Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence

TL;DR

This paper reviews the theory of pathwise weak solutions for fully nonlinear PDEs with multiplicative rough time dependence, focusing on well-posedness and applications in stochastic and deterministic settings.

Contribution

It provides a comprehensive overview of the well-posedness of pathwise solutions for complex nonlinear PDEs with rough time dependence, including new insights into their applications.

Findings

Established well-posedness of pathwise solutions for nonlinear PDEs with rough time dependence.

Connected weak solutions to classical viscosity and entropy solutions in regular cases.

Discussed applications to stochastic control and conservation laws.

Abstract

The notes are an overview of part of the theory of pathwise weak solutions to two classes of scalar fully nonlinear first- and second-order degenerate parabolic partial differential equations with multiplicative rough time dependence, a special case being Brownian. These are Hamilton-Jacobi-Isaacs-Bellman and quasilinear divergence form equations including multi-dimensional scalar conservation laws. If the time dependence is `regular', the weak solutions are respectively the viscosity and entropy/kinetic solutions. The results presented here are about the wellposedness of the solutions. Some concrete applications are also discussed. The material for the first class of problems are part of the ongoing development of the theory in collaboration with P.-L. Lions.