On interweaving relations

TL;DR

This paper introduces interweaving relations as a strengthened form of intertwining relations in Markov processes, enabling transfer of ergodic and mixing properties through a randomized delay framework.

Contribution

It develops a new classification scheme for Markov semigroups using interweaving relations, extending the transfer of properties and providing several examples and transformations.

Findings

Interweaving relations generalize classical intertwining relations.

They enable transfer of ergodic and mixing properties.

Examples include Laguerre, Jacobi, and hypoelliptic Ornstein-Uhlenbeck semigroups.

Abstract

Interweaving relations are introduced and studied here in a general Markovian setting as a strengthening of usual intertwining relations between semigroups, obtained by adding a randomized delay feature. They provide a new classification scheme of the set of Markovian semigroups which enables to transfer from a reference semigroup and up to an independent warm-up time, some ergodic, analytical and mixing properties including the $\varphi$-entropy convergence to equilibrium, the hyperboundedness and when the warm-up time is deterministic the cut-off phenomena. We also present several useful transformations that preserve interweaving relations. We provide a variety of examples of interweaving relations ranging from classical, discrete, and non-local Laguerre and Jacobi semigroups to degenerate hypoelliptic Ornstein-Uhlenbeck semigroups and some non-colliding particle systems