  We present a perturbation approach to calculate the short-time propagator, or
transition density, of the one-dimensional Fokker-Planck equation, to in
principle arbitrary order in the time increment. Our approach preserves
probability exactly and allows us to evaluate expectation values of analytical
observables to in principle arbitrary accuracy; to showcase this, we derive
perturbation expansions for the moments of the spatial increment, the
finite-time Kramers-Moyal coefficients, and the mean medium entropy production
rate. For an explicit multiplicative-noise system with available analytical
solution, we validate all our perturbative results. Throughout, we compare our
perturbative results to those obtained from the widely used Gaussian
approximation of the short-time propagator; we demonstrate that this Gaussian
propagator leads to errors that can be many orders of magnitude larger than
those resulting from our perturbation approach. Potential applications of our
results include parametrizing diffusive stochastic dynamics from observed time
series, and sampling path spaces of stochastic trajectories numerically.
