Stochastic Monotonicity and Duality for One-Dimensional Markov Processes

The theory of monotonicity and duality is developed for general one-dimensional Feller processes, extending the approach from [1]. Moreover it is shown that local monotonicity conditions (conditions on the Lévy kernel) are sufficient to prove the well-posedness of the corresponding Markov semigroup and process, including unbounded coefficients and processes on the half-line.