  We develop a novel Monte Carlo strategy for the simulation of the
Boltzmann-BGK model with both low-collisional and high-collisional regimes
present. The presented solution to maintain accuracy in low-collisional regimes
and remove exploding simulation costs in high-collisional regimes uses
hybridized particles that exhibit both kinetic behaviour and diffusive
behaviour depending on the local collisionality. In this work, we develop such
a method that maintains the correct mean, variance, and correlation of the
positional increments over multiple time steps of fixed step size for all
values of the collisionality, under the condition of spatial homogeneity during
the time step. In the low-collisional regime, the method reverts to the
standard velocity-jump process. In the high-collisional regime, the method
collapses to a standard random walk process. We analyze the error of the
presented scheme in the low-collisional regime for which we obtain the order of
convergence in the time step size. We furthermore provide an analysis in the
high-collisional regime that demonstrates the asymptotic-preserving property.
