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We study stochastic monotonicity and propagation of order for Markovprocesses with respect to stochastic integral orders characterized by cones offunctions 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 anoperator $B$ with positive resolvent such that $\Phi A - B \Phi$ is a positiveoperator for a large enough class of functions implies stochastic monotonicity.This establishes a technique for proving stochastic monotonicity andpropagation of order that can be applied in a wide range of settings includingvarious orders for diffusion processes with or without boundary conditions andorders for discrete interacting particle systems.

A generator approach to stochastic monotonicity and propagation of order