arXiv subjects

Marco Cirant

Publications and source records attributed to Marco Cirant.

At least 19 recordsLinked to original sources

arXiv2026

On the unimportance of distant players in sparse stochastic differential network games

We study stochastic differential games with $N$ players, where interactions are determined by sequences of graphs in which the number of neighbours of each node remains bounded as $N$ grows, such as chain graphs or lattices. Our main goal is to quantify the phenomenon of the "unimportance of distant players" in such a large population, sparse regime: we show that, in order to determine the optimal trajectory in open-loop strategies of a given player with an arbitrarily small error, it suffices to consider a reduced game involving only the players at a certain distance in the graph, assigning arbitrary trajectories to the farther ones. Our main result provides an explicit non-asymptotic estimate in terms of the graph distance, valid independently of the time horizon $T$, under suitable convexity and monotonicity assumptions on the costs. Similar results are obtained for games in distributed strategies.

math.AP↗

arXiv2025

The local Turnpike Property in Mean Field Control and Games with quadratic Hamiltonian

We study the local stability properties of solutions to ergodic and discounted mean field games systems, as the time horizon $T \to +\infty$, around stationary equilibria, when the Hamiltonian is quadratic. We replace the usual monotonicity of the coupling term with a weaker, local assumption on the stationary equilibrium (that need not be unique), stemming from a second-order strict positivity condition. This new stability assumption, together with a symmetry property of the system, allows us to derive an exponential turnpike property for those solutions that are close to the stationary one, whenever the spatial domain $\Omega$ is either the flat torus $\mathbb{T}^n$ or $\mathbb{R}^n$. Finally, through a fixed-point argument, we establish the actual existence of stable solutions, both on the finite horizon $[0,T]$ and on the infinite horizon, in the periodic setting $\Omega=\mathbb{T}^n$, provided that the initial (and terminal) data are close enough to the stationary equilibrium.

math.AP↗

arXiv2025

Uniqueness of solutions to MFG systems with large discount

We prove that solutions to a class of Mean Field Game systems with discount are unique provided that the discount factor is large enough, and the Lagrangian term is (proportionally) small enough. This identifies an asymptotic uniqueness regime that falls outside the usual ones involving monotonicity.

math.AP↗

arXiv2025

A non-asymptotic approach to stochastic differential games with many players under semi-monotonicity

We consider stochastic differential games with a large number of players, with the aim of quantifying the gap between closed-loop, open-loop and distributed equilibria. We show that, under two different semi-monotonicity conditions, the equilibrium trajectories are close when the interactions between the players are weak. Our approach is non-asymptotic in nature, in the sense that it does not make use of any a priori identification of a limiting model, like in mean field game (MFG) theory. The main technical step is to derive bounds on solutions to systems of PDE/FBSDE characterizing the equilibria that are independent of the number of players. When specialized to the mean field setting, our estimates yield quantitative convergence results for both open-loop and closed-loop equilibria without any use of the master equation. In fact, our main bounds hold for games in which interactions are much sparser than those of MFGs, and so we can also obtain some "universality" results for MFGs, in which we show that games governed by dense enough networks converge to the usual MFG limit. Finally, we use our estimates to study a joint vanishing viscosity and large population limit in the setting of displacement monotone games without idiosyncratic noise.

math.PR↗

arXiv2025

Convergence rates for the vanishing viscosity approximation of Hamilton-Jacobi equations: the convex case

We study the speed of convergence in $L^\infty$ norm of the vanishing viscosity process for Hamilton-Jacobi equations with uniformly or strictly convex Hamiltonian terms with superquadratic behavior. Our analysis boosts previous findings on the rate of convergence for this procedure in $L^p$ norms, showing rates in sup-norm of order $\mathcal{O}(\epsilon^\beta)$, $\beta\in(1/2,1)$, or $\mathcal{O}(\epsilon|\log\epsilon|)$ with respect to the vanishing viscosity parameter $\epsilon$, depending on the regularity of the initial datum of the problem and convexity properties of the Hamiltonian. Our proofs are based on integral methods and avoid the use of techniques based on stochastic control or the maximum principle.

