  We review and extend the formalism introduced by Peliti, that maps a Markov
process to a path-integral representation. After developing the mapping, we
apply it to some illustrative examples: the simple decay process, the
birth-and-death process, and the Malthus-Verhulst process. In the first two
cases we show how to obtain the exact probability generating function using the
path integral. We show how to implement a diagrammatic perturbation theory for
processes that do not admit an exact solution. Analysis of a set of coupled
Malthus-Verhulst processes on a lattice leads, in the continuum limit, to a
field theory for directed percolation and allied models.
