Branching random walk solutions to the Wigner equation
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The stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator Θ_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 Θ_V=Θ^+_V-Θ^-_V, i.e., treating Θ_V as the difference of two positive operators Θ^±_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 Θ^±_V, instead of Θ_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 Θ_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.
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