  This research presents a universal mapping between two key concepts in the
study of stochastic processes: nonequilibrium steady-states under confinement
and extreme value statistics, focusing on one-dimensional systems. The mapping
holds irrespectively of the statistics of the noise driving the dynamics. This
result is based on a novel approach that provides an exact trajectory-wise
solution for first-order dynamics near a hard-wall boundary. I first
demonstrate the applicability of this method by efficiently reproducing known
results from Brownian motion theory, such as the distribution of the running
maximum, and by deriving new ones, such as the survival probability below a
specific curve with conditioning on the endpoint value. I then extend the
analysis to non-Markovian processes, focusing on run-and-tumble particles. The
mapping to a steady-state allows to compute several quantities of interest,
such as the long-time probability that run-and-tumble and Brownian particles do
not cross each other.
