The comonotone flow of a stochastically monotone Feller process on the real line

Abstract

We show that any stochastically monotone Feller semigroup on R can be extended by a consistent family of order-preserving Feller semigroups on the successive powers of R. We exhibit a specific such family, which is uniquely characterized by a maximality property with respect to the super-modular order on Rn. A consequence is that, in this fairly general setting, there always exists a coupling between n c{\`a}dl{\`a}g versions of the underlying Markov process starting from n distinct initial positions, which do not cross one another.