$\sigma$-finite invariant densities for eventually conservative Markov operators

Hisayoshi Toyokawa

doi:10.3934/dcds.2020144

Article Contents

2020, Volume 40, Issue 5: 2641-2669. Doi: 10.3934/dcds.2020144

This issue Previous Article Characterization of minimizable Lagrangian action functionals and a dual Mather theorem Next Article Multiple positive solutions for a Schrödinger logarithmic equation

$\sigma$-finite invariant densities for eventually conservative Markov operators

Hisayoshi Toyokawa

Department of Mathematics, Hokkaido University, Sapporo, Hokkaido, 060-0810, Japan

Received: March 1, 2019

Revised: December 1, 2019

Published online: May 1, 2020

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Abstract

We establish equivalent conditions for the existence of an integrable or locally integrable fixed point for a Markov operator with the maximal support. Maximal support means that almost all initial points will concentrate on the support of the invariant density under the iteration of the process. One of the equivalent conditions for the existence of a locally integrable fixed point is weak almost periodicity of the jump operator with respect to some sweep-out set. This result includes the case of the existence of an absolutely continuous $ \sigma $-finite invariant measure when we consider a nonsingular transformation on a probability space. Weak almost periodicity implies the Jacobs-Deleeuw-Glicksberg splitting and we show that constrictive Markov operators which guarantee the spectral decomposition are typical weakly almost periodic operators.

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Mathematics Subject Classification: Primary: 37A30, 37H99; Secondary: 47A35, 37D25.

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