  We present a variety of results analyzing the behavior of a class of
stochastic processes --- referred to as Stochastic Hybrid Systems (SHSs) --- in
or near equilibrium, and determine general conditions on when the moments of
the process will, or will not, be well-behaved. We also study the potential for
finite-time blowups for these processes, and exhibit a set of random recurrence
relations that govern the behavior for long times. In addition, we present a
connection between these recurrence relations and some classical expressions in
number theory.
