BRANCHING RANDOM WALK SOLUTIONS TO THE WIGNER EQUATION

We analyze the stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator Theta(V) with an antisymmetric kernel as the generator of two branches of jump processes. 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(+/-)(V), each of which characterizes the transition of states for one branch of particles. Despite v 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(+/-)(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 a decay property, we turn to the stationary phase approximation for Theta(V), which captures essential contributions from the stationary phase points as well as from the near-cancellation of positive and negative weights. The resulting branching random walk solutions are then proved to asymptotically solve the Wigner equation, but they gain a substantial reduction in variances, thereby ameliorating the sign problem. Numerical experiments in 4-D phase space validate our theoretical findings.