Results 1  10
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17
Generalized Mehler semigroups and applications
, 1994
"... We construct and study generalized Mehler semigroups (p t ) t#0 and their associated Markov processes M. The construction methods for (p t ) t#0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert space H can be extend ..."
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Cited by 26 (5 self)
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We construct and study generalized Mehler semigroups (p t ) t#0 and their associated Markov processes M. The construction methods for (p t ) t#0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert space H can be extended to some larger Hilbert space E, with the embedding H # E being HilbertSchmidt. The same analytic extension results are applied to construct strong solutions to stochastic differential equations of type dX t = CdW t + AX t dt (with possibly unbounded linear operators A and C on H) on a suitably chosen larger space E. For Gaussian generalized Mehler semigroups (p t ) t#0 with corresponding Markov process M, the associated (nonsymmetric) Dirichlet forms (E , D(E)) are explicitly calculated and a necessary and sufficient condition for path regularity of M in terms of (E , D(E)) is proved. Then, using Dirichlet form methods it is shown that M weakly solves the above stochastic differential ...
Analysis and geometry on configuration spaces: The Gibbsian case
, 1998
"... Using a natural "Riemanniangeometrylike" structure on the configuration space \Gamma over IR d , we prove that for a large class of potentials OE the corresponding canonical Gibbs measures on \Gamma can be completely characterized by an integration by parts formula. That is, if r \Ga ..."
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Cited by 20 (5 self)
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Using a natural "Riemanniangeometrylike" structure on the configuration space \Gamma over IR d , we prove that for a large class of potentials OE the corresponding canonical Gibbs measures on \Gamma can be completely characterized by an integration by parts formula. That is, if r \Gamma is the gradient of the Riemannian structure on \Gamma one can define a corresponding divergence div \Gamma OE such that the canonical Gibbs measures are exactly those measures ¯ for which r \Gamma and div \Gamma OE are dual operators on L 2 (\Gamma; ¯). One consequence is that for such ¯ the corresponding Dirichlet forms E \Gamma ¯ are defined. In addition, each of them is shown to be associated with a conservative diffusion process on \Gamma with invariant measure ¯. The corresponding generators are extensions of the operator \Delta \Gamma OE := div \Gamma OE r \Gamma . The diffusions can be characterized in terms of a martingale problem and they can be considered as a Brown...
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 17 (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.
(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 ...
Strong uniqueness for a class of infinite dimensional Dirichlet operators and applications to stochastic quantization
, 1997
"... Strong and Markov uniqueness problems in L² for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L^psetting is discussed. As a direct application essential self adjointness and strong uniqueness ..."
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Cited by 7 (1 self)
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Strong and Markov uniqueness problems in L² for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L^psetting is discussed. As a direct application essential self adjointness and strong uniqueness in L^p is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume ae R².
Tagged particle process in continuum with singular interactions
, 2008
"... We study the dynamics of a tagged particle in an infinite particle environment. Such processes have been studied in e.g. [GP85], [DMFGW89] and [Osa98]. I.e., we consider the heuristic system of stochastic differential equations ∞X dξ(t) = ∇φ(yi(t))dt + √ 2 dB1(t), t ≥ 0, (TP) dyi(t) = − i=1 P∞ j= ..."
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Cited by 5 (1 self)
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We study the dynamics of a tagged particle in an infinite particle environment. Such processes have been studied in e.g. [GP85], [DMFGW89] and [Osa98]. I.e., we consider the heuristic system of stochastic differential equations ∞X dξ(t) = ∇φ(yi(t))dt + √ 2 dB1(t), t ≥ 0, (TP) dyi(t) = − i=1 P∞ j=1 j=i ∇φ(yi(t) − yj(t)) − ∇φ(yi(t)) − P∞ j=1 ∇φ(yj(t)) + √ 2 d(Bi+1(t) − B1(t)), t ≥ 0