Note on the physical meaning of averaged amplitudes and intensities in parametric instabilities driven by a stochastic pump

The problem of the physical meaning of averaged amplitudes and intensities in parametric instabilities driven by a stochastic pump is addressed. The physical meaning of an averaged quantity is understood as its ability to yield a good estimate of the median of this quantity. In the limit of a purely temporal problem with a Gaussian white noise pump, rigorous results can be obtained which show that no physical meaning can be bestowed a priori either on the averaged amplitudes or on the averaged intensities. A conjecture is proposed which enlarges this result to the general case. The statistics of the wave amplitudes is obtained by constructing a tractable weak solution to the parametric problem.