Absolute continuity of solutions to reaction-diffusion equations with multiplicative noise

2019/05/21 by Carlo Marinelli, Marinelli, Carlo, Lluís Quer-Sardanyons +1
Computer Science · Economics, Econometrics and Finance · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #math.AP #math.PR

paper · pdf · doi:10.48550/arxiv.1905.08739

17 pages

arxiv created 2019/05/21 · openalex publication_date 2019/05/21 · arxiv updated 2019/05/22 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L²(G), where G is an open bounded domain in ℝ^d with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

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