Robert Neel

Columbia University

**Abstract**: It is well known that, on a compact Riemannian
manifold, minus t times the logarithm of the heat kernel converges uniformly
to the energy function as t goes to zero. Malliavin and Stroock have used
stochastic calculus to show that this limit commutes with spatial derivatives
away from the cut locus, but one expects more complicated behavior at the cut
locus. In this talk we will give formulas for the small time asymptotics of the
gradient and the Hessian of the logarithm of the heat kernel which are valid
everywhere on the manifold and which admit an appealing probabilistic
interpretation. We will also show how these formulas can be used to study both
the pointwise and the distributional limits of derivatives of the logarithm of
the heat kernel.