Realizable monotonicity for continuous-time Markov processes

Abstract

We formalize and analyze the notions of stochastic monotonicity and realizable monotonicity for Markov Chains ill continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which stochastic monotonicity and realizable monotonicity coincide in continuous-time but not in discrete-time. (C) 2010 Elsevier B.V. All rights reserved

Euler hydrodynamics for attractive particle systems in random environment

“…Property (11) is usually called attractiveness. Condition (8) implies the stronger complete monotonicity property ( [13,9]), that is, existence of a monotone Markov coupling for an arbitrary number of processes with generator (2), see (27) below; we also say that the process is strongly attractive.…”

Constructive Euler Hydrodynamics for One-Dimensional Attractive Particle Systems

“…Property ( 17) implies (2), that is, attractiveness. But it is more powerful: it implies the complete monotonicity property ( [24,20]), that is, existence of a monotone Markov coupling for an arbitrary number of processes with generator (4), which is necessary in our proof of strong hydrodynamics for general initial profiles. The coupled process can be defined by a Markov generator, as in [19] for two components, that is…”