  In this paper we consider a particular version of the random walk with
restarts: random reset events which bring suddenly the system to the starting
value. We analyze its relevant statistical properties like the transition
probability and show how an equilibrium state appears. Formulas for the
first-passage time, high-water marks and other extreme statistics are also
derived: we consider counting problems associated naturally to the system.
Finally we indicate feasible generalizations useful for interpreting different
physical effects.
