Existence and Smoothness of the Density for the Stochastic Continuity Equation
- 29 Downloads
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.
- 1.Beck, L., Flandoli, F., Gubinelli, M., Maurelli, M.: Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014). Preprint available on arXiv:1401.1530
- 7.Gess, B., Smith, S.: Stochastic continuity equations with conservative noise (2017). arXiv:1710.04906
- 9.Kunita, H.: First order stochastic partial differential equations. In: Stochastic Analysis, vol. 32, pp. 249–269. Katata Kyoto, North-Holland Mathematics Library (1984)Google Scholar
- 10.Lions, P.I.: Mathematical Topics in Fluid Dynamics, vol. I: Incompressible Models. Oxford Lecture Series in Mathematics and Its Applications, p. 3. Oxford University Press, Oxford (1996)Google Scholar
- 11.Lions, P.I.: Mathematical Topics in Fluid Dynamics, Vol. II: Compressible Models. Oxford Lecture Series in Mathematics and Its Applications, p. 10. Oxford University Press, Oxford (1996)Google Scholar