Perron’s method for pathwise viscosity solutions

We use Perron’s method to construct viscosity solutions of fully non-linear 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 non-trivially 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.