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.