Gradient and Schrödinger perturbations of transition densitites

Stochastic Analysis Seminar Series

We consider time-space integrability conditions on a drift term (for gradient pert.) and a potential (for Schrödinger pert.) under which we obtain certain estimates of the perturbations of  general transition densities.

We prove an optimal 4G Theorem for the Gaussian kernel, the inequality which is a nontrivial extension of the so called 3G Theorem ([2], [1]) and a counterpart of the 3P Theorem ([3], [4]) (as well known, 3P fails in its primary form for the Gaussian kernel). Furthermore, a new method of estimating Schrödinger perturbations applies to the Gaussian kernel.


[1] K. Bogdan, W. Hansen, and T. Jakubowski. Localization and Schrödinger perturbations of kernels

Preprint (arXiv), 2012

[2] K. Bogdan and T. Jakubowski. Estimates of heat kernel of fractional Laplacian perturbed by gradient

operators. Comm. Math. Phys., 271(1):179–198, 2007.

[3] M. Cranston, E. Fabes, and Z. Zhao. Conditional gauge and potential theory for the Schrödinger

operator. Trans. Amer. Math. Soc., 307(1):171–194, 1988.

[4] W. Hansen. Harnack inequalities for Schrödinger operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci.

(4), 28(3):413–470, 1999.



Karol Szczypkowski (Wroclaw University of Technology, Poland)

Monday, November 26, 2012 - 15:45
to 16:45