Hessian formulas and estimates for parabolic Schrödinger operators

Abstract

We study the Hessian of the fundamental solution to the parabolic problem for weighted Schrödinger operators of the form $\frac{1}{2}\Delta +\nabla h-V$ proving a second order Feynman-Kac formula and obtaining Hessian estimates. For manifolds with a pole, we use the Jacobian determinant of the exponential map to offset the volume growth of the Riemannian measure and use the semi-classical bridge as a delta measure at $y_0$ to obtain exact Gaussian estimates. These estimates are in terms of bounds on $Ric-2Hess(h)$, on the curvature operator, and on the cyclic sum of the gradient of the Ricci tensor.

61 pages

References in corpus (1)

• On the Semi-Classical Brownian Bridge Measure

Cited by in corpus (2)

• Identifying Constant Curvature Manifolds, Einstein Manifolds, and Ricci Parallel Manifolds
• Doubly Damped Stochastic Parallel Translations and Hessian Formulas