  The main results of the article are short time estimates and asymptotic
estimates for the first two order derivatives of the logarithmic heat kernel of
a complete Riemannian manifold. We remove all curvature restrictions and also
develop several techniques.
  A basic tool developed here is intrinsic stochastic variations with
prescribed second order covariant differentials, allowing to obtain a path
integration representation for the second order derivatives of the heat
semigroup $P_t$ on a complete Riemannian manifold, again without any
assumptions on the curvature. The novelty is the introduction of an
$\epsilon^2$ term in the variation allowing greater control. We also construct
a family of cut-off stochastic processes adapted to an exhaustion by compact
subsets with smooth boundaries, each process is constructed path by path and
differentiable in time, furthermore the differentials have locally uniformly
bounded moments with respect to the Brownian motion measures, allowing to
by-pass the lack of continuity of the exit time of the Brownian motions on its
initial position.
