Renormalization of a 1d quadratic Schrödinger model with additive noise

Abstract

The study is devoted to the interpretation and well-posedness of the stochastic NLS model (ı∂t−Δ)u=|u|2+B˙,u0=0,t∈R,x∈T, 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. Our investigations then focus on the rough regime corresponding to the condition 74<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.