Existence and Smoothness of the Density for the Stochastic Continuity Equation
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We consider the stochastic continuity equation driven by Brownian motion. We use the techniques of the Malliavin calculus to show that the law of the solution has a density with respect to the Lebesgue measure. We also prove that the density is Hölder continuous and satisfies some Gaussian-type estimates.
KeywordsContinuity equation Brownian motion Malliavin calculus method of characteristics existence and estimates of the density
Mathematics Subject ClassificationPrimary 60F05 Secondary 60H05 91G70
C. Olivera and C. Tudor acknowledge partial support from the CNRS-FAPESP Grant 267378. C. Olivera is partially supported by FAPESP by the Grants 2017/17670-0 and 2015/07278-0.
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