Preprintgreen2014

Controlled viscosity solutions of fully nonlinear rough PDEs

Massimiliano Gubinelli · Samy Tindel · Iván Torrecilla

16 citations0/0 files storedarXiv (Cornell University)

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Stochastic processes and financial applications Nonlinear Differential Equations Analysis Nonlinear Partial Differential Equations

Abstract

We propose a definition of viscosity solutions to fully nonlinear PDEs driven by a rough path via appropriate notions of test functions and rough jets. These objects will be defined as controlled processes with respect to the driving rough path. We show that this notion is compatible with the seminal results of Lions and Souganidis and with the recent results of Friz and coauthors on fully non-linear SPDEs with rough drivers.

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References

2004Controlling rough paths 2006Rough evolution equations 2010A (rough) pathwise approach to a class of non-linear stochastic partial differential equations 2013Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths 2002Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs 2013Rough path stability of (semi-)linear SPDEs 2016Stochastic control with rough paths 2015Pathwise Taylor expansions for random fields on multiple dimensional paths

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