Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms

Abstract

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.