Positivity and polynomial decay of energies for square-field operators

TL;DR

This paper proves positivity and polynomial decay of energies for square-field operators associated with Markov generators, using interpolation and convexity techniques, and establishes conditions for energy decay rates.

Contribution

It introduces a general framework for positivity of square-field operators and their iterations, leading to decay results for associated energies in Markov processes.

Findings

Positivity of square-field operators and their iterations.

Hierarchy of energy functionals decays to zero.

Polynomial decay of energies under normality conditions.

Abstract

We show that for a general Markov generator the associated square-field (or carr\'e du champs) operator and all their iterations are positive. The proof is based on an interpolation between the operators involving the generator and their semigroups, and an interplay between positivity and convexity on Banach lattices. Positivity of the square-field operators allows to define a hierarchy of quadratic and positive energy functionals which decay to zero along solutions of the corresponding evolution equation. Assuming that the Markov generator satisfies an operator-theoretic normality condition, the sequence of energies is log-convex. In particular, this implies polynomial decay in time for the energy functionals along solutions.