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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...
Extending Markov processes in weak duality by Poisson point processes of excursions
 in: Stochastic Analysis and Applications The Abel Symposium 2005 (Eds) F.E. Benth
, 2007
"... Let a be a nonisolated point of a topological space E. Suppose we are given standard processes X 0 and � X 0 on E0 = E \ {a} in weak duality with respect to a σfinite measure m on E0 which are of no killings inside E0 but approachable to a. We first show that their extensions X and � X to E admitt ..."
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Cited by 5 (0 self)
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Let a be a nonisolated point of a topological space E. Suppose we are given standard processes X 0 and � X 0 on E0 = E \ {a} in weak duality with respect to a σfinite measure m on E0 which are of no killings inside E0 but approachable to a. We first show that their extensions X and � X to E admitting no sojourn at a and keeping the weak duality are uniquely determined by the approaching probabilities of X 0, � X 0 and m up to a nonnegative constant δ0 representing the killing rate of X at a. We then construct, starting from X 0, such X by piecing together returning excursions around a and a possible nonreturning excursion including the instant killing. This extends a recent result by M. Fukushima and H. Tanaka [16] which treats the case where X 0, X are msymmetric diffusions and X admits no sojourn nor killing at a. Typical examples of jump type symmetric Markov processes and nonsymmetric diffusions on Euclidean domains are given at the end of the paper. 1
Markov processes with identical bridges
 Electron. J. Probab
, 1998
"... Let X and Y be timehomogeneous Markov processes with common state space E, and assume that the transition kernels of X and Y admit densities with respect to suitable reference measures. We show that if there is a time t> 0 such that, for each x ∈ E, the conditional distribution of (Xs)0≤s≤t, given ..."
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Let X and Y be timehomogeneous Markov processes with common state space E, and assume that the transition kernels of X and Y admit densities with respect to suitable reference measures. We show that if there is a time t> 0 such that, for each x ∈ E, the conditional distribution of (Xs)0≤s≤t, given X0 = x = Xt, coincides with the conditional distribution of (Ys)0≤s≤t, given Y0 = x = Yt, then the infinitesimal generators of X and Y are related by L Y f = ψ −1 L X (ψf) − λf, where ψ is an eigenfunction of L X with eigenvalue λ ∈ R. Under an additional continuity hypothesis, the same conclusion obtains assuming merely that X and Y share a “bridge ” law for one triple (x, t, y). Our work entends and clarifies a recent result of I. Benjamini and S. Lee.
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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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...