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Functional Inequalities for the Decay of Sub-Markov Semigroups
Authors:Wang  Feng-Yu
Affiliation:(1) Department of Mathematics, Beijing Normal University, Beijing, 100875, P.R. China
Abstract:A general functional inequality is introduced to describe various decays of semigroups. Our main result generalizes the classical one on the equivalence of the L2-exponential decay of a sub-Markov semigroup and the Poincaré inequality for the associated Dirichlet form. Conditions for the general inequality to hold are presented. The corresponding isoperimetric inequality is studied in the context of diffusion and jump processes. In particular, Cheeger's inequality for the principal eigenvalue is generalized. Moreover, our results are illustrated by examples of diffusion and jump processes.
Keywords:functional inequality  sub-Markov semigroup  isoperimetric inequality
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