Compact embeddings, Poincar´e
inequality, Log-Sobolev inequality.
AMS Classiﬁcation. Primary 31C25, Secondary 47D07, 60H10, 47D06, 60J60.
Financial support of the SFB 343 (Bielefeld), DFG/RFFI–Project No. 436 RUS 113/343/1, EU-TMR-
Project No. ERB-FMRX-CT96-0075, and the Russian Foundation of Fundamental Research (Grant No. 97-
01-00932) is gratefully acknowledged.