2019/06/30 by Christian Hamster, Hermen Jan Hupkes · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Economics, Econometrics and Finance · Mathematics · #Artificial intelligence #Brownian motion #Computer science #Diffusion and Search Dynamics #Diffusion process #Geometric Brownian motion #Innovation diffusion #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Multiplicative function #Multiplicative noise #Noise (video) #Nonlinear Dynamics and Pattern Formation #Partial differential equation #Physics #Reaction–diffusion system #Statistical physics #Statistics #Stochastic differential equation #Stochastic partial differential equation #Stochastic process #Stochastic processes and financial applications #Term (time) #White noise #Wiener process #math.AP #msc:35K57 #msc:35R60
paper · pdf · doi:10.1016/j.physd.2019.132233
openalex publication_date 2019/10/22 · arxiv created 2020/03/06 · arxiv updated 2020/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Inspired by applications, we consider reaction-diffusion equations on ℝ that are stochastically forced by a small multiplicative noise term that is white in time, coloured in space and invariant under translations. We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in [Hamster & Hupkes 2017] and [Hamster & Hupkes 2018] for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed. We illustrate our approach by two fully worked-out examples, which highlight the predictive power of our expansions.