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.