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.

Large Deviations of Kac's Conservative Particle System and Energy Non-Conserving Solutions to the Boltzmann Equation: A Counterexample to the Predicted Rate Function

“…The analysis is somewhat different in the case where the dynamics are driven by diffusive rather than jump noise, see the discussion in Léonard [31]. In this context let us mention the recent works [3,10,11,40].…”

Exponential tightness of a family of Skorohod integrals

“…In general, such a term is often obtained from the solution of a second-order stochastic differential equations in random environment or in the setting of fast-slow second-order system; see e.g., [10].…”

“…But this approach is no longer valid without the regularity of the random environment and also we can not cancel out effectively the large factor 1 ε 2 to provide estimates in probability; see the details in [1,2,9]. Another approach in [10] is to decompose λ(ξ(s)) into two parts, one of them is adapted and the other is "controllable". However, this approach requires the decay of the derivative of λ to control such decomposition.…”