Backward error analysis of stochastic Poisson integrators

TL;DR

The paper tackles backward error analysis for stochastic Poisson integrators applied to Poisson SDEs of the form $dy=B(y)\nabla\mathcal{H}(y)\,dt+\sum_{r=1}^m B(y)\nabla H_r(y)\circ dW_r(t)$. By leveraging Wong–Zakai approximations, it derives stochastic modified equations that retain the same Poisson structure and identifies a conserved random Hamiltonian along the modified flow. It proves long-time error/boundedness results for structure-preserving integrators, showing that modified Hamiltonians $\tilde{H}$ and $\overline{H}$ remain close to initial values under suitable assumptions, and supports the theory with numerical experiments on canonical stochastic Hamiltonian systems and a stochastic Maxwell–Bloch model. This work extends strong backward error analysis to stochastic Poisson problems with non-constant $B(y)$ and multiple noise terms, offering principled guidance for designing and assessing long-time structure-preserving integrators in stochastic settings.

Abstract

We address our attention to the numerical time discretization of stochastic Poisson systems via Poisson integrators. The aim of the investigation regards the backward error analysis of such integrators to reveal their ability of being structure-preserving, for long times of integration. In particular, we first provide stochastic modified equations suitable for such integrators and then we rigorously study them to prove accurate estimates on the long-term numerical error along the dynamics generated by stochastic Poisson integrators, with reference to the preservation of the random Hamiltonian conserved along the exact flow of the approximating Wong-Zakai Poisson system. Finally, selected numerical experiments confirm the effectiveness of the theoretical analysis.

Figures (3)

• Figure 1: Hamiltonian deviations, over single trajectories, computed by applying the stochastic midpoint method \ref{['exp1']} applied to the canonical Hamiltonian system with $m=3$ and Hamiltonians given by \ref{['test1']}, for $\sigma_1=0.01, \sigma_2=0.02, \sigma_3=0.03$ and for selected values of $h$. Here, $y_0 = [1, \ 2]^{\intercal}$.
• Figure 2: Numerical errors $\overline{H}(y_n)-\overline{H}(y_0)$, over single trajectories, with $\overline{H}$ given in \ref{['test2']}-\ref{['ran_ham1']}, computed by applying the stochastic midpoint method \ref{['exp2']} applied to the canonical Hamiltonian system with $m=2$ and Hamiltonians given by \ref{['test2']}, for $\sigma_1=\sigma_2=0.01$ and for selected values of $h$. Here, $y_0=[0, \ 1]^{\intercal}$.
• Figure 3: Numerical errors $\overline{H}(y_n)-\overline{H}(y_0)$, over single trajectories, with $\overline{H}$ given in \ref{['ran_ham1']}-\ref{['exp3']}, computed by method \ref{['poi_int_mb']} applied to the stochastic Maxwell-Block system \ref{['stoc_mb']}-\ref{['B0']}, for $\sigma_1=\sigma_2=0.01$ and for $h=0.1$, over $[0,10^5]$. Here, $y_0=[0, \ 0, \ 1]^{\intercal}$. We observe that the plot displays only 1000 equidistant step points overall those considered in the implementation, for the sake of clarity of the final figure.

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Example 2.1
• Example 2.2
• Example 2.3
• Theorem 2.2
• Corollary 2.1
• Definition 2.3
• Remark 2.1