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

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We consider nonlinear parabolic evolution equations of the form ∂ₜu=F(t,x,Du,D²u), subject to noise of the form H(x,Du) ∘ dB where H is linear in Du and ∘ dB denotes the Stratonovich differential of a multidimensional Brownian motion. Motivated by the essentially pathwise results of [Lions, P.-L. and Souganidis, P.E.; Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 9] we propose the use of rough path analysis [Lyons, T. J.; Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 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,...).