  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.
