Hill's Equation with Random Forcing Parameters: The Limit of Delta
  Function Barriers
  This paper considers random Hill's equations in the limit where the periodic
forcing function becomes a Dirac delta function. For this class of equations,
the forcing strength $q_k$, the oscillation frequency $\af_k$, and the period
are allowed to vary from cycle to cycle. Such equations arise in astrophysical
orbital problems in extended mass distributions, in the reheating problem for
inflationary cosmologies, and in periodic Schr{\"o}dinger equations. The growth
rates for solutions to the periodic differential equation can be described by a
matrix transformation, where the matrix elements vary from cycle to cycle.
Working in the delta function limit, this paper addresses several coupled
issues: We find the growth rates for the $2 \times 2$ matrices that describe
the solutions. This analysis is carried out in the limiting regimes of both
large $q_k \gg 1$ and small $q_k \ll 1$ forcing strength parameters. For the
latter case, we present an alternate treatment of the dynamics in terms of a
Fokker-Planck equation, which allows for a comparison of the two approaches.
Finally, we elucidate the relationship between the fundamental parameters
$(\af_k,q_k)$ appearing in the stochastic differential equation and the matrix
elements that specify the corresponding discrete map. This work provides
analytic -- and accurate -- expressions for the growth rates of these
stochastic differential equations in both the $q_k \gg1 $ and the $q_k \ll 1$
limits.
