If the situation was chaotic (which I would presume a system with many random perturbations to be) the expectation is for the repeated small changes to have a significant impact.
My guess would be this would be true only if they were coherent, and the resonant frequency were some multiple of the Planck length. The many small random perturbations would largely cancel each other out and the remainder would be insignificant (not energetic enough) to meet the lowest energy requirements to nudge a photon by even a miniscule amount.