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

2018/10/08 by Athreya, Siva R., Borkar, Vivek S., Kumar, K. Suresh +1 · 1 citation
#60G35 #60J60 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1810.03585

Abstract

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by dX^ε_t amp;=amp; b(X^ε_t, Y^ε_t)dt + ε^αdB_t, dY^ε_t amp;=amp; - (1)/(ε) ∇_yU(X^ε_t, Y^ε_t)dt + (s(ε))/(√(ε)) dW_t, where B_t, W_t are independent Brownian motions on \mathbb R^d and \mathbb R^m respectively, b : ℝ^d × ℝ^m → ℝ^d, U : ℝ^d × ℝ^m → 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 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.

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