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(Nonsymmetric) Dirichlet Operators On L¹: Existence, Uniqueness And Associated Markov Processes
"... Let L be a nonsymmetric operator of type Lu = P a ij @ i @ j u + P b i @ i u on an arbitrary subset U ae R d . We analyse L as an operator on L 1 (U; ¯) where ¯ is an invariant measure, i.e., a possibly infinite measure ¯ satisfying the equation L ¯ = 0 (in the weak sense). We explicitel ..."
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Cited by 9 (2 self)
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Let L be a nonsymmetric operator of type Lu = P a ij @ i @ j u + P b i @ i u on an arbitrary subset U ae R d . We analyse L as an operator on L 1 (U; ¯) where ¯ is an invariant measure, i.e., a possibly infinite measure ¯ satisfying the equation L ¯ = 0 (in the weak sense). We explicitely construct, under mild regularity assumptions, extensions of L generating subMarkovian C0 semigroups on L 1 (U; ¯) as well as associated diffusion processes. We give sufficient conditions on the coefficients so that there exists only one extension of L generating a C0 semigroup and apply the results to prove uniqueness of the invariant measure ¯. Our results imply in particular that if ' 2 H 1;2 loc (R d ; dx), ' 6= 0 dxa.e., the generalized Schrödinger operator (\Delta + 2' \Gamma1 r' \Delta r;C 1 0 (R d )) has exactly one extension generating a C0 semigroup if and only if the Friedrich's extension is conservative. We also give existence and uniqueness results for ...
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...
Existence and Uniqueness of invariant measures: an approach via sectorial forms
 Universitat Bielefeld, SFB 343, Preprint 97
, 1997
"... We prove existence and uniqueness for invariant measures of strongly continuous semigroups on L 2 (X ; ¯), where X is a (possibly infinitedimensional) space. Our approach is purely analytic based on the theory of sectorial forms. The generators covered are e.g. small perturbations (in the sense ..."
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Cited by 4 (1 self)
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We prove existence and uniqueness for invariant measures of strongly continuous semigroups on L 2 (X ; ¯), where X is a (possibly infinitedimensional) 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 (ae \Delta dx) on R d (resp. a Riemannian manifold) into L 2 (ae dx). Probabilistic consequences are also briefly discussed. AMS Subject Classification Primary: 31 C 25 Secondary: 47 D 07, 60 H 10, 47 D 06, 60 J 60 Key words and phrases: invariant measures, sectorial forms, compact embeddings, Poincare inequality, LogSobolev inequality Running head: Invariant measures for semigroups 1) Department of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia 2) Fakultat fur Mathematik, U...
Drift Transformations of Symmetric Diffusions, and Duality
, 2006
"... Starting with a symmetric Markov diffusion process X (with symmetry measure m and L 2 (m) infinitesimal generator A) and a suitable core C for the Dirichlet form of X, we describe a class of derivations defined on C. Associated with each such derivation B is a drift transformation of X, obtained thr ..."
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Cited by 1 (1 self)
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Starting with a symmetric Markov diffusion process X (with symmetry measure m and L 2 (m) infinitesimal generator A) and a suitable core C for the Dirichlet form of X, we describe a class of derivations defined on C. Associated with each such derivation B is a drift transformation of X, obtained through Girsanov’s theorem. The transformed process X B is typically nonsymmetric, but we are able to show that if the “divergence ” of B is positive, then m is an excessive measure for X B, and the L 2 (m) infinitesimal generator of X B is an extension of f ↦ → Af + B(f). The methods used are mainly probabilistic, and involve the notions of even and odd continuous additive functionals, and Nakao’s stochastic divergence.