density flow in dynamical networks via mean field games current distributed routing control algorithms for dynamic networks model networks using the time evolution of density at network edges while the routing control algorithm ensures edge density to converge to a wardrop equilibrium which was characterized by an equal traffic density on all used paths we rearrange the density model to recast the problem within the framework of mean field games in doing that we illustrate an extended state space solution approach and we study the stochastic case where the density evolution is driven by a brownian motion further we investigate the case where the density evolution is perturbed by a bounded adversarial disturbance for both the stochastic and the worst case scenarios we provide conditions for the density to converge to a pre assigned set moreover we analyze such conditions from two different perspectives repeated games with vector payoffs and inclusion theory

approximations of two dimensional mean field games with nonsymmetric controls abstract the numerical methods are presented for solving economic problems formulated in the mean field game mfg form the mean field equilibrium is a solution of the coupled system of two parabolic partial differential equations the fokker planck kolmogorov equation and the hamilton jacobi bellman one the description focuses on the discrete approximation of these equations and on the application of the mfg theory directly at discrete level this approach results in an efficient algorithm for finding the corresponding grid control functions contrary to other difference schemes here the semi lagrangian approximation is applied which improves properties of a discrete problem of this type this implies the fast convergence of an iterative algorithm for the minimization of the cost functional the constructed algorithms are implemented to the problem of carbon dioxide pollution

non coercive first order mean field games abstract we study first order evolutive mean field games where the hamiltonian are non coercive this situation occurs for instance when some directions are forbidden to the generic player at some points we establish the existence of a weak solution of the system via a vanishing viscosity method and mainly we prove that the evolution of the population s density is the push forward of the initial density through the flow characterized almost everywhere by the optimal trajectories of the control problem underlying the hamilton jacobi equation as preliminary steps we need to prove that the optimal trajectories for the control problem are unique at least for a e starting points and that the corresponding unique optimal control has a feedback expression in terms of the intrinsic gradient of the value function

a priori estimates of smoothness of solutions to difference bellman equations with linear and quasi linear operators a priori estimates for finite difference approximations for the first and second order derivatives are obtained for solutions of parabolic equations described in the title

value iteration algorithm for mean field games abstract in the literature existence of mean field equilibria has been established for discrete time mean field games under both the discounted cost and the average cost optimality criteria in this paper we provide a value iteration algorithm to compute stationary mean field equilibrium for both the discounted cost and the average cost criteria whose existence proved previously we establish that the value iteration algorithm converges to the fixed point of a mean field equilibrium operator then using this fixed point we construct a stationary mean field equilibrium in our value iteration algorithm we use q functions instead of value functions

nonequidimensional value distribution theory and meromorphic connections this is a sequel to the paper si2 1 in whichwe introduced the use ofmeromorphic connections to handle nonequidimensional value distribution theory the meromorphic connection is chosen so that the divisor has zero second fundamental form with respect to the meromorphic connection when we have more than one divisor they must have zero second fundamental form with respect to the same meromorphic connection such a condition is so stringent that it is difficult to find examples to which the method can be applied and which cannot be handled by other means one example is a collection of fermat curves in p2 of the form aowg atw a2wg 0 for the same homogeneous coordinate system wo wt w2 shift man i sh2 told me that for such a collection of fermat curves defect relations can be obtained by using a technique of h cartan c2 see 9 whereas the general case of allowing fermat curves with respect to different homogeneous coordinates cannot be handled by any of the known methods the only known result on the defect of general divisors is obtained by using the veronese map and is given in shl in the following form let f c pn be a holomorphic curve such that f c is not contained in any algebraic hypersurface in pn let sj be a collection of algebraic hypersurfaces of degree d in p the sum of the defects of f for the

the covering spectrum of a compact length space we define a new spectrum for compact length spaces and riemannian manifolds called the covering spectrum which roughly measures the size of the one dimensional holes in the space more specifically the covering spectrum is a set of real numbers 0 which identify the distinct covers of the space we investigate the relationship between this covering spectrum the length spectrum the marked length spectrum and the laplace spectrum we analyze the behavior of the covering spectrum under gromov hausdorff convergence and study its gap phenomenon

linear quadratic n person and mean field games with ergodic cost we consider stochastic differential games with n players linear gaussian dynamics in arbitrary state space dimension and long time average cost with quadratic running cost admissible controls

