We study the Cauchy problem for the parabolic equation ∂
∂t = L and the
h-Brownian motion which is the Markov process with the weighted Laplacian 1
2 ∆h :=
1
2 ∆ +∇h where ∆ the Laplace-Beltrami operator on M, and h, V real valued functions
on M and L is the weighted Schrodinger operator ¨ L = 1
2 ∆ + ∇h − V .
We first obtain new geometric criteria for the gradient stochastic differential equation
(SDE) with generator 1
2 ∆h: non-explosion, strong 1-completeness, moment bounds,
and exponential integrability. We then study the linearisation problem associated with
the gradient SDE, introduce also a doubly damped stochastic parallel transport on tensors, involving only geometric quantities. Together with the stochastic damped transport this allows to obtain a new Hessian formula for the weighted heat semi-group,
obtained with a hybrid formula Hess(P
h
t
f)(v2, v1) = E [∇df(Wt(v2), Wt(v1))] +
E
h
df(W
(2)
t
(v1, v2))i
, and a corresponding formula for e
tL. These formulae are then
used for obtaining path integration formula for Hess P
h,V
t
f(v1, v2), a 2nd order Feynman - Kac formula, based on path integration, not involving any derivatives of f or V.
With these intrinsic second order Feynman-Kac formula, global estimates are obtained for these semi-groups, their derivatives, and that of their fundamental solutions are
obtained. These estimates are in terms of bounds on Ric −2 Hess h, on the curvature
operator, and on the cyclic sum of the gradient of the Ricci tensor.
Finally, for manifolds with a pole, we prove that the Hessian of the fundamental
solution is the product of an exact Gaussian term with a term involving the semi-classical
bridge, the latter is further estimated to lead to Hessian estimates. Precise estimates are
then obtained for the derivatives of the logarithmic heat kernels.