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

摘要

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by where B-t, W-t are independent Brownian motions on R-d and R-m respectively, b : R-d xR(m) -> R-d, U : R-d xR(m) -> R and s : (0, infinity) -> (0, infinity). We impose regularity assumptions on b, U and let 0 0, we characterize all weak limit points of X-epsilon, as epsilon -> 0, as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of U(x, center dot) at its global minima we characterize all limit points as Filippov solutions to the differential equation.