math.AP↗

arXiv2024

Long Time Behavior and Stabilization for Displacement Monotone Mean Field Games

This paper is devoted to the study of the long time behavior of Nash equilibria in Mean Field Games within the framework of displacement monotonicity. We first show that any two equilibria defined on the time horizon $[0,T]$ must be close as $T \to \infty$, in a suitable sense, independently of initial/terminal conditions. The way this stability property is made quantitative involves the $L^2$ distance between solutions of the associated Pontryagin system of FBSDEs that characterizes the equilibria. Therefore, this implies in particular the stability in the 2-Wasserstein distance for the two flows of probability measures describing the agent population density and the $L^2$ distance between the co-states of agents, that are related to the optimal feedback controls. We then prove that the value function of a typical agent converges as $T \to \infty$, and we describe this limit via an infinite horizon MFG system, involving an ergodic constant. All of our convergence results hold true in a unified way for deterministic and idiosyncratic noise driven Mean Field Games, in the case of strongly displacement monotone non-separable Hamiltonians. All these are quantitative at exponential rates.

math.OC↗

arXiv2024

A priori estimates and large population limits for some nonsymmetric Nash systems with semimonotonicity

We address the problem of regularity of solutions $u^i(t, x^1, \dots, x^N)$ to a family of semilinear parabolic systems of $N$ equations, which describe closed-loop equilibria of some $N$-player differential games with Lagrangian having quadratic behaviour in the velocity variable, running costs $f^i(x)$ and final costs $g^i(x)$. By global (semi)monotonicity assumptions on the data $f=(f^i)_{1 \leq i \leq N}$ and $g=(g^i)_{1 \leq i \leq N}$, and assuming that derivatives of $f^i, g^i$ in directions $x^j$ are of order $1/N$ for $j \neq i$, we prove that derivatives of $u^i$ enjoy the same property. The estimates are uniform in the number of players $N$. Such a behaviour of the derivatives of $f^i, g^i$ arise in the theory of Mean Field Games, though here we do not make any symmetry assumption on the data. Then, by the estimates obtained we address the convergence problem $N \to \infty$ in a "heterogeneous'' Mean Field framework, where players all observe the empirical measure of the whole population, but may react differently from one another. We also discuss some results on the joint $N \to \infty$ and vanishing viscosity limit.

math.AP↗

arXiv2024

Critical Mass Phenomena and Blow-up behavior of Ground States in stationary second order Mean-Field Games systems with decreasing cost

This paper is devoted to the study of Mean-field Games (MFG) systems in the mass critical exponent case. We firstly establish the optimal Gagliardo-Nirenberg type inequality associated with the potential-free MFG system. Then, under some mild assumptions on the potential function, we show that there exists a critical mass $M^*$ such that the MFG system admits a least energy solution if and only if the total mass of population density $M$ satisfies $M<M^*$. Moreover, the blow-up behavior of energy minimizers are captured as $M\nearrow M^*$. In particular, given the precise asymptotic expansions of the potential, we establish the refined blow-up behavior of ground states as $M\nearrow M^*.$ While studying the existence of least energy solutions, we establish new local $W^{2,p}$ estimates of solutions to Hamilton-Jacobi equations with superlinear gradient terms.

math.AP↗

arXiv2024

Some remarks on Linear-quadratic closed-loop games with many players

We identify structural assumptions which provide solvability of the Nash system arising from a linear-quadratic closed-loop game, with stable properties with respect to the number of players. In a setting of interactions governed by a sparse graph, both short-time and long-time existence of a classical solution for the Nash system set in infinitely many dimensions are addressed, as well as convergence to the solution to the respective ergodic problem as the time horizon goes to infinity; in addition, equilibria for the infinite-dimensional game are shown to provide $ε$-Nash closed-loop equilibria for the $N$-player game. In a setting of generalized mean-field type (where the number of interactions is large but not necessarily symmetric), directly from the $N$-player Nash system estimates on the value functions are deduced on an arbitrary large time horizon, which should pave the way for a convergence result as $N$ goes to infinity.

math.OC↗

arXiv2023

Stationary equilibria and their stability in a Kuramoto MFG with strong interaction

Recently, R. Carmona, Q. Cormier, and M. Soner proposed a Mean Field Game (MFG) version of the classical Kuramoto model, which describes synchronization phenomena in a large population of rational interacting oscillators. The MFG model exhibits several stationary equilibria, but the characterization of these equilibria and their ability to capture dynamic equilibria in long time remains largely open. In this paper, we demonstrate that, up to a phase translation, there are only two possible stationary equilibria: the incoherent equilibrium and the self-organizing equilibrium, given that the interaction parameter is sufficiently large. Furthermore, we present some local stability properties of the self-organizing equilibrium.

math.AP↗

arXiv2023

Comparison principles for nonlinear potential theories and PDEs with fiberegularity and sufficient monotonicity

