Partial differential equations driven by rough paths are studied.We return to the investigations of [Caruana, Friz and Oberhauser: A (rough) pathwise approach to a class of non- linear SPDEs, Annales de l’Institut Henri Poincaré/Analyse Non Linéaire 2011, 28, pp. 27–46],motivated by the Lions–Souganidis theory of viscosity solutions for SPDEs.We continue and complement the previous (uniqueness) results with general existence and regularity statements. Much of this is transformed to questions for deterministic parabolic partial differential equations in viscosity sense. On a technical level, we establish a refined parabolic theorem of sums which may be useful in its own right.
Diehl, J., Friz, P. K., & Oberhauser, H. (2014). Regularity theory for rough partial differential equations and parabolic comparison revisited. In Springer Proceedings in Mathematics and Statistics (Vol. 100, pp. 203–238). Springer New York LLC. https://doi.org/10.1007/978-3-319-11292-3_8
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