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Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms
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  • Published: September 1991

Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms

  • S. Albeverio1 &
  • M. Röckner2 

Probability Theory and Related Fields volume 89, pages 347–386 (1991)Cite this article

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Summary

Using the theory of Dirichlet forms on topological vector spaces we construct solutions to stochastic differential equations in infinite dimensions of the type

$$dX_t = dW_t + \beta (X_t )dt$$

for possibly very singular drifts β. Here (X t ) t ≧0 takes values in some topological vector spaceE and (W t ) t ≧0 is anE-valued Brownian motion. We give applications in detail to (infinite volume) quantum fields where β is e.g. a renormalized power of a Schwartz distribution. In addition, we present a new approach to the case of linear β which is based on our general results and second quantization. We also prove new results on general diffusion Dirichlet forms in infinite dimensions, in particular that the Fukushima decomposition holds in this case.

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Authors and Affiliations

  1. Institut für Mathematik, Ruhr-Universität Bochum, W-4630, Bochum 1, Federal Republic of Germany

    S. Albeverio

  2. Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, W-5300, Bonn 1, Federal Republic of Germany

    M. Röckner

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  1. S. Albeverio
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Albeverio, S., Röckner, M. Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Th. Rel. Fields 89, 347–386 (1991). https://doi.org/10.1007/BF01198791

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  • Received: 04 December 1989

  • Revised: 09 April 1991

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01198791

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Keywords

  • Differential Equation
  • Vector Space
  • Stochastic Process
  • Brownian Motion
  • Probability Theory
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