Results 1  10
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17
Analysis and Geometry on Configuration Spaces
, 1997
"... In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X ..."
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Cited by 50 (8 self)
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In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X. This geometry is "nonflat" even if X = IR d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient r \Gamma , divergence div \Gamma , and LaplaceBeltrami operator H \Gamma = \Gammadiv \Gamma r \Gamma are constructed. The associated volume elements, i.e., all measures ¯ on \Gamma X w.r.t. which r \Gamma and div \Gamma become dual operators on L 2 (\Gamma X ; ¯), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on \Gamma X . The corresponding Dirichlet...
(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 ...
Stochastic calculus of generalized Dirichlet forms and applications to stochastic differential equations in infinite dimensions
"... this paper we systematically develop a stochastic calculus for generalized Dirichlet forms (cf. [St1]). In particular, we show Fukushima's decomposition of additive functionals and an Itotype formula in this framework. The class of generalized Dirichlet forms is much larger than the wellstudied cl ..."
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Cited by 7 (3 self)
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this paper we systematically develop a stochastic calculus for generalized Dirichlet forms (cf. [St1]). In particular, we show Fukushima's decomposition of additive functionals and an Itotype formula in this framework. The class of generalized Dirichlet forms is much larger than the wellstudied class of symmetric and coercive Dirichlet forms as in [FOT 94] resp. [MR 92] and time dependent Dirichlet forms as in [O 92]. It contains examples of an entirely new kind (cf. Section 6, [St1]). Therefore, the results obtained in this paper lead to extensions of the corresponding results in the "classical" theories. In particular the proofs are "locally" completely different (cf. e.g. Theorem 2.3 and Theorem 2.5; though for the reader's convenience we tried to follow the line of argument in [FOT 94] as closely as possible). This difference has several reasons: First of all we do not assume any sector condition; in certain cases we have to handle Equasilowersemicontinuous
Wellposedness and asymptotic behaviour of nonautonomous linear evolution equations
 A. Lorenzi, B. Ruf (Eds.): “Evolution Equations, Semigroups and Functional Analysis,” Birkhäuser
, 2002
"... We review results on the existence and the long term behaviour of nonautonomous linear evolution equations. Emphasis is put on recent results on the asymptotic behaviour using a semigroup approach. ..."
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Cited by 6 (4 self)
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We review results on the existence and the long term behaviour of nonautonomous linear evolution equations. Emphasis is put on recent results on the asymptotic behaviour using a semigroup approach.
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...
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.
Vector fields on mapping spaces and related Dirichlet forms and diffusions
 OSAKA J. MATH
, 1996
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Dirichlet Forms on InfiniteDimensional "manifoldLike" State Spaces: A Survey of Recent Results and Some Prospects for the Future
, 1996
"... We give a (to some extent pedagogical) survey on recent results about Dirichlet forms on infinitedimensional "manifoldlike" state spaces including path and loop spaces as well as spaces of measures. The latter are associated with interacting FlemingViot processes resp. infinite particle syste ..."
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Cited by 3 (0 self)
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We give a (to some extent pedagogical) survey on recent results about Dirichlet forms on infinitedimensional "manifoldlike" state spaces including path and loop spaces as well as spaces of measures. The latter are associated with interacting FlemingViot processes resp. infinite particle systems. Also some new results, further developing the Dirichlet form approach to infinite particle systems, are enclosed. Finally, a brief summary of other research activities in the theory of Dirichlet forms is given and some prospects for the future are indicated.
Kolmogorov equations in infinite dimensions: wellposedness and regularity of solutions, with applications to stochastic generalized Burgers equations, to appear on The Annals of Probab
"... Abstract. We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time ..."
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Cited by 2 (0 self)
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Abstract. We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of StroockVaradhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic NavierStokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.
Probabilistic Representations and Hyperbound Estimates for Semigroups
, 1998
"... In this paper, we study lower order perturbations of a symmetric second order differential operator generating a hypercontractive semigroup. We give a probabilistic representation of the in general not subMarkovian semigroup associated with the perturbed operator and prove that the perturbed semigr ..."
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In this paper, we study lower order perturbations of a symmetric second order differential operator generating a hypercontractive semigroup. We give a probabilistic representation of the in general not subMarkovian semigroup associated with the perturbed operator and prove that the perturbed semigroup is also hypercontractive under some exponential integrability conditions on the coefficients. 1 Introduction Let X be a locally convex Hausdorff topological vector space which is Souslinean and let (H; !; ?H ) be a Hilbert space continuously and densely embedded into X . Let ¯ be a probability measure on (X; B(X)), where B(X) stands for the Borel oealgebra. The starting point of this paper is a second order symmetric differential operator Lu(x) = 1 2 div(A(x)ru(x)) (1) on L 2 (X; ¯) determined by a quadratic form (E ; D(E)), which satisfies a LogSobolev inequality. This is equivalent to the associated semigroup being hypercontractive (see e.g. [8]). Consider now a lower order pertu...