# Free boundary regularity and support propagation in mean field games and optimal transport

Sebastian Munoz

Wednesday, December 20, 2023

2:00-3:00 PM

Off Campus Location

This talk presents recent findings on the regularity of first-order mean

field game systems with a local coupling. We focus on systems where the initial

density is a compactly supported function on the real line. Our results show that

the solution is smooth in regions where the density is strictly positive and that

the density itself is globally continuous. Additionally, the speed of propagation

is determined by the behavior of the cost function for small densities. When the

coupling is entropic, we demonstrate that the support of the density propagates

with infinite speed. On the other hand, when f(m) = mθ with θ > 0, we prove

that the speed of propagation is finite. In this case, we establish that under

a natural non-degeneracy assumption, the free boundary is strictly convex and

enjoys C1,1

regularity. We also establish sharp estimates on the speed of support

propagation and the rate of long-time decay for the density. Our methods are based on analyzing a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

field game systems with a local coupling. We focus on systems where the initial

density is a compactly supported function on the real line. Our results show that

the solution is smooth in regions where the density is strictly positive and that

the density itself is globally continuous. Additionally, the speed of propagation

is determined by the behavior of the cost function for small densities. When the

coupling is entropic, we demonstrate that the support of the density propagates

with infinite speed. On the other hand, when f(m) = mθ with θ > 0, we prove

that the speed of propagation is finite. In this case, we establish that under

a natural non-degeneracy assumption, the free boundary is strictly convex and

enjoys C1,1

regularity. We also establish sharp estimates on the speed of support

propagation and the rate of long-time decay for the density. Our methods are based on analyzing a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

Building: | Off Campus Location |
---|---|

Location: | Virtual |

Event Type: | Workshop / Seminar |

Tags: | Mathematics |

Source: | Happening @ Michigan from Financial/Actuarial Mathematics Seminar - Department of Mathematics, Department of Mathematics |