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43
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 97 (13 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...
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 wellstudi ..."
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Cited by 19 (8 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
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 19 (2 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.
2002a) Wellposedness and asymptotic behaviour of nonautonomous linear evolution equations
 Ruf (Eds.): “Evolution Equations, Semigroups and Functional Analysis,” Birkhäuser
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Kolmogorov equations in infinite dimensions: wellposedness and regularity of solutions, with applications to stochastic generalized Burgers equations
, 2008
"... 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 solu ..."
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Cited by 16 (0 self)
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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.
(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 ...
Generalized covariation and extended Fukushima decompositions for Banach valued processes. Application to windows of Dirichlet processes.
, 2011
"... This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of Métivier and Pellaumail which is quite restrictive. We make use of the notion of χcovariation which is a generalized notion of covariation for p ..."
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Cited by 8 (6 self)
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This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of Métivier and Pellaumail which is quite restrictive. We make use of the notion of χcovariation which is a generalized notion of covariation for processes with values in two Banach spaces B1 and B2. χ refers to a suitable subspace of the dual of the projective tensor product of B1 and B2. We investigate some C 1 type transformations for various classes of stochastic processes admitting a χquadratic variation and related properties. If X 1 and X 2 admit a χcovariation, F i: Bi → R, i = 1,2 are of class C 1 with some supplementary assumptions then the covariation of the real processes F 1 (X 1) and F 2 (X 2) exist. A detailed analysis will be devoted to the socalled window processes. Let X be a real continuous process; the C([−τ,0])valued process X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ,0], is called window process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. In fact we aim to generalize the following properties valid for B = R. If X = X is a real valued Dirichlet process and F: B → R of class C 1 (B) then F(X) is still a Dirichlet process. If X = X is a weak Dirichlet process with finite quadratic variation,
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 L²(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 s ..."
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Cited by 8 (2 self)
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We prove existence and uniqueness for invariant measures of strongly continuous semigroups on L²(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 L²(ae dx). Probabilistic consequences are also briefly discussed.
Markov processes associated with Lpresolvents and appli cations to stochastic differential equations on Hilbert space
"... Abstract. We give general conditions on a generator of a C0semigroup (resp. of a C0resolvent) on Lp(E,µ), p ≥ 1, where E is an arbitrary (Lusin) topological space and µ a σfinite measure on its Borel σalgebra, so that it generates a sufficiently regular Markov process on E. We present a general ..."
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Cited by 7 (4 self)
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Abstract. We give general conditions on a generator of a C0semigroup (resp. of a C0resolvent) on Lp(E,µ), p ≥ 1, where E is an arbitrary (Lusin) topological space and µ a σfinite measure on its Borel σalgebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given.