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

Abstract

We consider non-linear parabolic evolution equations of the form ${\partial }_{t}u=F(t,x,Du,D^{2}u)$, subject to noise of the form $H(x,Du)\circ dB$ where H is linear in Du and $\circ dB$ 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 Sér. 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, …).