MATHEMATICAL ANALYSIS WITH APPLICATIONS Seminar

 

PROGRAM

---

Plan rada Seminar iz matematičke analize sa primenama za NOVEMBAR 2023.

Registracija za učešće na seminaru je dostupna na sledećem linku:
https://miteam.mi.sanu.ac.rs/asset/LKGS835QQhi4x5uGw
Ukoliko ste već registrovani, predavanja možete pratiti na sledećem linku (nakon sto se ulogujete):
https://miteam.mi.sanu.ac.rs/asset/5dTX9wSbARfwfpNY5
Neulogovani korisnici mogu pratiti prenos predavanja na ovom linku (ali ne mogu postavljati pitanja osim putem chata i ne ulaze u evidenciju prisustva):
https://miteam.mi.sanu.ac.rs/call/dpesCPawfHWm6LN3L/4omy3gZWp8F_Q4RjQSiJO9g-PWGPlbZs_TYO3M6m00K

Četvrtak, 16.11.2023. u 16:00, Kneza Mihaila 36, sala 301f i Online
Miłosz Krupski, University of Wrocław
FULLY NONLINEAR MEAN FIELD GAMES
In the original formulation of mean field games (Lasry–Lions, Huang–Caines–Malham'e, 2006) agents control the drift of a Wiener process describing their movement and collect gains based on their position within the aggregate population. This can be seen as the ability to control the spatial direction of their advancement, to find and follow the optimal location.
By converting the problem to a PDE form, we obtain a coupled system of a Hamilton–Jacobi–Bellman and a Fokker–Planck–Kolmogorov equation
\begin{equation}
\begin{cases}
-\partial_tu=\Delta u+F(\nabla u)+f(m) & \text{on }[0,T]\times\mathbb R^d,\\
u(T)=\mathfrak{g}(m(T)) & \text{on }\mathbb R^d,\\
\partial_tm=\Delta m+\operatorname{div}(\nabla H(\nabla u)m) & \text{on }[0,T]\times\mathbb R^d,\\
m(0)=m_0 & \text{on }\mathbb R^d.
\end{cases}
\end{equation}
In the talk I will explain how the “undisturbed” movement of agents may instead be governed by a general L´evy process (one or multidimensional). Then, due to the stochastic nature of this setting, the control of direction is no longer directly available. The agents can only control the time rate at which they choose to disperse. In case of self-similar processes (like drift), this approach can still be reinterpreted as spatial control.
Such model justifies the study of a much broader class of mean field games. In the most general form the Hamilton–Jacobi–Bellman equation is fully nonlinear and the Fokker–Planck equation may be strongly degenerate
\begin{equation}
\begin{cases}
-\partial_tu=F(\mathcal Lu)+\mathfrak f(m) & \text{on }[0,T]\times\mathbb R^d,\\
u(T)=\mathfrak{g}(m(T)) & \text{on }\mathbb R^d,\\
\partial_tm=\mathcal L^*(F'(\mathcal Lu)m) & \text{on }[0,T]\times\mathbb R^d,\\
m(0)=m_0 & \text{on }\mathbb R^d.
\end{cases}
\end{equation}
Establishing the well-posedness of solutions requires a revision of classical proofs which involves some novel technical observations, as well as some innovative approaches to problems specific to this setting.
Based on joint work with Indranil Chowdhury (Kanpur) and Espen Jakobsen (Trondheim).

---