  We formulate a short-time expansion for one-dimensional Fokker-Planck
equations with spatially dependent diffusion coefficients, derived from
stochastic processes with Gaussian white noise, for general values of the
discretization parameter $0 \leq \alpha \leq 1$ of the stochastic integral. The
kernel of the Fokker-Planck equation (the propagator) can be expressed as a
product of a singular and a regular term. While the singular term can be given
in closed form, the regular term can be computed from a Taylor expansion whose
coefficients obey simple ordinary differential equations. We illustrate the
application of our approach with examples taken from statistical physics and
biophysics. Further, we show how our formalism allows to define a class of
stochastic equations which can be treated exactly. The convergence of the
expansion cannot be guaranteed independently from the discretization parameter
$\alpha$.
