  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.
