A generator approach to stochastic monotonicity and propagation of order

TL;DR

This paper introduces a new functional analytic method using the generator and resolvent of Markov processes to establish stochastic monotonicity and propagation of order across various settings.

Contribution

It develops a novel technique based on the generator and resolvent operators to prove stochastic monotonicity for a broad class of Markov processes.

Findings

Provides a new operator-based criterion for stochastic monotonicity.

Applicable to diffusion processes with different boundary conditions.

Extends to discrete interacting particle systems.

Abstract

We study stochastic monotonicity and propagation of order for Markov processes with respect to stochastic integral orders characterized by cones of functions satisfying $\Phi f \geq 0$ for some linear operator $\Phi$. We introduce a new functional analytic technique based on the generator $A$ of a Markov process and its resolvent. We show that the existence of an operator $B$ with positive resolvent such that $\Phi A - B \Phi$ is a positive operator for a large enough class of functions implies stochastic monotonicity. This establishes a technique for proving stochastic monotonicity and propagation of order that can be applied in a wide range of settings including various orders for diffusion processes with or without boundary conditions and orders for discrete interacting particle systems.