Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field

We investigate solutions to the equation $\partial_t{\cal E} - {\cal D}\Delta {\cal E} = \lambda S^2{\cal E}$, where $S(x,t)$ is a Gaussian stochastic field with covariance $C(x-x',t,t')$, and $x\in {\mathbb R}^d$. It is shown that the coupling $\lambda_{cN}(t)$ at which the $N$-th moment $<{\cal E}^N(x,t)>$ diverges at time $t$, is always less or equal for ${\cal D}>0$ than for ${\cal D}=0$. Equality holds under some reasonable assumptions on $C$ and, in this case, $\lambda_{cN}(t)=N\lambda_c(t)$ where $\lambda_c(t)$ is the value of $\lambda$ at which $<\exp\lbrack \lambda\int_0^tS^2(0,s)ds\rbrack>$ diverges. The ${\cal D}=0$ case is solved for a class of $S$. The dependence of $\lambda_{cN}(t)$ on $d$ is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, ${\cal D}\to i{\cal D}$, the case of interest for backscattering instabilities in laser-plasma interaction.