arXiv research

Let $X$be a complex hyperelliptic curve of genus two equipped with the canonical metric $ds^2$. We study mean field equations on complex hyperelliptic curves and show that the Gaussian curvature function of $(X,ds^2)$ determines an explicit solution to a mean field equation.

New inequality criterion for a mean field equation on spheres.

The paper connects PSO and CBO methods using stochastic modeling and mean-field limits.

Mean-field approximations simplify insurance liability calculations.

In his lectures at College de France, P.L. Lions introduced the concept of Master equation, see [5] for Mean Field Games. It is introduced in a heuristic fashion, from the system of partial differential equations, associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in…

Study on the smoothness of solutions to a specific type of stochastic differential equation.

Novel approach to Nash equilibrium in mean-field stochastic games with operator resolvents.

Proves existence of solution to Lichnerowicz equation on non-CMC manifolds.

Revises mean-field theory of Santa Fe model using kinetic theory.

Recently, variational approximations such as the mean field approximation have received much interest. We extend the standard mean field method by using an approximating distribution that factorises into cluster potentials. This includes undirected graphs, directed acyclic graphs and junction trees. We derive generaliz…

Let $Ω$ be an annulus. We prove that the mean field equation $-Δψ=\frac{e\sp{-βψ}}{\int\sbΩe\sp{-βψ}} $ admits a solution with zero boundary for $β\in (-16π,-8π)$. This is a supercritical case for the Moser-Trudinger inequality.

Mean field game theory studies the behavior of a large number of interacting individuals in a game theoretic setting and has received a lot of attention in the past decade (Lasry and Lions, Japanese journal of mathematics, 2007). In this work, we derive mean field game partial differential equation systems from determi…

Develops a dynamic mean field theory for reinforcement learning.

Study on LOB dynamics using mean-field game theory.

In this paper, we prove that the even solution of the mean field equation $Δu=λ(1-e^u)$ on $S^2$ must be axially symmetric when $4<λ\leq 8$. In particular, zero is the only even solution for $λ=6$. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.

Algorithm learns interaction kernels for particle systems from data.

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density up…

We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.

Study timelike surfaces with parallel mean curvature in Minkowski 4-space.

This paper extends transfer operator theory to McKean-Vlasov equations.

Paper studies a generalized mean field equation on closed Riemann surfaces.

Study on Langevin dynamics convergence rates and their application to GAN training.

We study the mean field games equations, consisting of the coupled Kolmogorov-Fokker-Planck and Hamilton-Jacobi-Bellman equations. The equations are complemented by initial and terminal conditions. It is shown that with some specific choice of data, this problem can be reduced to solving a quadratically nonlinear syste…

The paper analyzes the mean field Langevin dynamics and its convergence rate.

Study on liquidation games with market drop-out, proving unique equilibria.

New sampling method uses gradient-free IPS with RKHS velocity field.

Proves equations for high-dimensional gradient-based methods from Gaussian data.

The paper proves existence of solutions for mean field equations on compact Riemann surfaces.

Study on market entry timing in stock liquidation with trading constraints.

Recent studies have suggested that the cognitive process of the human brain is realized as probabilistic inference and can be further modeled by probabilistic graphical models like Markov random fields. Nevertheless, it remains unclear how probabilistic inference can be implemented by a network of spiking neurons in th…

Improved disability insurance model with collective health claims.

Given a regular bounded domain $Ω\subset\R{2m}$, we describe the limiting behavior of sequences of solutions to the mean field equation of order $2m$, $m\geq 1$, $(-Δ)^m u=ρ\frac{e^{2mu}}{\int_Ωe^{2mu}dx}\quad\text{in}Ω,$ under the Dirichlet boundary condition and the bound $0<ρ\leq C$. We emphasize the connection wi…

Modeling pollution from competing firms using mean-field games.

We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack in the case of closed manifolds and establish existence and non-existence results for the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds.

Mean field game with defaultable agents and systemic risk quantified.

Study shows finite agent equilibrium converges to mean-field limit in asset pricing.

In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -Δ_{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only const…

Study on mean field games with singular controls and their applications.

Study optimizes portfolio liquidation strategies with complex market impacts.

New neural networks learn mappings between probability measures and functions.

We review recent quantitative results on the approximation of mean field diffusion equations by large systems of interacting particles, obtained by optimal coupling methods. These results concern a larger range of models, more precise senses of convergence and links with the long time behaviour of the systems to be con…

In this paper, we give an algebraic construction of the solution to the following mean field equation $Δψ+e^ψ=4π\sum_{i=1}^{2g+2}δ_{P_{i}},$ on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots,P_{2g+2}\}$ is the set of Weierstrass points on $X.$ Furt…

Model analyzes competitive pricing strategies in large markets of perishable products.

The paper proves functions related to mean field equations on surfaces are Morse functions under certain conditions.

Deep Galerkin Method estimates value function for mean-field control problem.

Developed LQ MFG theory with common noise, proving existence and uniqueness.

Solves a game between brokers and informed traders using stochastic differential equations.

Study uses Mean Field Game to analyze Bitcoin mining hashpower dynamics.