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 by $$\begin{aligned} dX^{\varepsilon }_t= & {} b(X^{\varepsilon }_t, Y^{\varepsilon }_t)dt + \varepsilon ^{\alpha }dB_t, \\ dY^{\varepsilon }_t= & {} - \frac{1}{\varepsilon } \nabla _yU(X^{\varepsilon }_t, Y^{\varepsilon }_t)dt + \frac{s(\varepsilon )}{\sqrt{\varepsilon }} dW_t, \end{aligned}$$where \(B_t, W_t\) are independent Brownian motions on \({\mathbb R}^d\) and \({\mathbb R}^m\) respectively, \(b : \mathbb {R}^d \times \mathbb {R}^m \rightarrow \mathbb {R}^d\), \(U : \mathbb {R}^d \times \mathbb {R}^m \rightarrow \mathbb {R}\) and \(s :(0,\infty ) \rightarrow (0,\infty )\). We impose regularity assumptions on b, U and let \(0< \alpha < 1.\) When \(s(\varepsilon )\) goes to zero slower than a prescribed rate as \(\varepsilon \rightarrow 0\), we characterize all weak limit points of \(X^{\varepsilon }\), as \(\varepsilon \rightarrow 0\), as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of \(U(x, \cdot )\) at its global minima we characterize all limit points as Filippov solutions to the differential equation.