A BASIC IDENTITY FOR KOLMOGOROV OPERATORS IN THE SPACE OF CONTINUOUS FUNCTIONS RELATED TO RDES WITH MULTIPLICATIVE NOISE

• 成果类型: Article
• 署名作者: Cerrai, Sandra; Da Prato, Giuseppe
• 署名单位: University System of Maryland; University of Maryland College Park; Scuola Normale Superiore di Pisa
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP853
• 发表日期: 2014
• 页码: 1297-1336
• 关键词: reaction-diffusion equations; systems; semigroups

摘要

We consider the Kolmogorov operator associated with a reaction diffusion equation having polynomially growing reaction coefficient and perturbed by a noise of multiplicative type, in the Banach space E of continuous functions. By analyzing the smoothing properties of the associated transition semigroup, we prove a modification of the classical identite du carre des champs that applies to the present non-Hilbertian setting. As an application of this identity, we construct the Sobolev space W-1,W-2 (E; mu), where mu is an invariant measure for the system, and we prove the validity of the Poincare inequality and of the spectral gap.