Among Markov processes that share the same invariant measure, it can be argued that reversible processes are the slowest to converge to their steady states.  We consider time-averages of the empirical current and density in such processes -- for large times these obey a large deviation principle.  By analysing these large deviation principles, we offer a geometrical interpretation of the slow convergence of reversible processes.  We discuss the consequences of these results for physical systems driven far from equilibrium, and for Markov Chain Monte Carlo sampling methods.

[Joint work with Marcus Kaiser and Johannes Zimmer, University of Bath
 Kaiser, RLJ and Zimmer, J Stat Phys 168, 259 (2017); and arXiv:1708.01453 ]