defect relations for holomorphic maps between spaces of different dimensions there is now a rather rich theory of defect relations for holomorphic maps between spaces of equal dimension developed by nevanlinna stoll griffiths and many other people however for the case of maps between spaces of unequal dimension essentially there is only the theory developed by weyl weyl and ahlfors when the target space is the complex projective space and their theory is built on the projective linear structure of the projective space a partial list of references for these results is given at the end of this paper in this paper we derive a defect relation for holomorphic maps between spaces of different dimensions where the target space can be an arbitrary compact complex algebraic manifold since the special case of maps from c to a compact complex manifold of complex dimension two contains already the methods and techniques needed for the general case in order to avoid unnecessarily complicated notations and to make the arguments more easily understood we give complete details only for this special case and indicate the easy modifications needed for the general case besides the usual techniques in value distribution theory of using the poincar6 lelong formula the calculus lemma of estimating the integrand by its integral and the concavity of logarithm a new key point in our argument is to introduce a meromorphic connection so that the second fundamental form of a divisor with respect to the connection is zero more precisely for the case of a meromorphic map f from c to a compact kihler surface m to consider the defect for a nonsingular zero set of a holomorphic section s of a positive holomorphic line bundle l we introduce a meromorphic connection i v form with the following two properties i for some hermitian holomorphic line bundle f over m and some holomorphic section 0 of f

minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field in this work we study the minimal time to steer a given crowd to a desired configuration the control is a vector field representing a perturbation of the crowd velocity localized on a fixed control set we will assume that there is no interaction between the agents we give a characterization of the minimal time both for microscopic and macroscopic descriptions of a crowd we show that the minimal time to steer one initial configuration to another is related to the condition of having enough mass in the control region to feed the desired final configuration the construction of the control is explicit providing a numerical algorithm for computing it we finally give some numerical simulations

stationary mean field games systems defined on networks we consider a stationary mean field games system constrained on a network according to the optimal control interpretation of the problem some transition conditions on the vertices are imposed we prove separately the well posedness for each of the two equations composing the system finally we prove existence and uniqueness of the solution of the mean field games system

a variational approach to the mean field planning problem abstract we investigate a first order mean field planning problem of the form t u h x d u f x m in 0 t r d t m m h p x d u 0 in 0 t r d m 0 m 0 m t m t in r d associated to a convex hamiltonian h with quadratic growth and a monotone interaction term f with polynomial growth we exploit the variational structure of the system which encodes the first order optimality condition of a convex dynamic optimal entropy transport problem with respect to the unknown density m and of its dual involving the maximization of an integral functional among all the subsolutions u of an hamilton jacobi equation combining ideas from optimal transport convex analysis and renormalized solutions to the continuity equation we will prove existence and at least partial uniqueness of a weak solution m u a crucial step of our approach relies on a careful analysis of distributional subsolutions to hamilton jacobi equations of the form t u h x d u under minimal summability conditions on and to a measure theoretic description of the optimality via a suitable contact defect measure finally using the superposition principle we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players

the second main theorem of hypersurfaces in the projective space we deal with a holomorphic map from the complex plane mathbb c to the n dimensional complex projective space mathbb p n mathbb c and prove the nevanlinna second main theorem for some families of non linear hypersurfaces in mathbb p n mathbb c this second main theorem implies the defect relation if the degree of the hypersurfaces are sufficiently large the defect of the map is smaller than one this means that holomorphic maps which omit the irreducible hypersurface of large degree is algebraically degenerate to prove the second main theorem we use a meromorphic partial projective connection which is totally geodesic with respect to these hypersurfaces a meromorphic partial projective connection is a family of locally defined meromorphic connections such which work as an entirely defined meromorphic connection under the wronskian operator by resolving the singularity and pulling back a meromorphic partial projective connection we also prove the second main theorem for singular hypersurfaces in mathbb p n mathbb c and prove the second main theorem for smooth hypersurfaces in mathbb p 2 mathbb c which are not normal crossing

defects for ample divisors of abelian varieties schwarz lemma and hyperbolic hypersurfaces of low degrees we prove that the defect vanishes for a holomorphic map f from the affine complex line to an abelian variety a and for an ample divisor d in a the proof uses the translational invariance of the zariski closure of the k jet space of the image of f and the theorem of riemann roch to construct a nonidentically zero meromorphic k jet differential whose pole divisor is dominated by a divisor equivalent to pd and which vanishes along the k jet space of d to order q with p q smaller than a prescribed small positive number then estimates involving the theta function with divisor d and the logarithmic derivative lemma are used we also prove a pointwise schwarz lemma which gives the vanishing of the pullback by a holomorphic map from the affine complex line to a compact complex manifold of a holomorphic jet differential vanishing on an ample divisor this pointwise schwarz lemma is a slight modification of a statement whose proof green and griffiths sketched in their alternative treatment of bloch s theorem on entire curves in abelian varieties the log pole case of the pointwise schwarz lemma is also given we construct examples of hyperbolic hypersurface whose degree is only 16 times the square of its dimension

