Model of Parametric Excitation by an Imperfect Pump

We examine the three-wave decay instability due to a monochromatic pump with a phase which undergoes a random walk with diffusion coefficient $D$. Analytic results are obtained for small and large $\frac{D}{{\gamma }_0}$ (where ${\gamma }_0$ is the growth rate for $D=0\mathrm{}$), which demonstrate, respectively, a fractional reduction of $\gamma$ by $\frac{D}{{\gamma }_0}$ and a growth rate $\frac{{{\gamma }_0}^2}{D}$. Results are also presented for intermediate values of $\frac{D}{{\gamma }_0}$.