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.

arXiv comment61 pages

References in corpus (5)

• Formulae for the derivatives of heat semigroups
• A Poincaré Inequality on Loop Spaces
• Geometric stochastic analysis on path spaces
• First Order Feynman-Kac Formula
• On the Semi-Classical Brownian Bridge Measure

Cited by in corpus (6)

• First Order Feynman-Kac Formula
• Identifying Constant Curvature Manifolds, Einstein Manifolds, and Ricci Parallel Manifolds
• Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $L^1$
• Doubly Damped Stochastic Parallel Translations and Hessian Formulas
• Some inequalities on Riemannian manifolds linking Entropy,Fisher information, Stein discrepancy and Wasserstein distance
• Derivatives of Feynman-Kac Semigroups