A class of Langevin equations with Markov switching involving strong damping and fast switching

Abstract

This work is devoted to a class of Langevin equations involving strong damping and fast Markov switching. Modeling using continuous dynamics and discrete events together with their interactions much enlarged the applicability of Langevin equations in a random environment. Strong damping and fast switching are characterized by the use of multiple small parameters, resulting in singularly perturbed systems. The motivation of our work stems from the reduction of complexity for complex systems. Under suitable conditions, it is established that the solutions of the Langevin equations satisfy a large deviations principle. Then, we apply our results to statistical physics problems of a small particle in time-inhomogeneous environment and low temperature. Some connections to other fields in physics are also given.

Exponential tightness of a family of Skorohod integrals

“…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].…”

“…By using the variation of parameter formula (see e.g., [2]), we can obtain explicitly the diffusion part of x ε . Dealing with this part requires the treatment of the family {F ε } ε>0 defined as in (4.1); see e.g., [2,9,10]. The non-adaptedness of e − 1 ε 2 1 0 λ(ξε(r))dr is a main challenge here because we cannot move it inside the stochastic integral in Itô's sense and estimates for martingales are no longer valid.…”

“…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 But this approach is no longer valid without the regularity of the random environment and also we cannot cancel out effectively the large factor 1 Exponential tightness of a family of Skorohod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".…”

“…In Polish space, the exponential tightness is a necessary condition for the large deviations principle (with inf-compact rate functions) and implies the large deviations relative compactness (i.e., every sub-sequence contains another sub-sequence satisfying the large deviations principle with some rate function); see e.g., [3]. Hence, this property has a crucial role in the large deviations theory, for example, see [4,8,9,13]. Moreover, since it provides a kind of "exponential tail estimates for the tightness", it is also very interesting in its own right in stochastic analysis.…”

Large Deviations Principles for Langevin Equations in Random Environment and Applications

“…When µ = ε 2 , this equation becomes (1.1). Much effort is devoted to the study of equation (1.1) and its applications; see e.g., [4,5,6,11,23] and references therein. While a time-homogeneous environment is usually used with the force field b not depending on any other random process, we consider a randomly-varying environment in this work.…”