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12
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 19 (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...
Quasiregular Dirichlet forms: Examples and counterexamples
, 1993
"... We prove some new results on quasiregular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of exam ..."
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Cited by 14 (7 self)
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We prove some new results on quasiregular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of examples including cases with possibly degenerate (sub)elliptic part, diffusions on loops spaces, and certain FlemingViot processes.
On the Local Property for Positivity Preserving Coercive Forms
, 1995
"... . We show that, under mild conditions, two wellknown definitions for the local property of a Dirichlet form are equivalent. We also show that forms that come from di#erential operators are local. 1991 AMS Subject Classification: 31C25 The purpose of this paper is to clarify the relationship between ..."
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Cited by 8 (1 self)
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. We show that, under mild conditions, two wellknown definitions for the local property of a Dirichlet form are equivalent. We also show that forms that come from di#erential operators are local. 1991 AMS Subject Classification: 31C25 The purpose of this paper is to clarify the relationship between two di#erent notions of locality that have appeared in the literature of Dirichlet forms. The first is a slightly modified version of the definition of locality found in the book of Bouleau and Hirsch [BH 91; Chapter I, Corollary 5.1.4], while the second comes from the book of Ma and Rockner [MR 92; Chapter V, Proposition 1.2]. But here we do not assume that the form satisfies any normal contraction property, but only that it is positivity preserving (see Definition 0.1 below). Let (E, F , m) be a measure space, and suppose (E , D(E)) is a densely defined, closed, bilinear form on L 2 (E, F , m). Following [MR 92], we call such a form (E , D(E)) coercive if E(u, u) # 0 for all u ...
Generalized Dirichlet forms and associated Markov processes
 C.R. Acad. Paris
, 1994
"... We prove that for a certain class of bilinear forms satisfying some regularity conditions which include quasiregular Dirichlet forms (cf. [3]) and time dependent Dirichlet forms (cf. [5]) as particular cases there exists an associated strong Markov process having nice sample path properties. These ..."
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Cited by 6 (2 self)
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We prove that for a certain class of bilinear forms satisfying some regularity conditions which include quasiregular Dirichlet forms (cf. [3]) and time dependent Dirichlet forms (cf. [5]) as particular cases there exists an associated strong Markov process having nice sample path properties. These forms, called generalized Dirichlet forms, are the sum of a coercive part and a perturbation (e.g. the time derivative in the time dependent case), so that in general neither the sector condition is fulfilled by the sum nor is the associated L 2 semigroup analytic. A wide variety of new examples can be treated in this extended framework of Dirichlet forms including fractional diffusion operators and transformations of time dependent Dirichlet forms by ffexcessive functions h (htransformations). Formes de Dirichlet g'en'eralis'ees et processus de Markov associ'es R'esum'e  Nous construisons des processus standard sp'eciaux associ'es `a certaines formes bilin'eaires qui satisfont `a quel...
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...
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.
Approximation of arbitrary Dirichlet processes by Markov chains
"... We prove that any Hunt process on a Hausdorff topological space associated with a Dirichlet form can be approximated by a Markov chain in a canonical way. This also gives a new and "more explicit" proof for the existence of Hunt processes associated with strictly quasiregular Dirichlet forms on gene ..."
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Cited by 1 (0 self)
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We prove that any Hunt process on a Hausdorff topological space associated with a Dirichlet form can be approximated by a Markov chain in a canonical way. This also gives a new and "more explicit" proof for the existence of Hunt processes associated with strictly quasiregular Dirichlet forms on general state spaces. AMS Subject Classification Primary: 31 C 25 Secondary: 60 J 40, 60 J 10, 60 J 45, 31 C 15 Key words: Dirichlet forms, Markov chains, Poisson processes, tightness, Hunt processes Running head: Approximation of Dirichlet processes 1) Institute of Applied Mathematics, Academia Sinica, Beijing 100080, China 2) Fakultat fur Mathematik, Universitat Bielefeld, Postfach 100131, 33501 Bielefeld, Germany 3) Faculty of Engineering, HSH, Skaregt 103, 5500 Haugesund, Norway 1 Introduction In the last few years the theory of Dirichlet forms on general (topological) state spaces has been used to construct and analyze a number of fundamental processes on infinitedimensional "manif...