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... Show more