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 equation perturbed by an additive white in time and colored in space noise of amplitude $\epsilon$. The initial datum gives rise to a soliton when $\epsilon=0$. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of $\epsilon^{-2}$. We prove exponential upper and lower bounds for the small noise limit of the probability that the exit time from a neighborhood of this randomly modulated soliton is less than $T$, of the same order in $\epsilon$ 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 the deterministic soliton solution. We are able to quantify the gain of eliminating the secular modes to better describe the persistence of the soliton.