Perron's method for pathwise viscosity solutions

TL;DR

This paper develops a method to construct viscosity solutions for complex nonlinear degenerate parabolic equations driven by rough paths, using Perron's method and comparison principles, applicable to multidimensional geometric rough noise.

Contribution

It introduces an intrinsic Perron-based approach for pathwise viscosity solutions that does not rely on smooth approximations, extending to multidimensional geometric rough paths.

Findings

Established existence of solutions via Perron's method.

Extended the comparison principle to pathwise equations.

Summarized classes of equations with proven solutions.

Abstract

We use Perron's method to construct viscosity solutions of fully nonlinear degenerate parabolic pathwise (rough) partial differential equations. This provides an intrinsic method for proving the existence of solutions that relies only on a comparison principle, rather than considering equations driven by smooth approximating paths. The result covers the case of multidimensional geometric rough path noise, where the noise coefficients depend nontrivially on space and on the gradient of the solution. Also included in this note is a discussion of the comparison principle and a summary of the pathwise equations for which one has been proved.