  We present the complete set of stochastic Verlet-type algorithms that can
provide correct statistical measures for both configurational and kinetic
sampling in discrete-time Langevin systems. The approach is a brute-force
general representation of the Verlet-algorithm with free parameter coefficients
that are determined by requiring correct Boltzmann sampling for linear systems,
regardless of time step. The result is a set of statistically correct methods
given by one free functional parameter, which can be interpreted as the
one-time-step velocity attenuation factor. We define the statistical
characteristics of both true on-site $v^n$ and true half-step
$u^{n+\frac{1}{2}}$ velocities, and use these definitions for each
statistically correct Stormer-Verlet method to find a unique associated
half-step velocity expression, which yields correct kinetic Maxwell-Boltzmann
statistics for linear systems. It is shown that no other similar, statistically
correct on-site velocity exists. We further discuss the use and features of
finite-difference velocity definitions that are neither true on-site, nor true
half-step. The set of methods is written in convenient and conventional
stochastic Verlet forms that lend themselves to direct implementation for,
e.g., Molecular Dynamics applications. We highlight a few specific examples,
and validate the algorithms through comprehensive Langevin simulations of both
simple nonlinear oscillators and complex Molecular Dynamics.
