Existence and Uniqueness of Invariant Measures: An Approach via Sectorial Forms

Existence and Uniqueness of Invariant Measures: An Approach via Sectorial Forms We prove the existence and uniqueness for invariant measures of strongly continuous semigroups on L 2 (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 H 1,2 (ρ . . . dx) on \bf R d (resp. a Riemannian manifold) into L 2 (ρ dx) . Probabilistic consequences are also briefly discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Existence and Uniqueness of Invariant Measures: An Approach via Sectorial Forms

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Publisher
Springer-Verlag
Copyright
Copyright © Inc. by 2000 Springer-Verlag New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459911005
Publisher site
See Article on Publisher Site

Abstract

We prove the existence and uniqueness for invariant measures of strongly continuous semigroups on L 2 (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 H 1,2 (ρ . . . dx) on \bf R d (resp. a Riemannian manifold) into L 2 (ρ dx) . Probabilistic consequences are also briefly discussed.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2008

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