Semigroup inequalities, stochastic domination, Hardy's inequality, and strong ergodicity

Carl Graham (1)

1 CMAP - Centre de Mathématiques Appliquées

For the classical Lp spaces of signed measures on the integers, we devise a framework in which bounds for a sub-Markovian semigroup of interest can be obtained, up to a constant factor, from bounds for another tractable semigroup that dominates stochastically the first one. The main tools are the Hardy inequality, the definition of related auxiliary Lp spaces suited to take advantage of the domination, and the proof that the norms are equivalent to the classical ones if the reference measure is quasi-geometrically decreasing. We illustrate the results using birth-death and single-birth processes.

Mots clés: semigroup inequalities; stochastic order; Hardy's inequality; strong ergodicity; exponential stability; spectral gap; birth-death and single-birth processes

Domaines: Probabilités [math.PR]

Carl Graham. Semigroup inequalities, stochastic domination, Hardy's inequality, and strong ergodicity. 2008. ⟨hal-00286007v1⟩