Laplacian Growth Models

Joint work with Duncan Dauvergne, Danny Nam and Dor Elboim.

Hydrodynamic limits of particle systems with selection are related to a class of free boundary problems (FBP) associated with second order parabolic PDE. This relation has been established rigorously for some selection models but remains open for others, where the difficulty lies in questions of regularity of the free boundary. We introduce a weak formulation that does not involve the notion of a free boundary, but reduces to a FBP when classical solutions exist. For the injection-branching-selection model of Brownian particles on the line with arbitrarily varying removal rates, the approach yields the characterization of limits in terms of a PDE beyond cases where classical solutions are known (or even expected) to exist. Results on a higher dimensional model of motionless branching particles will also be described, as well as a formal relation to the Stefan FBP and open problems.

Supercooled Stefan problems describe the evolution of the boundary between the solid and liquid phases of a substance, where the liquid is assumed to be cooled below its freezing point. Following the methodology of Delarue, Nadtochiy and Shkolnikov, we construct solutions to the one-phase one-dimensional supercooled Stefan problem through a certain McKean-Vlasov equation, which allows to define global solutions even in the presence of blow-ups. Solutions to the McKean-Vlasov equation arise as mean-field limits of particle systems interacting through hitting times, which is important for systemic risk modeling. During the presentation we will illustrate two of our main results. The first one consists in a general tightness theorem for the Skorokhod M1-topology which applies to processes that can be decomposed into a continuous and a monotone part. The second one concerns the propagation of chaos for a perturbed version of the particle system for general initial conditions.

In this talk, we will explore a Brownian perturbation of the supercooled Stefan problem which arises naturally as a model of contagion in financial markets, conditional on the market risk that remains after full diversification. In line with the work of Delarue, Nadtochiy, and Skholnikov, we first present a probabilistic formulation of the problem. Starting from there, we then give a simple outline of a succinct argument for whether or not blow-ups will occur with probability one, depending on the initial specification. Finally, we turn to the financial application, looking at how the problem may serve as the basis for a forward-looking systemic risk indicator, aiming to capture the current market sentiment, and briefly discussing ongoing empirical work in this direction.

We study physical solutions to the three-dimensional Stefan-Gibbs-Thomson Problems under radial symmetry, of which the existence has been established in [Stefan problem with surface tension: global existence of physical solutions under radial symmetry, Nadtochiy and Shkolnikov 2023]. In this talk, I will focus on the proof that the physical solution is unique, the first such result when the free boundary is not flat, or when two phases are present. The main argument relies on a detailed analysis of the hitting probabilities for a three-dimensional Brownian motion, as well as on a novel convexity property of the free boundary obtained by comparison techniques. We also establish a variety of regularity estimates for the free boundary and for the temperature function in the course of the proof. This talk is based on joint work with Sergey Nadtochiy and Mykhaylo Shkolnikov.

A numerical method for the computation of free boundaries when a stochastic representation is available will be discussed.  It is based on an algorithm which we call deep empirical risk minimization developed by E, Han and Jentzen.  Their approach applies generally to many stochastic optimal control problems.   In the presence of free boundaries, it has to be modified to account for training based on hitting times.  In this talk, I  outline how this is achieved for the classical problems of optimal stopping or the obstacle problem, and for the Stefan problem for the water-ice interfaces.  For the Stefan problem, we use the recent stochastic representations, the notion of physical probabilistic solutions, and level-sets parameterized by deep neural networks on the numerical side.