Hessian Formulas and Estimates for Parabolic Schrödinger Operators
Xue-Mei Li · Journal of Stochastic Analysis · 2021
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 Schrödinger 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(, and a corresponding formula for e tL .These formulae are then used for obtaining path integration formula for Hess P h,V t f (v 1 , v 2 ), 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.