The stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator $\Theta_V$ with an anti-symmetric kernel as {the generator of two branches of jump processes}, are analyzed. All existing branching random walk solutions are formulated based on the Hahn-Jordan decomposition $\Theta_V=\Theta^+_V-\Theta^-_V$, i.e., treating $\Theta_V$ as the difference of two positive operators $\Theta^\pm_V$, each of which characterizes the transition of states for one branch of particles. Despite the fact that the first moments of such models solve the Wigner equation, we prove that the bounds of corresponding variances grow exponentially in time with the rate depending on the upper bound of $\Theta^\pm_V$, instead of $\Theta_V$. In other words, the decay of high-frequency components is totally ignored, resulting in a severe {numerical sign problem}. {To fully utilize such decay property}, we have recourse to the stationary phase approximation for $\Theta_V$, which captures essential contributions from the stationary phase points as well as the near-cancelation of positive and negative weights. The resulting branching random walk solutions are then proved to asymptotically solve the Wigner equation, but {gain} a substantial reduction in variances, thereby ameliorating the sign problem. Numerical experiments in 4-D phase space validate our theoretical findings.

中文翻译：

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Wigner 方程的分支随机游走解

分析了 Wigner 方程的随机解，该方程解释了具有反对称核的非局部振荡积分算子 $\Theta_V$ 作为{跳跃过程的两个分支的生成器}。所有现有的分支随机游走解决方案都是基于 Hahn-Jordan 分解 $\Theta_V=\Theta^+_V-\Theta^-_V$ 制定的，即将 $\Theta_V$ 视为两个正算子 $\Theta^ \pm_V$，每一个都表征了一个粒子分支的状态转变。尽管这些模型的一阶矩求解 Wigner 方程，但我们证明相应方差的边界随时间呈指数增长，取决于 $\Theta^\pm_V$ 的上限，而不是 $\Theta_V$ . 换句话说，完全忽略高频分量的衰减，导致严重的{数字符号问题}。{为了充分利用这种衰减特性}，我们求助于 $\Theta_V$ 的固定相位近似值，它捕获了固定相位点的基本贡献以及正负权重的近似抵消。然后证明得到的分支随机游走解可以渐近地求解 Wigner 方程，但{获得}方差的显着减少，从而改善了符号问题。4-D 相空间中的数值实验验证了我们的理论发现。然后证明得到的分支随机游走解可以渐近地求解 Wigner 方程，但{获得}方差的显着减少，从而改善了符号问题。4-D 相空间中的数值实验验证了我们的理论发现。然后证明得到的分支随机游走解可以渐近地求解 Wigner 方程，但{获得}方差的显着减少，从而改善了符号问题。4-D 相空间中的数值实验验证了我们的理论发现。