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

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Abstract

We consider non-linear parabolic evolution equations of the form δtu=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 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, ...). © 2010 Elsevier Masson SAS. All rights reserved.

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Caruana, M., Friz, P. K., & Oberhauser, H. (2011). A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Annales de l’Institut Henri Poincare (C) Analyse Non Lineaire, 28(1), 27–46. https://doi.org/10.1016/j.anihpc.2010.11.002

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