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, …).

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