Some conditional FKG inequalities for Markov chains and random processes

This paper is interested in proving correlation inequalities of the FKG-type for a broad range of random processes. The two pivotal tools which yield these correlation inequalities are that many discrete-time Markov chains, even under a very general conditioning, satisfy the FKG inequality, and that such inequalities are preserved through weak convergences. In particular, we prove that L\'evy processes, Bessel processes and several conditioned Brownian processes have positive correlations.