Stochastic partial differential equations: a rough path view

Abstract

We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on RPDEs, our definition gives honest meaning to RPDEs as integral equation, based on which we are able to obtain existence, uniqueness and stability results. The case of weak "rough" forward equations, may be seen as robustification of the (measure-valued) Zakai equation in the rough path sense. Feynman-Kac representation for RPDEs, in formal analogy to similar classical results in SPDE theory, play an important role.

References in corpus (3)

• Non-standard approximations of the Ito-map
• Robust filtering: Correlated noise and multidimensional observation
• Controlled viscosity solutions of fully nonlinear rough PDEs

Cited by in corpus (11)

• The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory
• Stochastic nonlinear Fokker-Planck equations
• Generalized Burgers equation with rough transport noise
• Rough nonlocal diffusions
• Rough semimartingales and $p$-variation estimates for martingale transforms
• Transport and continuity equations with (very) rough noise
• Rough linear transport equation with an irregular drift
• L2 convergence of smooth approximations of Stochastic Differential Equations with unbounded coefficients
• Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering
• Weak solutions of rough path SDE's via Girsanov
• Backward stochastic differential equations with Young drift