The infinitesimal generator of the stochastic Burgers equation

• 成果类型: Article
• 署名作者: Gubinelli, Massimiliano; Perkowski, Nicolas
• 署名单位: University of Bonn; University of Bonn; Free University of Berlin
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00996-5
• 发表日期: 2020
• 页码: 1067-1124
• 关键词: strong uniqueness; hilbert-spaces; kpz equation; sdes; RENORMALIZATION; fluctuations; QUANTIZATION; driven; noise; limit

摘要

We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domainmust have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for energy solutions of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427-471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially L-2-ergodic, and that the stochastic Burgers equation on the real line is ergodic.