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Abstract: We study a class of linear first and second order partial differentialequations driven by weak geometric $p$-rough paths, and prove the existence ofa unique solution for these equations. This solution depends continuously onthe driving rough path. This allows a robust approach to stochastic partialdifferential equations. In particular, we may replace Brownian motion by moregeneral Gaussian and Markovian noise. Support theorems and large deviationstatements all became easy corollaries of the corresponding statements of thedriving process. In the case of first order equations with Gaussian noise, wediscuss the existence of a density with respect to the Lebesgue measure for thesolution.



Author: Michael Caruana, Peter Friz

Source: https://arxiv.org/







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