  Diffusion models are a new class of generative models that revolve around the
estimation of the score function associated with a stochastic differential
equation. Subsequent to its acquisition, the approximated score function is
then harnessed to simulate the corresponding time-reversal process, ultimately
enabling the generation of approximate data samples. Despite their evident
practical significance these models carry, a notable challenge persists in the
form of a lack of comprehensive quantitative results, especially in scenarios
involving non-regular scores and estimators. In almost all reported bounds in
Kullback Leibler (KL) divergence, it is assumed that either the score function
or its approximation is Lipschitz uniformly in time. However, this condition is
very restrictive in practice or appears to be difficult to establish.
  To circumvent this issue, previous works mainly focused on establishing
convergence bounds in KL for an early stopped version of the diffusion model
and a smoothed version of the data distribution, or assuming that the data
distribution is supported on a compact manifold. These explorations have lead
to interesting bounds in either Wasserstein or Fortet-Mourier metrics. However,
the question remains about the relevance of such early-stopping procedure or
compactness conditions. In particular, if there exist a natural and mild
condition ensuring explicit and sharp convergence bounds in KL.
  In this article, we tackle the aforementioned limitations by focusing on
score diffusion models with fixed step size stemming from the
Ornstein-Ulhenbeck semigroup and its kinetic counterpart. Our study provides a
rigorous analysis, yielding simple, improved and sharp convergence bounds in KL
applicable to any data distribution with finite Fisher information with respect
to the standard Gaussian distribution.
