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# Particle approximation of one-dimensional Mean-Field-Games with local interactions

• *Corresponding author: Serikbolsyn Duisembay

M. Di Francesco was supported by KAUST during his visit in 2020

S. Duisembay, D. A. Gomes and R. Ribeiro were partially supported by King Abdullah University of Science and Technology (KAUST) baseline funds and KAUST OSR-CRG2021-4674

• We study a particle approximation for one-dimensional first-order Mean-Field-Games (MFGs) with local interactions with planning conditions. Our problem comprises a system of a Hamilton-Jacobi equation coupled with a transport equation. As we deal with the planning problem, we prescribe initial and terminal distributions for the transport equation. The particle approximation builds on a semi-discrete variational problem. First, we address the existence and uniqueness of a solution to the semi-discrete variational problem. Next, we show that our discretization preserves some previously identified conserved quantities. Finally, we prove that the approximation by particle systems preserves displacement convexity. We use this last property to establish uniform estimates for the discrete problem. We illustrate our results for the discrete problem with numerical examples.

Mathematics Subject Classification: Primary: 91A16, 49N80; Secondary: 65C35.

 Citation: • • Figure 1.  Periodic case: $\Omega = \mathbb{T}$, $g(m) = m^2/2$ (hence, $G(r) = r^2/6$) and $V(x,t)$ is given by (32). (A) Optimal trajectories of the $N = 50$ particles minimizing (31). (B) Exact and approximate CDFs for $N = 50$ at $t = T/2$

Figure 2.  $\Omega = \mathbb{R}$, $g(m) = m$ (hence, $G(r) = r/2$) and $V(x,t)$ is given by (34). (A) Optimal trajectories of the $N = 50$ particles minimizing (31). (B) Exact and approximate CDFs for $N = 50$ at $t = T/2$

Figure 3.  $\Omega = \mathbb{R}$, $g(m) = m$ (hence, $G(r) = r/2$) and $V(x,t)$ is given by (35). (A) Optimal trajectories of the $N = 50$ particles minimizing (31). (B) Exact and approximate CDFs for $N = 50$ at $t = T/2$

Figure 4.  $L(v) = v^2/2$. (A) Optimal trajectories of the particles. (B) Discrete conserved quantity $\sum_{i = 1}^{N} L'\left(\frac{x_i^{m+1}-x_i^m}{\Delta t}\right)R_i$. (C) Discrete approximation of the semi-discrete quantity $\sum_{i = 1}^N (L'(u_i)u_i-L(u_i)-G(R_i))R_i$ given by (21)

Figure 5.  Displacement convexity illustration for $\sum_{i = 1}^{N} U(R_i(t)) (x_i(t) - x_{i-1}(t))$ with $U(z) = e^{-z}$, $N = 5, \Delta t = \frac{1}{20}$

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