The stochastic Zakharov system in dimension $d \geq 4$: Local well-posedness and regularization by noise for scattering

TL;DR

This paper establishes local well-posedness and noise-induced regularization for the stochastic Zakharov system in high dimensions, demonstrating global existence and scattering for large initial data with high probability.

Contribution

It introduces a novel functional framework and new estimates to prove well-posedness and regularization effects for the stochastic Zakharov system beyond the energy space.

Findings

Proved local well-posedness in the full deterministic regularity regime.

Established noise-induced global existence and scattering for large data.

Developed new trilinear estimates leveraging temporal regularity of Brownian motions.

Abstract

In this paper, we develop the well-posedness theory and uncover the noise-regularization effect on scattering for the stochastic Zakharov system in dimensions $d \geq 4$ and beyond the energy space. Our focus is particularly directed at the large data regime, where the global existence and long-time dynamics of the deterministic Zakharov system remain largely open. We prove the local well-posedness of the stochastic system in the full deterministic regularity regime and establish a blow-up alternative at the endpoint regularity, which implies the persistence of regularity in the full well-posedness regime. Furthermore, we prove that for any large initial data, with high probability, non-conservative noise yields global and scattering solutions. Our proof introduces a tailored functional framework. To establish local well-posedness, we employ a refinement of adapted Fourier restriction and lateral Strichartz spaces, which allows us to control both the nonlinear interactions and the critical first-order derivative perturbations arising from rescaling transforms. To achieve the noise-regularization effect, we augment this setting with maximal function spaces. We derive new trilinear estimates for the stochastic wave nonlinearity that are crucial for the global dynamics by fully exploiting the temporal regularity of geometric Brownian motions in scaling-(sub)critical Besov spaces.