Branching Random Walk Solutions to the Wigner Equation

Abstract

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.