As a phase space language for quantum mechanics, the Wigner function approaches bear a close analogy to classical mechanics and have been drawing growing attention, especially in simulating quantum many-body systems. In this talk, we will summarize recent progress in numerical methods for the time dependent Wigner equation, among which a stochastic algorithm based on signed particle Monte Carlo method has also proven to be a promising tool within different physical applications. However, there has been so far no detailed mathematical analysis. To this end, we will construct a firm mathematical foundations for it, and explore the inherent relation between the Wigner equation and the stochastic process. We start from the discussion on two typical truncations of the nonlocal term, i.e., the k-truncated and y-truncated models. After introducing an auxiliary function, the (truncated) Wigner equation is reformulated into the integral formulation as well as its adjoint correspondence, both of which can be regarded as the renewal-type equations and have transparent stochastic interpretation. We prove that the moment of a branching process happens to be the solution for the adjoint equation, which connects rigorously the Wigner quantum dynamics to the stochastic branching process, and thus a sound mathematical framework for the Wigner Monte Carlo methods is established. More interestingly, with the help of the rigorous connection, we find that a constant auxiliary function performs better from the point of view of both theoretical and numerical aspects. Particularly, we obtain a simple but analytical calculation formula for the growth rate of particle number. Typical numerical experiments validate our theoretical findings, demonstrate the accuracy, the efficiency and thus the computability of the Wigner branching process.
 Sihong Shao, Yunfeng Xiong, A computable branching process for the Wigner quantum dynamics, arXiv:1603.00159, 2016.
 Yunfeng Xiong, Zhenzhu Chen, Sihong Shao, An advective-spectral-mixed method for time-dependent many-body Wigner simulations, arXiv:1602.08853, 2016.
 Sihong Shao, Jean Michel Sellier, Comparison of deterministic and stochastic methods for time-dependent Wigner simulations, Journal of Computational Physics 300 (2015) 167.
 Sihong Shao, Tiao Lu, Wei Cai, Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport, Communications in Computational Physics 9 (2011) 711.