Stochastic monotonicity and duality for one-dimensional Markov processes

TL;DR

This paper develops a comprehensive theory of monotonicity and duality for one-dimensional Feller processes, establishing conditions for well-posedness even with unbounded coefficients and on the half-line.

Contribution

It introduces local monotonicity conditions based on the Lévy kernel that ensure well-posedness of the Markov process and semigroup.

Findings

Established a general theory of monotonicity and duality for Feller processes.

Proved local monotonicity conditions suffice for well-posedness.

Extended results to processes with unbounded coefficients and on the half-line.

Abstract

The theory of monotonicity and duality is developed for general one-dimensional Feller processes. Moreover it is shown that local monotonicity conditions (conditions on the L\'evy kernel) are sufficient to prove the well-posedness of the corresponding Markov semigroup and process, including unbounded coefficients and processes on the half-line.