2022/04/06 by Nicolas Schaeffer, Schaeffer, Nicolas
Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2204.02808
openalex publication_date 2022/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we study a d-dimensional stochastic quadratic nonlinear Schrödinger equation (SNLS), driven by a fractional derivative (of order -α<0) of a space-time white noise: \ i∂t u-Δu= ρ2 |u|2 + ⟨ ∇ ⟩-αW , t∈ [0,T] , x∈ ℝd ,
u0 = ϕ ,. where ρ:ℝd → ℝ is a smooth compactly-supported function. When α< (d)/(2), the stochastic convolution is a function of time with values in a negative-order Sobolev space and the model has to be interpreted in the Wick sense by means of a time-dependent renormalization. When 1≤ d ≤ 3, combining both the classical Strichartz estimates and a deterministic local smoothing, we establish the local well-posedness of (SNLS) for a small range of α, in the spirit of \citeSchaeffer1. Then, we revisit our arguments and establish multilinear smoothing on the second order stochastic term. This allows us to improve our local well-posedness result for some α. We point out that this is the first result concerning a Schrödinger equation on ℝd driven by such an irregular noise and whose local well-posedness results from both a stochastic multilinear smoothing and a deterministic local one combined with Strichartz inequalities.