CENTRAL LIMIT THEOREM FOR GIBBS MEASURES ON PATH SPACES INCLUDING LONG RANGE AND SINGULAR INTERACTIONS AND HOMOGENIZATION OF THE STOCHASTIC HEAT EQUATION

• 成果类型: Article
• 署名作者: Mukherjee, Chiranjib
• 署名单位: University of Munster
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1727
• 发表日期: 2022
• 页码: 2028-2062
• 关键词: brownian occupation measures; mean-field interaction; Local Time; uniqueness; polaron; regime

摘要

We consider a class of Gibbs measures defined with respect to increments {omega(t) - omega(s)}(s 0) and unbounded (singular) interactions (including singularities of the form x bar right arrow 1/vertical bar x vertical bar(p) in d >= 3 or x bar right arrow delta(0)(x) in d = 1) attached to the space variables. These assumptions on the interaction seem to be sharp and cover quantum mechanical models like the Nelson model and the polaron problem with ultraviolet cut off (both carrying bounded spatial interactions with power law decay in time) as well as the Frohlich polaron with a short range interaction in time but carrying Coulomb singularity in space. In this set up, we develop a unified approach for proving a central limit theorem for the rescaled process of increments for any coupling parameter and obtain an explicit expression for the limiting variance, which is strictly positive. As a further application, we study the solution of the multiplicative-noise stochastic heat equation in spatial dimensions d >= 3. When the noise is mollified both in time and space, we show that the averages of the diffusively rescaled solutions converge pointwise to the solution of a diffusion equation whose coefficients are homogenized in this limit.