Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions

Abstract

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described bywhere Bt, Wt are independent Brownian motions on R d and R m respectively, b :. We impose regularity assumptions on b, U and let 0 < α < 1. When s(ε) goes to zero slower than a prescribed rate as ε → 0, we characterize all weak limit points of X ε , as ε → 0, as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of U (x, ·) at its global minima we characterize all limit points as Filippov solutions to the differential equation.Laplace's principle. Consequently, for this class of U , we show that every limit point is a generalized Filippov solution to (3) driven by a vector field (see Theorem 1.6). In Section 1.1 we state the model, assumptions made, and the two results precisely and in Section 1.2 we discuss examples of U that satisfy the required assumptions.The factor 1 ε in the drift term in (2), intuitively suggests that the Y ε process is the "fast moving" process as ε → 0 and that the "slow moving" process X ε will see an averaging of Y in this limit. The study of averaging principle in various dynamical systems dates back to the work of Khasminskii and others, summarized in, e.g., Freidlin and Wentzell [FW12], Kabanov and Pergamenshchikov [KP03]. The dynamical systems considered there involve a "slow process" X ε as a solution to an ordinary differential equation (i.e. (1) with no B t term) coupled with the fast process Y ε given by a stochastic differential equation with absence of small noise (i.e. (2) with s(ε) = 1). In this setting, under further assumptions on b, U , the averaging principle leading to characterization of limit points, normal deviations, and large deviations from the averaging principle are detailed in [FW12, Chapter 7]. The ground work for this lies in understanding the long-term behavior of solutions to (2) (for fixed ε > 0), and is laid out in [FW12, Chapters 4-6]. We shall rely on this foundation in prescribing assumptions for U in our main results.Large deviations and generalizations to "full dependence" systems were considered in the works of Veretennikov in [Ver13,Ver94,Ver99]. Motivated by questions from homogenization, [Ver00] considered the fast process (2) with s(ε) = 1 but with presence of small noise for the slow process (i.e. (1) with α = 1 2 ) and established a large deviation principle (LDP) for X ε as ε → 0. One can characterize the limit points of X ε as ε → 0 as the set where the rate function is equal to zero (see [Ver00, Remark 3]). In [Lip96], Liptser considered the joint distribution of the slow process and of the empirical process associated with the fast variable in the one-dimensional setting and derived an LDP. This was recently generalized to multidimensional and full dependence systems by Puhalskii in [Puh16]. The diffusions driving the slow and the fast processes in [Puh16] do not have to be uncorrelated.

Backward multivalued McKean-Vlasov SDEs and associated variational inequalities

“…Very recently, the author [24] improved the result in [10] to the case of general A 1 , A 2 and θ = 1 2 . For the system (1), as far as we know, there are not related average principle results. However, these systems have appeared as models for many fields, such as biology and chemistry (cf.…”

“…Freidlin-Wentzell LDPs have been established for many equations, such as multiscale SDEs ( [12,18,21,23,37,38]), multiscale multivalued SDEs ( [22,24]), multivalued McKean-Vlasov SDEs ( [2,13]) and multiscale McKean-Vlasov SDEs ([4, 6, 14, 16]). For multiscale multivalued McKean-Vlasov stochastic systems like (1), as far as we know, there are not related LDP results. Therefore, we study the LDP for the system (1) with θ = 1 2 .…”

“…If A 1 = ∂ϕ, A 2 = ∂ψ, where ϕ, ψ are lower semicontinuous convex functions on R n , R m , respectively, and ∂ϕ, ∂ψ are corresponding subdifferential operators (cf. Example 2.1), θ = 0 and b 1 , σ 1 , b 2 , σ 2 are independent of the distribution in the system (1), for different σ 1 Chen and Wu [10] established corresponding average principles. Very recently, the author [24] improved the result in [10] to the case of general A 1 , A 2 and θ = 1 2 .…”

“…Example 2.1), θ = 0 and b 1 , σ 1 , b 2 , σ 2 are independent of the distribution in the system (1), for different σ 1 Chen and Wu [10] established corresponding average principles. Very recently, the author [24] improved the result in [10] to the case of general A 1 , A 2 and θ = 1 2 . For the system (1), as far as we know, there are not related average principle results.…”

“…However, these systems have appeared as models for many fields, such as biology and chemistry (cf. [3,17]), which motivates us to study the system (1). First of all, we establish the well-posedness for general multivalued McKean-Vlasov stochastic systems under non-Lipschitz conditions.…”