Abstract
In this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter varepsilon such that, for every fixed value of the slow variable, the fast dynamics are sufficiently chaotic with ergodic invariant measure. Convergence of the slow process to the solution of a homogenized stochastic differential equation (SDE) in the limit varepsilon to zero, with explicit formulas for drift and diffusion coefficients, has so far only been obtained for the case that the fast dynamics evolve independently. In this paper we give sufficient conditions for the convergence of the first moments of the slow variable in the coupled case. Our proof is based upon a new method of stochastic regularization and functional-analytical techniques combined via a double limit procedure involving a zero-noise limit as well as considering varepsilon to zero. We also give exact formulas for the drift and diffusion coefficients for the limiting SDE. As a main application of our theory, we study weakly-coupled systems, where the coupling only occurs in lower time scales.
Highlights
Many natural processes can be modeled by systems with two clearly separated sets of variables: a set of variables which evolve rapidly in time and a set of slowly varying variables; see [30] for many examples and techniques in fast-slow systems
Let (X ε(t; ξ, η), Y ε(t; ξ, η)) denote the solution of the ordinary differential equations (ODEs) (1.3) for any ε > 0 and let C0(Rd ) denote the space of continuous functions vanishing at infinity, i.e., as x → ∞
One possible way to weaken this assumption is to consider systems that are not coupled in the strongest possible sense, but for which the coupling occurs in smaller time scales
Summary
Many natural processes can be modeled by systems with two clearly separated sets of variables: a set of variables which evolve rapidly in time (for instance, within milliseconds) and a set of slowly varying variables (for instance, variables for which change is observed after hundreds of years); see [30] for many examples and techniques in fast-slow systems. When the slow variables start to evolve under the influence of the fast dynamics, one observes induced fluctuations In this setting, the method of reduction to a single slow equation is usually called homogenization. A main dynamical assumption is to require ergodicity for the fastest scale, i.e., the ODE y = g(y), y ∈ Rm, generates a flow φt : Rm → Rm with a compact invariant set ⊂ Rm and ergodic invariant probability measure μ supported on Another intrinsic part of this setup is the centering condition b(x, y) dμ(y) = 0, for all x ∈ Rd. Systems of the form (1.1) are called skew products, because they are not coupled but instead the fast variables yε can be described by a separate dynamical system on. The specific proofs will need limits of the respective integrals for the coefficients such that mixing assumptions have to be made; this is the price we pay to show such results for the coupled case
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
