2023/04/06 by Aurélien Deya, Reika Fukuizumi, Deya, Aurélien +3
Economics, Econometrics and Finance · Physics and Astronomy · #Analysis of PDEs (math.AP) #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2304.03114
openalex publication_date 2023/04/06 · openalex created_date 2023/04/09 · openalex updated_date 2026/07/28
The study is devoted to the interpretation and wellposedness of the stochastic NLS model (\imath ∂t-Δ)u=|u|2+B, u0=0, t∈ ℝ, x∈ \mathbbT, where B stands for a space-time fractional noise with index H=(H0,H1) in a subset of (0,1)2. We first establish that in the situation where 0<2H0+H1≤ 2, the equation cannot be interpreted in a (classical) functional sense. \indent Our investigations then focus on the rough regime corresponding to the condition \frac74<2H0+H1≤ 2. In this specific case, we exhibit an explicit renormalization procedure allowing to restore the (local) convergence of the approximated solutions. We follow a pathwise-type approach emphasizing the distinction between the stochastic objects at the core of the dynamics and the general deterministic machinery.