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On Uniqueness of Invariant Measures for Finite and Infinite Dimensional Diffusions
, 1998
"... We prove uniqueness of "invariant measures", i.e., solutions to the equation L ¯ = 0 where L = \Delta +B \Delta r on R n with B satisfying some mild integrability conditions and ¯ is a probability measure on R n . This solves an open problem posed by S.R.S. Varadhan in 1980. The same conditio ..."
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Cited by 6 (2 self)
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We prove uniqueness of "invariant measures", i.e., solutions to the equation L ¯ = 0 where L = \Delta +B \Delta r on R n with B satisfying some mild integrability conditions and ¯ is a probability measure on R n . This solves an open problem posed by S.R.S. Varadhan in 1980. The same conditions are shown to imply that the closure of L on L 1 (¯) generates a strongly continuous semigroup having ¯ as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L 1 (¯) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on B. The special case when B is a gradient of a function (i.e., the "symmetric case") is in particular studied and conditions are identified ensuring that L ¯ = 0 implies that L is symmetric on L 2 (¯) resp. L ¯ = 0 has a unique solution. We also prove infinite dimensional analogues of the latter two results and a ne...
L_pAnalysis Of Finite And Infinite Dimensional Diffusion Operators
"... This paper consists of lectures given at the C.I.M.E. summer school on Kolmogorov equations held at Cetraro in 1998. The purpose of these lectures was to present an approach to Kolmogorov equations in infinite dimensions which is based on an L p ()analysis of the corresponding diffusion operator ..."
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Cited by 4 (1 self)
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This paper consists of lectures given at the C.I.M.E. summer school on Kolmogorov equations held at Cetraro in 1998. The purpose of these lectures was to present an approach to Kolmogorov equations in infinite dimensions which is based on an L p ()analysis of the corresponding diffusion operators w.r.t. suitably chosen measures. The main ideas and aims are explained, and an as complete as possible presentation is given of what has been achieved in this respect over the last few years.
Invariant Measures for Nonlinear SPDE's: Uniqueness and Stability
 Archivum Math
, 1998
"... . The paper presents a review of some recent results on uniqueness of invariant measures for stochastic di#erential equations in infinitedimensional state spaces, with particular attention paid to stochastic partial di#erential equations. Related results on asymptotic behaviour of solutions like erg ..."
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Cited by 1 (0 self)
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. The paper presents a review of some recent results on uniqueness of invariant measures for stochastic di#erential equations in infinitedimensional state spaces, with particular attention paid to stochastic partial di#erential equations. Related results on asymptotic behaviour of solutions like ergodic theorems and convergence of probability laws of solutions in strong and weak topologies are also reviewed. AMS Subject Classification. 60H15 Keywords. Stochastic evolution equations, invariant measures, ergodic theorems, stability 1 Introduction The aim of the present paper is to review some recent results on uniqueness of invariant measures (that is, strictly stationary solutions) for nonlinear stochastic evolution equations (or, more generally, for stochastic di#erential equations in infinitedimensional state spaces). Related asymptotic and ergodic properties of solutions like convergence of their probability laws to the invariant measure and ergodic theorems are also discussed. The...