Probability Theory and Related Fields

, Volume 89, Issue 3, pp 347–386 | Cite as

Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms

  • S. Albeverio
  • M. Röckner


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.


Differential Equation Vector Space Stochastic Process Brownian Motion Probability Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • S. Albeverio
    • 1
  • M. Röckner
    • 2
  1. 1.Institut für MathematikRuhr-Universität BochumBochum 1Federal Republic of Germany
  2. 2.Institut für Angewandte MathematikUniversität BonnBonn 1Federal Republic of Germany

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