Partial differential equations driven by rough paths

We study a class of linear first and second order partial differential equations driven by weak geometric p-rough paths, and prove the existence of a unique solution for these equations. This solution depends continuously on the driving rough path. This allows a robust approach to stochastic partial differential equations. In particular, we may replace Brownian motion by more general Gaussian and Markovian noise. Support theorems and large deviation statements all become easy corollaries of the corresponding statements of the driving process. In the case of first order equations with Gaussian noise, we discuss the existence of a density with respect to the Lebesgue measure for the solution.