2024/06/06 by Mark van den Bosch, Bosch, Mark van den, Hermen Jan Hupkes +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2406.04232
openalex publication_date 2024/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations. In the deterministic setting, multidimensional stability of planar waves on the whole space \mathbb Rd has been studied by many. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. This is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.