Related papers: Second order mean field games with degenerate diff…
The cornerstone underpinning deep learning is the guarantee that gradient descent on an objective converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, where there are multiple…
We consider a beam equation in presence of a leading degenerate operator which is not in divergence form. We impose clamped conditions where the degeneracy occurs and dissipative conditions at the other endpoint. We provide some conditions…
We prove the local boundedness of the solutions to degenerate second order partial differential equations of Kolmogorov type with measurable coefficients in divergence form, under minimal integrability assumption on the lower order…
In this study, we employ the established Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, along with the telescoping series method, to establish an observability inequality for the…
In this paper, using variational approaches, we investigate the first order planning problem arising in the theory of mean field games. We show the existence and uniqueness of weak solutions of the problem in the case of a large class of…
We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$ \partial_tu+\text{div}\big(f(u)\big)=\mathcal{L}[b(u)] $$ on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion…
In this paper we describe the long time behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of corresponding diffusion processes.
For a class of finite horizon first order mean field games and associated N-player games, we give a simple proof of convergence of symmetric N-player Nash equilibria in distributed open-loop strategies to solutions of the mean field game in…
We study a degenerate second order mean field game (MFG) system in a Hilbert space $H$ which couples a Fokker--Planck equation describing the evolution of probability measures on $H$ with a Hamilton--Jacobi--Bellman (HJB) equation for the…
In this paper, we develop an analytical framework for the partial differential equation underlying the consensus-based optimization model. The main challenge arises from the nonlinear, nonlocal nature of the consensus point, coupled with a…
Partial order reductions have been successfully applied to model checking of concurrent systems and practical applications of the technique show nontrivial reduction in the size of the explored state space. We present a theory of partial…
This paper investigates a linear-quadratic mean field games problem with common noise, where the drift term and diffusion term of individual state equations are coupled with both the state, control, and mean field terms of the state, and we…
Our main contributions include proving sufficient conditions for the existence of solution to a second order problem with nonzero nonlocal initial conditions, and providing a comprehensive analysis using fundamental solutions and…
Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods…
A cross-diffusion system describing ion transport through biological membranes or nanopores in a bounded domain with mixed Dirichlet-Neumann boundary conditions is analyzed. The ion concentrations solve strongly coupled diffusion equations…
We consider mildly degenerate Kirchhoff equations with a small parameter and a weak dissipation term. We prove the existence of global solutions when the parameter is small with respect to the size of initial data. Then we provide…
We establish existence and uniqueness of solutions to evolutive fractional Mean Field Game systems with regularizing coupling, for any order of the fractional Laplacian $s\in(0,1)$. The existence is addressed via the vanishing viscosity…
We investigate linear parabolic equations in divergence form with singular coefficients and non-smooth boundary data. When the diffusion, drift, or potential terms, as well as the initial or boundary conditions, are distributions rather…
The policy iteration method is a classical algorithm for solving optimal control problems. In this paper, we introduce a policy iteration method for Mean Field Games systems, and we study the convergence of this procedure to a solution of…
Coordination games with explicit spatial or relational structure are of interest to economists, ecologists, sociologists, and others studying emergent global properties in collective behavior. When assemblies of individuals seek to…