  The paper presents new simple sharp bounds for transition density functions
for time-homogeneous diffusions processes. The bounds are obtained under mild
conditions on the drift and diffusion coefficients, extending and substantially
improving previous results in the literature which were limited to drifts
satisfying a linear growth condition. They lead to an asymptotic expression for
the transition density as the transition time approaches zero. While the focus
is on the one-dimensional case, an extension to multiple dimensions is
discussed. Results are illustrated by numerical examples.
