A (rough) pathwise approach to a class of non-linear stochastic partial differential equations

We consider non-linear parabolic evolution equations of the form partial derivative(t)u = F(t, x, Du, D(2)u), subject to noise of the form H(x, Du) o dB where H is linear in Du and odB denotes the Stratonovich differential of a multi-dimensional Brownian motion. Motivated by the essentially pathwise results of [P.-L. Lions, P.E. Souganidis, Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Ser. I Math. 326 (9) (1998) 1085-1092] we propose the use of rough path analysis [T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215-310] in this context. Although the core arguments are entirely deterministic, a continuity theorem allows for various probabilistic applications (limit theorems, support, large deviations, ...). (c) 2010 Elsevier Masson SAS. All rights reserved.