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

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by \begin{eqnarray*} 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{eqnarray*} 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 $s(\epsilon )$ goes to zero slower than a prescribed rate as $\epsilon \to 0$, we characterize all weak limit points of $X^{\epsilon }$, 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.