Longest excursion of stochastic processes in nonequilibrium systems.

We consider the excursions, i.e., the intervals between consecutive zeros, of stochastic processes that arise in a variety of nonequilibrium systems and study the temporal growth of the longest one l_{max}(t) up to time t. For smooth processes, we find a universal linear growth l_{max}(t) approximately Q_{infinity}t with a model dependent amplitude Q_{infinity}. In contrast, for nonsmooth processes with a persistence exponent theta, we show that l_{max}(t) has a linear growth if theta < theta_{c} while l_{max}(t) approximately t;{1-psi} if theta > theta_{c}. The amplitude Q_{infinity} and the exponent psi are novel quantities associated with nonequilibrium dynamics. This behavior is obtained by exact analytical calculations for renewal and multiplicative processes and numerical simulations for other systems such as the coarsening dynamics in Ising model as well as the diffusion equation with random initial conditions.