  The Donsker-Varadhan rate function for occupation-time fluctuations has been
seen numerically to exhibit monotone return to stationary nonequilibrium [Phys.
Rev. Lett. 107, 010601 (2011)]. That rate function is related to dynamical
activity and, except under detailed balance, it does not derive from the
relative entropy for which the monotonicity in time is well understood. We give
a rigorous argument that the Donsker-Varadhan function is indeed monotone under
the Markov evolution at large enough times with respect to the relaxation time,
provided that a "normal linear-response" condition is satisfied.