discrete homotopies and the fundamental group abstract we generalize and strengthen the theorem of gromov that the fundamental group of any compact riemannian manifold of diameter at most d has a set of generators g 1 g k of length at most 2 d and relators of the form g i g m g j in particular we obtain an explicit bound for the number k of generators in terms of the number of short loops at every point and the number of balls required to cover a given semi locally simply connected geodesic space as a corollary we obtain a fundamental group finiteness theorem new even for riemannian manifolds that replaces the curvature and volume conditions of anderson and the 1 systole bound of shen wei by more general geometric hypothesis implied by these conditions this theorem in turn is a special case of a theorem for arbitrary compact geodesic spaces proved using the method of discrete homotopies introduced by the first author and v n berestovskii central to the proof is the notion of homotopy critical spectrum introduced in this paper as a natural consequence of discrete homotopy methods this spectrum is closely related to the sormani wei covering spectrum which is a subset of the classical length spectrum studied by de verdiere and duistermaat guillemin it is completely determined including multiplicity by special closed geodesics called essential circles

diffusion processes on graphs stochastic differential equations large deviation principle ito s rule is established for the diffusion processes on the graphs we also consider a family of diffusions processes with small noise on a graph large deviation principle is proved for these diffusion processes and their local times at the vertices

universal covers for hausdorff limits of noncompact spaces we prove that if y is the gromov hausdorff limit of a sequence of complete manifolds m n i with a uniform lower bound on ricci curvature then y has a universal cover

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

phase diagram of a mean field game abstract mean field games were introduced by j m lasry and p l lions in the mathematical community and independently by m huang and co workers in the engineering community to deal with optimization problems when the number of agents becomes very large in this article we study in detail a particular example called the seminar problem introduced by o gueant j m lasry and p l lions in 2010 this model contains the main ingredients of any mean field game but has the particular feature that all agents are coupled only through a simple random event the seminar starting time that they all contribute to form in the mean field limit this event becomes deterministic and its value can be fixed through a self consistent procedure this allows for a rather thorough understanding of the solutions of the problem through both exact results and a detailed analysis of various limiting regimes for a sensible class of initial configurations distinct behaviors can be associated to different domains in the parameter space for this reason the seminar problem appears to be an interesting toy model on which both intuition and technical approaches can be tested as a preliminary study toward more complex mean field game models

classical solvability of linear parabolic equations on networks on etudie la resolubilite classique des problemes aux valeurs limites et initiales pour des equations paraboliques lineaires generales sur des reseaux c 8 finis

weak solutions to fokker planck equations and mean field games we deal with systems of pdes arising in mean field games theory where viscous hamilton jacobi and fokker planck equations are coupled in a forward backward structure we consider the case of local coupling when the running cost depends on the pointwise value of the distribution density of the agents in which case the smoothness of solutions is mostly unknown we develop a complete weak theory proving that those systems are well posed in the class of weak solutions for monotone couplings under general growth conditions and for superlinear convex hamiltonians as a key tool we prove new results for fokker planck equations under minimal assumptions on the drift through a characterization of weak and renormalized solutions the results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned

the vanishing viscosity limit for hamilton jacobi equations on networks abstract for a hamilton jacobi equation defined on a network we introduce its vanishing viscosity approximation the elliptic equation is given on the edges and coupled with kirchhoff type conditions at the transition vertices we prove that there exists exactly one solution of this elliptic approximation and mainly that as the viscosity vanishes it converges to the unique solution of the original problem

linear elliptic difference inequalities with random coefficients we prove various pointwise estimates for solutions of linear ellip tic difference inequalities with random coefficients these estimates include discrete versions of the maximum principle of aleksandrov and harnack in equalities and holder estimates of krylov and safonov for elliptic differential operators with bounded coefficients

