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Construction of Diffusions on Configuration Spaces
"... We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Ga ..."
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We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Gamma E ; ¯) for a large class of measures ¯ on \Gamma E (without assuming an integration by parts formula) generalizing all corresponding results known so far. Subsequently, we prove that under mild conditions the Dirichlet forms E \Gamma ¯ are quasiregular, and that hence there exist associated diffusions on \Gamma E , provided E is a complete separable metric space and \Gamma E is equipped with a suitable topology, which is the vague topology if E is locally compact. We discuss applications to the case where E is a finite dimensional manifold yielding an existence result on diffusions on \Gamma E which was already announced in [AKR96a, AKR96b], resp. used in [AKR98, AKR97b]. Furthermore...
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...
Stochastic Cohomology of the Frame Bundle of the Loop Space
, 1997
"... We study the differential forms over the frame bundle of the based loop space. They are stochastics in the sense that we put over this frame bundle a probability measure. In order to understand the curvatures phenomena which appear when we look at the Lie bracket of two horizontal vector fields, we ..."
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We study the differential forms over the frame bundle of the based loop space. They are stochastics in the sense that we put over this frame bundle a probability measure. In order to understand the curvatures phenomena which appear when we look at the Lie bracket of two horizontal vector fields, we impose some regularity assumptions over the kernels of the differential forms. This allows us to define an exterior stochastic differential derivative over these forms.