Exit Problems Related to the Persistence of Solitons for the Korteweg-de Vries Equation with Small Noise

We consider two exit problems for the Korteweg-de Vries equationperturbed by an additive white in time and colored in space noise of amplitude. The initial datum gives rise to a soliton when = 0. It has been provedrecently that the solution remains in a neighborhood of a randomly modulatedsoliton for times at least of the order of ??2. We prove exponential upper andlower bounds for the small noise limit of the probability that the exit timefrom a neighborhood of this randomly modulated soliton is less than T, of thesame order in and T. We obtain that the time scale is exactly the right one.We also study the similar probability for the exit from a neighborhood of thedeterministic soliton solution. We are able to quantify the gain of eliminatingthe secular modes to better describe the persistence of the soliton.