Renormalized Solutions to Stochastic Continuity Equations with Rough Coefficients

2017/10/17 by Samuel Punshon‐Smith, Samuel Punshon-Smith, Punshon-Smith, Samuel
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications #math.AP #math.PR

paper · pdf · doi:10.48550/arxiv.1710.06041

42 pages

arxiv created 2017/10/17 · openalex publication_date 2017/10/17 · arxiv updated 2017/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the stochastic continuity equation associated to an Itô diffusion with irregular drift and diffusion coefficients. We give regularity conditions under which weak solutions are renormalized in the sense of DiPerna/Lions, and prove well-posedness in L^p. As an application, we give a new proof of renormalizability (hence uniqueness) of weak solutions to the stochastic continuity equation when the diffusion matrix is constant and the drift only belongs to L^q_tL^p, where (2)/(q) + (n)/(p) <1, without resorting to the regularity of the stochastic flow or a duality method.

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