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

Journal article

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, ...).

Authors: Caruana, M; Friz, P; Oberhauser, H

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• Volume: 28
• Issue: 1
• Pages: 27-46
• Publication date: 2010-11-09
• Acceptance date: 2010-11-04
• DOI: 10.1016/j.anihpc.2010.11.002
• ISSN: 0294-1449
• Keywords: Parabolic viscosity PDEs; Journal Article; Rough path theory; Stochastic PDEs; SBTMR