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Regularization by Noise in One-Dimensional Continuity Equation

Abstract

A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the vector field has low regularity, in which the classical DiPerna-Lions-Ambrosio theory of uniqueness of distributional solutions does not apply. We solve partially the open problem that is the case when the vector-field has random dependence. In addition, we prove a stability result for the solutions.

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Acknowledgements

Christian Olivera is partially supported by FAPESP by the grants 2017/17670-0. and 2015/07278-0.

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Correspondence to Christian Olivera.

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Olivera, C. Regularization by Noise in One-Dimensional Continuity Equation. Potential Anal 51, 23–35 (2019). https://doi.org/10.1007/s11118-018-9700-z

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Keywords

  • Stochastic partial differential equation
  • Continuity equation
  • Regularization by noise
  • Itô-Wentzell-Kunita formula
  • Low regularity

Mathematics Subject Classification (2010)

  • 60H15
  • 35R60
  • 35F10
  • 60H30