Resonance-based integrators for stochastic Schrödinger equations. Convergence and long-time error bounds
This work provides the first long-time error bounds for low-regularity integrators for stochastic dispersive equations, benefiting numerical analysts and practitioners simulating stochastic Schrödinger equations over long times.
The paper develops resonance-based integrators for stochastic Schrödinger equations with additive noise, achieving first-order accuracy in H^σ for solutions in H^{σ+1} (improving from H^{σ+2}) and long-time error bounds of O(ε^2τ) for linear and O(ε^2τ^δ) for cubic nonlinear cases up to times O(ε^{-2}).
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 $H^σ$ for solutions in $H^{σ+1}$, improving the classical $H^{σ+2}$ 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 $O(\varepsilon^2τ)$. 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 $O(\varepsilon^2τ^δ)$, for any $δ<1$, 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.