Moderate Deviations for the Langevin Equation with Strong Damping

Abstract

In this paper, we establish a moderate deviation principle for the Langevin dynamics with strong damping. The weak convergence approach plays an important role in the proof. Keyword: Stochastic Langevin equation Large deviations Moderate deviations.MSC: 60H10 60F10.

Exponential tightness of a family of Skorohod integrals

“…ε } ε>0 , which is in fact a family of Skorohod integrals. By applying our results, we can obtain the exponential tightness of {F(1) ε } ε>0 under certain conditions. First, it is readily seen that λ(ξε(r))dr g(ξ ε (s))ds ≤ cε 2 .ECP 27 (2022), paper 1.Exponential tightness of a family of Skorohod integralsOn the other hand, we haveD t e − 1 ε λ(ξε(r))dr g(ξ ε (s)) = λ(ξε(r))dr g(ξ ε (s))D t λ(ξ ε (r))dr + e − λ(ξε(r))dr g (ξ ε (s))D t ξ ε (s) = − ε −2 e − λ(ξε(r))dr g(ξ ε (s)) (ξ ε (r))D t ξ ε (r)dr + e − λ(ξε(r))dr g (ξ ε (s))D t ξ ε (s).…”

“…Moreover, λ(x) ≥ κ 0 > 0, ∀x. In many problems in mathematical physics such as Langevin equations, stochastic acceleration, we need to deal with this family and establish its tightness (to obtain the limit behavior, the large deviations principle, the averaging principle, etc); see e.g., [1,2,9] and references therein. In general, such a term is often related to 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].…”

“…For example, if we consider the case where ξ ε (t) is continuously differentiable (or piecewise continuously differentiable), Cerrai and Freidlin [2] have tried to interpret in the pathwise sense. But this approach is no longer valid without the regularity of the random environment and also we cannot cancel out effectively the large factor 1 Exponential tightness of a family of Skorohod integrals to provide estimates in probability; see the details in [1,2,9]. Another approach in [10] is to decompose λ(ξ ε (s)) into two parts, one of them is adapted and the other is "controllable".…”

“…ε } ε>0 . The challenge, which we now should focus more on, is to handle the family {F Therefore, by applying Theorem 1.2, under certain condition on Dξ ε L 2 (Ω,L 2 ([0,1] 2 )) , we can obtain the exponential tightness for {F (1) ε } ε>0 with some speed v 1 (ε). To be clear, let us state an explicit result as the following theorem, which follows immediately from Theorem 1.2.…”

“…Theorem 4.1. Assume that the family of random environments ξ ε is such that there are constants κ 1 > 1/2, κ 2 ≥ 0 satisfying Dξ ε L p (Ω,L 2 ([0,1] 2 )) ≤ cε κ1 p κ2 , ∀ε > 0, p ≥ 1, for some universal constant c. The family {F (1) ε } ε>0 is exponentially tight with the speed v 1 (ε) = ε α for any α satisfying α < κ 1 − 1/2 5 + κ 2 .…”