Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions
Applied Mathematics and Optimization
We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by dXtε=b(Xtε,Ytε)dt+εαdBt,dYtε=-1ε∇yU(Xtε,Ytε)dt+s(ε)εdWt,where Bt, Wt are independent Brownian motions on Rd and Rm respectively, b: Rd× Rm→ Rd, U: Rd× Rm→ R and s: (0 , ∞) → (0 , ∞). 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 Xε, 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.
Athreya, Siva R.; Borkar, Vivek S.; Kumar, K. Suresh; and Sundaresan, Rajesh, "Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions" (2021). Journal Articles. 1961.
Open Access, Green