Resonance-based integrators for stochastic Schr\"odinger equations. Convergence and long-time error bounds

TL;DR

This paper introduces resonance-based low-regularity numerical integrators for stochastic Schrödinger equations, achieving improved convergence and long-time error bounds, especially in low-regularity and long-time regimes.

Contribution

It extends the regularity-compensation oscillation technique to stochastic equations, providing the first long-time error bounds for low-regularity integrators in stochastic dispersive PDEs.

Findings

Proves strong and almost sure convergence with first-order accuracy in low regularity.

Establishes uniform moment bounds and long-time error estimates up to O(ε^{-2}) timescales.

Supports theoretical results with numerical experiments.

Abstract

We develop resonance-based low-regularity numerical integrators for stochastic Schr"odinger equations with additive $Q$-Wiener noise, covering both the linear equation with rough potential and the cubic nonlinear case. For the linear problem, we prove strong and almost sure convergence, achieving first-order accuracy in $H^\sigma$ for solutions in $H^{\sigma+1}$, improving the classical $H^{\sigma+2}$ requirement. In a regime of $O(\varepsilon^2)$ potentials and $O(\varepsilon)$ noise, we establish uniform moment bounds up to times $O(\varepsilon^{-2})$ and construct a non-resonant scheme with long-time error $O(\varepsilon^2\tau)$. For the cubic case, we derive analogous pathwise convergence results at low regularity. In the weakly nonlinear stochastic regime, we obtain long-time pathwise errors of size $O(\varepsilon^2\tau^\delta)$, for any $\delta<1$, up to times $O(\varepsilon^{-2})$. The analysis relies on a novel extension of the regularity-compensation oscillation (RCO) technique to the stochastic setting, overcoming the loss of temporal regularity induced by stochastic convolutions and yielding an $O(\varepsilon^2)$ improvement in long-time error bounds. To the best of our knowledge, this is the first work establishing long-time error bounds for low-regularity integrators for stochastic dispersive equations. Numerical experiments support the theory.