  These notes are based on the lectures that I gave (virtually) at the Bruneck
Summer School in 2021 on first-passage processes and some applications of the
basic theory. I begin by defining what is a first-passage process and
presenting the connection between the first-passage probability and the
familiar occupation probability. Some basic features of first passage on the
semi-infinite line and a finite interval are then discussed, such as splitting
probabilities and first-passage times. I also treat the fundamental connection
between first passage and electrostatics. A number of applications of
first-passage processes are then presented, including the hitting probability
for a sphere in greater than two dimensions, reaction rate theory and its
extension to receptors on a cell surface, first-passage inside an infinite
absorbing wedge in two dimensions, stochastic hunting processes in one
dimension, the survival of a diffusing particle in an expanding interval, and
finally the dynamics of the classic birth-death process.
