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

By M. Caruana, Peter K. Friz, Harald Oberhauser. 2009

108 citations recorded by the source corpus.

We consider nonlinear 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 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 S\'{e}r. 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, ...).

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Citations in the corpus

A priori estimates for rough PDEs with application to rough conservation laws (2016, methodology): “Note that this rough parabolic model (when z stands for a p-rough path, wih p ∈ [2, 3)) has already been considered in the literature (see for instance [8]).”
Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency (2014, methodology): “1) in the recent past, with several motivations for its study: • It is one of the basic stochastic partial differential equations (PDEs) one might wish to solve, either by extending Itô’s theory [16, 41] or by pathwise techniques [10, 22].”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, unknown): “with  fully non-linear but  = (1    ) linear (with respect to the derivatives of ); see [120] (also for exercise 17.”
Malliavin Calculus for Fractional Delay Equations (2009, methodology): “thanks to rough paths techniques, either by smart transformations allowing to use limiting arguments [3, 4, 8] or by an adaptation of the rough paths formalism to evolution equations [6, 7, 10].”