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The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 18 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Dirichlet Forms And Markov Processes: A Generalized Framework Including Both Elliptic And Parabolic Cases
"... We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dir ..."
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Cited by 5 (3 self)
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We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dirichlet forms (cf. [Fu2], [M/R]) as well as time dependent Dirichlet forms (cf. [O1]) as special cases and also many new examples. Among these are, e.g. transformations of time dependent Dirichlet forms by ffexcessive functions h (htransformations), Dirichlet forms with time dependent linear drift and fractional diffusion operators. One of the main results is that we identify an analytic property of these forms which ensures the existence of associated strong Markov processes with nice sample path properties, and give an explicit construction for such processes. This construction extends previous constructions of the processes in the elliptic and the parabolic cases, is, in particular, c...
First Order Perturbations Of Dirichlet Operators: Existence And Uniqueness
, 1996
"... We study perturbations of type B \Delta r of Dirichlet operators (L 0 ; D(L 0 )) associated with Dirichlet forms of type E 0 (u; v) = 1=2 R hru; rviH d¯ on L 2 (E; ¯) where E is a finite or infinite dimensional Banach space E. Here H denotes a Hilbert space densely and continuously embed ..."
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Cited by 5 (1 self)
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We study perturbations of type B \Delta r of Dirichlet operators (L 0 ; D(L 0 )) associated with Dirichlet forms of type E 0 (u; v) = 1=2 R hru; rviH d¯ on L 2 (E; ¯) where E is a finite or infinite dimensional Banach space E. Here H denotes a Hilbert space densely and continuously embedded in E. Assuming quasiregularity of (E 0 ; D(E 0 )) we show that there always exists a closed extension of Lu := L 0 u + hB; ruiH that generates a subMarkovian C 0 semigroup of contractions on L 2 (E; ¯) (resp. L 1 (E; ¯)), if B 2 L 2 (E; H;¯) and R hB; ruiH d¯ 0; u 0. If D is an appropriate core for (L 0 ; D(L 0 )) we show that there is only one closed extension of (L; D) in L 1 (E; ¯) generating a strongly continuous semigroup. In particular we apply our results to operators of type \Delta H +B \Delta r, where \Delta H denotes the GrossLaplacian on an abstract Wiener space (E; H; fl) and B = \Gammaid E + v, where v takes values in the CameronMartin s...
Perturbation of symmetric Markov processes
, 2005
"... We present a pathspace integral representation of the semigroup associated with the quadratic form obtained by a lower order perturbation of the L2infinitesimal generator L of a general sym, where D is a metric Markov process. An illuminating concrete example for L is ∆D − (−∆) s D bounded Euclid ..."
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Cited by 5 (4 self)
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We present a pathspace integral representation of the semigroup associated with the quadratic form obtained by a lower order perturbation of the L2infinitesimal generator L of a general sym, where D is a metric Markov process. An illuminating concrete example for L is ∆D − (−∆) s D bounded Euclidean domain in Rd, s ∈]0, 1[, ∆D is the Laplacian operator in D with zero Dirichlet boundary condition and −(−∆) s D is the fractional Laplacian operator in D with zero exterior condition. The strong Markov process corresponding to L is a Lévy process that is the sum of Brownian motion in R d and an independent symmetric (2s)stable process in R d killed upon exiting domain D. This probabilistic representation is a combination of FeynmanKac and Girsanov formulas. Crucial to the development is to use the extension of Nakao’s stochastic integral for zeroenergy additive functionals and the associated Itô formula, both of which were recently developed in [3].
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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Convergence Of Operator Semigroups Generated By Elliptic Operators
"... . Consider a sequence of operator semigroups i T (n) t j t?0 , n 2 N, whose generators are elliptic and are (informally) of type L (n) = P d i;j=1 @ i i a (n) ij @ j + d (n) i j \Gamma P d i=1 b (n) i @ i \Gamma c (n) on a possibly unbounded open set U ae R d with measur ..."
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. Consider a sequence of operator semigroups i T (n) t j t?0 , n 2 N, whose generators are elliptic and are (informally) of type L (n) = P d i;j=1 @ i i a (n) ij @ j + d (n) i j \Gamma P d i=1 b (n) i @ i \Gamma c (n) on a possibly unbounded open set U ae R d with measurable coefficients. Under weak assumptions on the coefficients we prove strong convergence of i T (n) t j t?0 , n 2 N, on L 2 (U ; dx) resp. the Sobolev space H 1;2 0 (U ; dx). In particular, this is done without assuming that b (n) i ; d (n) i ; c (n) , are bounded in L 1 (U ; dx) uniformly in n. 1. Introduction and main results Let U ae R d , d 3, U open (not necessarily bounded), and let dx denote Lebesgue measure on U . Below all functions are supposed to be realvalued. Let a (n) ij , b (n) i , d (n) i , c (n) 2 L 1 loc (U ; dx), 1 i; j d, n 2 N [ f1g satisfying the following conditions: (1.1) There exists ffi 2]0; 1[ such that for all n 2 N [ f1g an...