Regularization by Noise in One-Dimensional Continuity Equation

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...

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Bibliographic Details
Published in:Potential analysis 2019-07, Vol.51 (1), p.23-35
Main Author: Olivera, Christian
Format: Article
Language:English
Subjects:
Cauchy problems
Continuity equation
Dependence
Fields (mathematics)
Functional Analysis
Geometry
Mathematics
Mathematics and Statistics
Potential Theory
Probability Theory and Stochastic Processes
Regularization
Uniqueness
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Summary: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.
ISSN:0926-2601
1572-929X
DOI:10.1007/s11118-018-9700-z