Rough stochastic PDEs

Abstract

In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too-high spatial roughness for classical analytical methods to apply. In fact, the class of SPDEs that we consider is genuinely ill-posed in the sense that different approximations to the nonlinearity may converge to different limits. Using rough path theory, a pathwise notion of solution to these SPDEs is formulated, and we show that this yields a well-posed problem that is stable under a large class of perturbations, including the approximation of the rough-driving noise by a mollified version and the addition of hyperviscosity. We also show that under certain structural assumptions on the coefficients, the SPDEs under consideration generate a reversible Markov semigroup with respect to a diffusion measure that can be given explicitly. © 2011 Wiley Periodicals, Inc.