  It is known that the distribution of nonreversible Markov processes breaking
the detailed balance condition converges faster to the stationary distribution
compared to reversible processes having the same stationary distribution. This
is used in practice to accelerate Markov chain Monte Carlo algorithms that
sample the Gibbs distribution by adding nonreversible transitions or
non-gradient drift terms. The breaking of detailed balance also accelerates the
convergence of empirical estimators to their ergodic expectation in the
long-time limit. Here, we give a physical interpretation of this second form of
acceleration in terms of currents associated with the fluctuations of empirical
estimators using the level 2.5 of large deviations, which characterises the
likelihood of density and current fluctuations in Markov processes. Focusing on
diffusion processes, we show that there is accelerated convergence because
estimator fluctuations arise in general with current fluctuations, leading to
an added large deviation cost compared to the reversible case, which shows no
current. We study the current fluctuation most likely to arise in conjunction
with a given estimator fluctuation and provide bounds on the acceleration,
based on approximations of this current. We illustrate these results for the
Ornstein-Uhlenbeck process in two dimensions and the Brownian motion on the
circle.
