This talk is concerned with the probabilistic methods for solving Stefan
free-boundary PDEs that allow for supercooling. The latter equations appear in
many models of fundamental physical and biological processes, such as: phase
transition (i.e., melting/freezing), phase segregation (e.g., aging of alloys), crystal
growth, neurons interaction, etc.. Despite their importance, to date, there is no
general existence and uniqueness theory for such equations due to the potential
singularity of their solutions, which makes it difficult to apply the classical analytical
methods. Recently, novel probabilistic methods, based on the analysis of
associated mean-field particle systems and McKean-Vlasov equations, were
successfully used to tackle these mathematical challenges yielding new well-
posedness results for certain types of Stefan equations. I will present an overview
of the recent results and will focus on the well-posedness of the Stefan equation
with surface tension. This talk is based on joint works with F. Delarue, M.
Shkolnikov, and X. Zhang.