Stefano Di Giovacchino

Jianbo Cui, Raffaele D'Ambrosio, Stefano Di Giovacchino et al.

We study strong temporal approximation of periodic stochastic Korteweg--de Vries equation driven by small additive \(Q\)-Wiener noise of amplitude \(\mathcal O(\varepsilon)\), \(0<\varepsilon\ll1\). Strong error analysis for temporal approximations of stochastic KdV is a challenging problem, due to the additional derivative term in the nonlinearity and thanks to the lack of suitable exponential moment bounds for the exact solutions. Exploiting the small-noise regime, we first decompose the solution into a deterministic KdV flow and a stochastic component; then we linearize the obtained stochastic equation and approximate the resulting equation by means of Fourier analytic techniques. Combining the small-noise linearization error, the discretization error of the linearized equation, and the deterministic temporal approximation error, we prove strong convergence rates of order \(\mathcal O(\max(\varepsilon^2,τ,\varepsilonτ^{1/2}))\) under \(H^1\)-regularity and \(\mathcal O(\max(\varepsilon^2,τ))\) under \(H^2\)-regularity, for the obtained approximation of the original stochastic KdV. To the best of our knowledge, these are the first explicit strong convergence rates shown for numerical time approximations of the stochastic KdV.

Stefano Di Giovacchino

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.