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Small noise limit for singularly perturbed diffusion Athreya, Siva


We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by \begin{eqnarray*} X^{\varepsilon}_t &=& x_0 + \int_0^t b(X^{\varepsilon}_s, Y^{\varepsilon}_s)ds + \varepsilon^{\alpha}B_t, \label{ex} \\ Y^{\varepsilon}_t &=& y_0 - \frac{1}{\varepsilon} \int_0^t \nabla_yU(X^{\varepsilon}_s, Y^{\varepsilon}_s)ds+ \frac{s(\varepsilon)}{\sqrt{\varepsilon}} W_t,\label{wye} \end{eqnarray*} where $x_0 \in {\mathbb R}^d, y_0 \in {\mathbb R}^m$, $B_t, W_t$ are independent Brownian motions, $b : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}^d$, $U : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}$, and $s : (0, \infty) \rightarrow (0, \infty).$ One observes that there is a time scale separation between $X$ and $Y$. Under suitable assumptions on $b, U$, for $0 < \alpha < \frac{1}{2}$, if $s(\epsilon) \rightarrow 0$ goes to zero at a prescribed slow enough rate then we establish all weak limits points of $X^{\epsilon}$, as $\epsilon \rightarrow 0$, as Fillipov solutions to a differential inclusion. This is joint work with V. Borkar, S. Kumar and R. Sundaresan.

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