Probability Theory and Related Fields

, Volume 89, Issue 3, pp 347–386

Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms


  • S. Albeverio
    • Institut für MathematikRuhr-Universität Bochum
  • M. Röckner
    • Institut für Angewandte MathematikUniversität Bonn

DOI: 10.1007/BF01198791

Cite this article as:
Albeverio, S. & Röckner, M. Probab. Th. Rel. Fields (1991) 89: 347. doi:10.1007/BF01198791


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 (Xt)t≧0 takes values in some topological vector spaceE and (Wt)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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© Springer-Verlag 1991