  We study the statistics of infima, stopping times and passage probabilities
of entropy production in nonequilibrium steady states, and show that they are
universal. We consider two examples of stopping times: first-passage times of
entropy production and waiting times of stochastic processes, which are the
times when a system reaches for the first time a given state. Our main results
are: (i) the distribution of the global infimum of entropy production is
exponential with mean equal to minus Boltzmann's constant; (ii) we find the
exact expressions for the passage probabilities of entropy production to reach
a given value; (iii) we derive a fluctuation theorem for stopping-time
distributions of entropy production. These results have interesting
implications for stochastic processes that can be discussed in simple colloidal
systems and in active molecular processes. In particular, we show that the
timing and statistics of discrete chemical transitions of molecular processes,
such as, the steps of molecular motors, are governed by the statistics of
entropy production. We also show that the extreme-value statistics of active
molecular processes are governed by entropy production, for example, the
infimum of entropy production of a motor can be related to the maximal
excursion of a motor against the direction of an external force. Using this
relation, we make predictions for the distribution of the maximum backtrack
depth of RNA polymerases, which follows from our universal results for
entropy-production infima.
