Strichartz and local smoothing estimates for stochastic dispersive equations with linear multiplicative noise

TL;DR

This paper establishes Strichartz and local smoothing estimates for a broad class of stochastic dispersive equations with linear multiplicative noise, enabling analysis of nonlinear problems and large deviation principles.

Contribution

It introduces pathwise estimates for stochastic dispersive equations with linear multiplicative noise, including Schrödinger and Airy equations, and applies these to nonlinear well-posedness and large deviations.

Findings

Proved pathwise Strichartz estimates for stochastic dispersive equations.

Established local smoothing estimates with integrability of constants.

Demonstrated large deviation principles for small noise asymptotics.

Abstract

We study a quite general class of stochastic dispersive equations with linear multiplicative noise, including especially the Schr\"odinger and Airy equations. The pathwise Strichartz and local smoothing estimates are derived here in both the conservative and nonconservative case. In particular, we obtain the P-integrability of constants in these estimates, where P is the underlying probability measure. Several applications are given to nonlinear problems, including local well-posedness of stochastic nonlinear Schr\"odinger equations with variable coefficients and lower order perturbations, integrability of global solutions to stochastic nonlinear Schr\"odinger equations with constant coefficients. As another consequence, we prove as well the large deviation principle for the small noise asymptotics.