Mean field type control problems lead to Hamilton–Jacobi–Bellman (HJB) equations on the space of probability measures. In this talk, we will present a probabilistic approach to constructing classical solutions through forward-backward stochastic differential equations (FBSDEs), allowing for degenerate and unbounded diffusion coefficients. Starting with linear dynamics, we will explain how convexity ensures the global well-posedness of the associated FBSDE system, and how variational equations establish the regularity of the value function needed for the HJB equation. We will then extend the framework to jump-diffusion models, where the associated HJB equation is an integro-partial differential equation. Finally, we will consider the case where a backward equation forms part of the state.

Ballistic deposition is a model of interface growth introduced by Vold in 1959,
which has remained largely mathematically intractable. It is defined as a process
of vertically falling blocks that stick to the top, the right or the left of growing
columns. We introduce a variant, the one-sided ballistic deposition model, in
which vertically falling blocks can only stick to the top or the upper right corner
of growing columns. We establish that in dimension d=2, strong KPZ universality
holds near the time axis, proving that the fluctuations of the height function there
are given by the Tracy-Widom GUE distribution. The proof is based on a graphical
construction of the process in terms of a last passage percolation model. This talk
is based on a joint work with Pablo Groisman, Santiago Saglietti and Sebastián
Zaninovich.

Mean field games (MFGs) and mean field control (MFC) provide a mathemat-
ical framework for modeling large controlled interacting particle systems, with
applications ranging from financial mathematics to biological neural networks. In
the first part of this talk, I will give an accessible introduction to MFGs and MFC.
I will then show how the infinite-dimensional case arises naturally in applications,
highlight its specific analytical challenges, and provide an overview of my recent
results.
In the second part of the talk, I will detail how we address these challenges in
recent joint work with A. Święch, where we investigate an MFG posed in a Hilbert
space and driven by degenerate noise. We study the associated MFG system
consisting of a Hamilton–Jacobi–Bellman (HJB) equation for the value function
coupled with a nonlinear Fokker–Planck (FP) equation for the distribution of the
particles, both governed by a degenerate Kolmogorov operator. This degeneracy
introduces significant analytical challenges. In particular, the HJB equation is
treated in the viscosity sense, while the FP equation is interpreted in a suitable
weak formulation. Proving the uniqueness of solutions to the FP equation becomes
particularly delicate. Under appropriate structural assumptions, we establish the
well-posedness of the MFG system.