Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift

• 成果类型: Article
• 署名作者: Fehrman, Benjamin; Gess, Benjamin
• 署名单位: University of Oxford; University of Bielefeld; Max Planck Society
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01207-3
• 发表日期: 2023
• 页码: 573-636
• 关键词: scalar conservation-laws; partial-differential-equations; entropy solutions; kinetic formulation; pathwise solutions; well-posedness; cauchy-problem; uniqueness; rough; principles

摘要

Large deviations of conservative interacting particle systems, such as the zero range process, about their hydrodynamic limit and their respective rate functions lead to the analysis of the skeleton equation; a degenerate parabolic-hyperbolic PDE with irregular drift. We develop a robust well-posedness theory for such PDEs in energy critical spaces based on concepts of renormalized solutions and the equation's kinetic form. We establish these properties by proving that renormalized solutions are equivalent to classical weak solutions, extending concepts of (DiPerna and Lions in Invent. Math. 98(3):511-547, 1989; Ambrosio in Invent. Math. 158(2):227-260, 2004) to the nonlinear setting.The relevance of the results toward large deviations in interacting particle systems is demonstrated by applications to the identification of l.s.c. envelopes of restricted rate functions, to zero noise large deviations for conservative SPDE, and to the G- convergence of rate functions. The first of these solves a long-standing open problem in the large deviations for zero range processes. The second makes rigorous an informal link between the non-equilibrium statistical mechanics approaches of macroscopic fluctuation theory and fluctuating hydrodynamics.