Large Deviations Principles for Langevin Equations in Random Environment and Applications

Abstract: In contrast to the study of Langevin equations in a homogeneous environment in the literature, the study on Langevin equations in randomly-varying environments is relatively scarce. Almost all the existing works require random environments to have a specific formulation that is independent of the systems. This paper aims to consider large deviations principles (LDPs) of Langevin equations involving a random environment that is a process taking value in a measurable space and that is allowed to interact with the systems, without specified formulation on the random environment. Examples and applications to statistical physics are provided. Our formulation of the random environment presents the main challenges and requires new approaches. Our approach stems from the intuition of the Smoluchowski-Kramers approximation. The techniques developed in this paper focus on the relation between the solutions of the second-order equations and the associate first-order equations. We obtain the desired LDPs by showing a family of processes enjoy the exponential tightness and local LDPs with an appropriate rate function.