We present some recent advances in the productive and symbiotic interplay between general potential theories (subharmonic functions associated to closed subsets $\mathcal{F} \subset \mathcal{J}^2(X)$ of the 2-jets on $X \subset \mathbb{R}^n$ open) and subsolutions of degenerate elliptic and parabolic PDEs of the form $F(x,u,Du,D^2u) = 0$. We will implement the monotonicity-duality method begun by Harvey and Lawson in 2009 (in the pure second order constant coefficient case) for proving comparison principles for potential theories where $\mathcal{F}$ has sufficient monotonicity and fiberegularity (in variable coefficient settings) and which carry over to all differential operators $F$ which are compatible with $\mathcal{F}$ in a precise sense for which the correspondence principle holds. We will consider both elliptic and parabolic versions of the comparison principle in which the effect of boundary data is seen on the entire boundary or merely on a proper subset of the boundary. Particular attention will be given to gradient dependent examples with the requisite sufficient monotonicity of proper ellipticity and directionality in the gradient. Example operators we will discuss include the degenerate elliptic operators of optimal transport in which the target density is strictly increasing in some directions as well as operators which are weakly parabolic in the sense of Krylov. Further examples, modeled on hyperbolic polynomials in the sense of Gårding give a rich class of examples with directionality in the gradient. Moreover we present a model example in which the comparison principle holds, but standard viscosity structural conditions fail to hold.

math.AP↗

arXiv2023

Ergodic Mean Field Games: existence of local minimizers up to the Sobolev critical case

We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.

math.AP↗

arXiv2022

Gradient estimates for quasilinear elliptic Neumann problems with unbounded first-order terms

This paper studies global a priori gradient estimates for divergence-type equations patterned over the $p$-Laplacian with first-order terms having polynomial growth with respect to the gradient, under suitable integrability assumptions on the source term of the equation. The results apply to elliptic problems with unbounded data in Lebesgue spaces complemented with Neumann boundary conditions posed on convex domains of the Euclidean space.

math.AP↗

arXiv2022

On the improvement of Hölder seminorms in superquadratic Hamilton-Jacobi equations

We show in this paper that maximal $L^q$-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superquadratic $γ$-growth in the gradient holds in the full range $ q > (N+2)\frac{γ-1}γ$. Our approach is based on new $\frac{γ-2}{γ-1}$-Hölder estimates, which are consequence of the decay at small scales of suitable nonlinear space and time Hölder quotients. This is obtained by proving suitable oscillation estimates, that also give in turn some Liouville type results for entire solutions.

math.AP↗

arXiv2022

Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms

We study the local Hölder regularity of strong solutions $u$ of second-order uniformly elliptic equations having a gradient term with superquadratic growth $γ> 2$, and right-hand side in a Lebesgue space $L^q$. When $q > N\frac{γ-1}γ$ ($N$ is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent $α_q > \frac{γ-2}{γ-1}$. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of $u$ in $L^q$. Our methods are based on blow-up techniques and a Liouville theorem.

math.AP↗

arXiv2021

On the Liouville property for fully nonlinear equations with superlinear first-order terms

We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is $F(x,D^2u)\geq H_i(x,u,Du)$ in $\mathbb{R}^N$, where $H_i$ has superlinear growth in the gradient variable. After a brief survey on the existing literature, we discuss the validity or the failure of the Liouville property in the model cases $H_1(u,Du)=u^q+|Du|^γ$, $H_2(u,Du)=u^q|Du|^γ$ and $H_3(x,u,Du)=\pm u^q|Du|^γ-b(x)\cdot Du$, where $q\geq0$, $γ>1$ and $b$ is a suitable velocity field. Several counterexamples and open problems are thoroughly discussed.

math.AP↗

arXiv2021

Long time behaviour and turnpike solutions in mildly non-monotone mean field games

We consider mean field game systems in time-horizon $(0,T)$, where the individual cost functional depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (i.e. the aggregation rate of the cost functions) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either the case of globally Lipschitz Hamiltonians or the case of quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we investigate the long time behavior of solutions and we give a complete description of the ergodic and long term properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon $(0,T)$, (ii) the convergence of the system in $(0,T)$ towards the system in $(0,\infty)$, (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. This way we extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.

math.AP↗

arXiv2020

Brake orbits and heteroclinic connections for first order Mean Field Games

We consider first order variational MFG in the whole space, with aggregative interactions and density constraints, such that the stationary states of the game are contained in two isolated compact sets of mass distributions with finite quadratic moments. Under general assumptions on the interaction potential, we provide a method for the construction of periodic in time solutions for the MFG, which oscillate among the two sets of static equilibria. Moreover, as the period increases to infinity, we show that these periodic solutions converge, in a suitable sense, to heteroclinic connections. As a model example, we consider a MFG system where the interactions are modeled via a Riesz-type aggregative potential with spatial preferences.

math.AP↗