STOCHASTIC DIFFERENTIAL-EQUATIONS IN INFINITE DIMENSIONS - SOLUTIONS VIA DIRICHLET FORMS

• 成果类型: Article
• 署名作者: ALBEVERIO, S; ROCKNER, M
• 署名单位: University of Bonn
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01198791
• 发表日期: 1991
• 页码: 347-386
• 关键词: global markov property; topological vector-spaces; quantum-field; euclidean fields; uniqueness

摘要

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 beta. Here (X(t))t greater-than-or-equal-to 0 takes values in some topological vector space E and (W(t))t greater-than-or-equal-to 0 is an E-valued Brownian motion. We give applications in detail to (infinite volume) quantum fields where beta is e.g. a renormalized power of a Schwartz distribution. In addition, we present a new approach to the case of linear beta 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.