Blow-up in the supercooled Stefan problem with noise: unstable states and discontinuity of the temperature

Following on from Christoph's talk last week, I will present a version of the supercooled Stefan problem with noise. I will start by discussing the physical intuition and then give a probabilistic representation of solutions. From there, I will identify a simple relationship between the initial heat profile and a single parameter for how the liquid solidifies, which, if violated, forces the temperature to develop a discontinuity in finite time with positive probability. On the other hand, when the relationship is satisfied, the temperature remains globally continuous with probability one. The work is part of a new preprint that should soon be available on arXiv.