i will prove aymptotic results (large deviation and laplace's method) for the laws of solutions of (formal stochastic) differential equations in the rough path sense. my result can be regarded as a "rough path version" of famous results for finite dimensional sdes. however, formulated on a general banach space, my results contain something new. examples include: (1) solutions of sdes on m-type 2 banach spaces, and (2) heat processes (or heat kernel measures) on loop spaces.