Logarithmic heat kernel estimates without curvature restrictions

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