Title: Probabilistic solutions to the supercooled Stefan problem

**Abstract: **The supercooled Stefan problem is a free boundary partial

differential equation, which for general initial conditions may

exhibit finite-time blowup of the derivative of the free boundary and

admit multiple solutions; such a problem is deemed ill-posed.

In recent years, a probabilistic reformulation of this problem has

given rise to a new notion of solution:

these so-called 'physical solutions' satisfy a McKean-Vlasov equation

with interaction through a hitting time, and the size of any possible

discontinuity of the free boundary is governed by the law of the

associated McKean-Vlasov diffusion. Following this discovery,

'minimal' solutions to this problem were introduced, which happen to

also be physical solutions when the initial condition is integrable.

Herein, we give an introduction to the above solution concepts,

indicating their similarities and possible differences. Moreover, we

address the problem of well-posedness for minimal solutions by

investigating the sensitivity of these solutions to changes in the

initial data. We show continuous dependency when the data is shifted

to the right, but possible discontinuity for shifts to the left.