Malliavin Calculus for the one-dimensional Stochastic Stefan Problem

Abstract

We consider the one-dimensional outer stochastic Stefan problem with reflection. The problem admits maximal solutions as long as the velocity of the moving boundary remains bounded, [3,9,10]. We apply Malliavin calculus to the transformed equation and first prove that its maximal solution u has continuous paths a.s. In the case of the unreflected problem, the previous enables the localization of a proper approximating sequence of the maximal solution. Then, we derive there locally the differentiability of maximal u in the Malliavin sense. The novelty of this work, apart from the derivation of continuity of the paths for the maximal solution with reflection, is that for the unreflected case we introduce a localization argument on maximal solutions and define efficiently the relevant sample space. More precisely, we prove the local (in the sample space) existence of the Malliavin derivative and, under a non-degeneracy condition on the noise coefficient, the absolute continuity of the law of the solution with respect to the Lebesgue measure.