Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients

This paper introduces path derivatives, in the spirit of Dupire’s functional Itô calculus, for controlled rough paths in rough path theory with possibly non-geometric rough paths. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients are less regular in the time variable, which is not covered in the existing literature. The results are useful for studying viscosity solutions of stochastic PDEs.