The previously developed methods for investigation of the above question include such advanced tools as Harnack inequalities and estimates of fundamental solutions. Here we adopt another approach that originates and that uses only very basic tools as capacities and volumes. As a consequence of this inequality, we obtain various interesting comparison inequalities for heat semigroups and heat kernels, which can be used for obtaining pointwise estimates of heat kernels. As an example of application, we present a new method of deducing sub-Gaussian upper bounds of the heat kernel from on-diagonal bounds and tail estimates.

D.Emelianov, B.Miller, E.Rubinovich. Information Set-Based Guidance Algorithm against a Decelerating Maneuvering Target // IEEE Trans. on Aerospace and Electronic Systems. Vol. 41, No. 1, January 2005, pp. 65-74.