  We study a class of graphs that represent local independence structures in
stochastic processes allowing for correlated error processes. Several graphs
may encode the same local independencies and we characterize such equivalence
classes of graphs. In the worst case, the number of conditions in our
characterizations grows superpolynomially as a function of the size of the node
set in the graph. We show that deciding Markov equivalence is coNP-complete
which suggests that our characterizations cannot be improved upon
substantially. We prove a global Markov property in the case of a multivariate
Ornstein-Uhlenbeck process which is driven by correlated Brownian motions.