synchronization of coupled oscillators is a game the purpose of this paper is to understand phase transition in noncooperative dynamic games with a large number of agents applications are found in neuroscience biology and economics as well as traditional engineering applications the focus of analysis is a variation of the large population linear quadratic gaussian lqg model of huang et al 2007 comprised here of a controlled n dimensional stochastic differential equation model coupled only through a cost function the states are interpreted as phase angles for a collection of heterogeneous oscillators and in this way the model may be regarded as an extension of the classical coupled oscillator model of kuramoto a deterministic pde model is proposed which is shown to approximate the stochastic system as the population size approaches infinity key to the analysis of the pde model is the existence of a particular nash equilibrium in which the agents opt out of the game setting their controls to zero resulting in the incoherence equilibrium methods from dynamical systems theory are used in a bifurcation analysis based on a linearization of the partial differential equation pde model about the incoherence equilibrium a critical value of the control cost parameter is identified above this value the oscillators are incoherent and below this value when control is sufficiently cheap the oscillators synchronize these conclusions are illustrated with results from numerical experiments

complete intersection varieties with ample cotangent bundles any smooth projective variety contains many complete intersection subvarieties with ample cotangent bundles of each dimension up to half its own dimension

convergence and the length spectrum abstract the author defines and analyzes the 1 k length spectra l 1 k m whose union over all k n is the classical length spectrum these new length spectra are shown to converge in the sense that lim k k 1 k m i l 1 k m 0 as m i m in the gromov hausdorff sense energy methods are introduced to estimate the shortest element of l 1 k as well as a concept called the minimizing index which may be used to estimate the length of the shortest closed geodesic of a simply connected manifold in any dimension a number of gap theorems are proven including one for manifolds m n with ricci n 1 and volume close to vol s n many results in this paper hold on compact length spaces in addition to riemannian manifolds

linear quadratic gaussian mixed games with continuum parametrized minor players we consider a mean field linear quadratic gaussian game with a major player and a large number of minor players parametrized by a continuum set the mean field generated by the minor players is approximated by a random process depending only on the initial state and the brownian motion of the major player and this leads to two limiting optimal control problems with random coefficients which are solved subject to a consistency requirement on the mean field approximation the set of decentralized strategies constructed from the limiting control problems has an varepsilon nash equilibrium property when applied to the large but finite population model

mean field games convergence of a finite difference method mean field type models describing the limiting behavior of stochastic differential games as the number of players tends to infty have been recently introduced by lasry and lions numerical methods for the approximation of the stationary and evolutive versions of such models have been proposed by the authors in previous works here convergence theorems for these methods are proved under various assumptions on the coupling operator

mean field games numerical methods for the planning problem mean field games describe the asymptotic behavior of differential games in which the number of players tends to infty here we focus on the optimal planning problem i e the problem in which the positions of a very large number of identical rational agents with a common value function evolve from a given initial spatial density to a desired target density at the final horizon time we propose a finite difference semi implicit scheme for the optimal planning problem which has an optimal control formulation the latter leads to existence and uniqueness of the discrete control problem we also study a penalized version of the semi implicit scheme for solving the resulting system of equations we propose a strategy based on newton iterations we describe some numerical experiments

the cut off covering spectrum we introduce the r cut off covering spectrum and the cut off covering spectrum of a metric space or a riemannian manifold the spectra measure the sizes of localized holes in the space and are defined using covering spaces called covers and r cut off covers they are investigated using 6 homotopies which are homotopies via grids whose squares are mapped into balls of radius on locally compact spaces we prove that these new spectra are subsets of the closure of the length spectrum we prove the r cut off covering spectrum is almost continuous with respect to the pointed gromov hausdorff convergence of spaces and that the cut off covering spectrum is also relatively well behaved this is not true of the covering spectrum defined in our earlier work which was shown to be well behaved on compact spaces we close by analyzing these spectra on riemannian manifolds with lower bounds on their sectional and ricci curvature and their limit spaces

discrete methods for fully nonlinear elliptic equations this paper exhibits and proves the stability of discrete approximations for the solution of the dirichlet problem for fully nonlinear uniformly elliptic partial differential equations in situations where classical solutions are not known to exist the resulting solutions are characterized in the viscosity sense of crandall and lions traps amer math soc 177 1983 pp 1 42 and our stability analysis depends on estimates for nonlinear difference equations of positive type which are consequences of earlier work on linear difference equations with random coefficients

strongly coupled elliptic equations related to mean field games systems abstract in this paper we study existence of solutions for the following elliptic problem related to mean field games systems div m x div a x u f in div m x u u a x u u p in 0 u on where p 0 0 1 and f 0 is a function in some lebesgue space

hausdorff convergence and universal covers we prove that if y is the gromov hausdorff limit of a sequence of compact manifolds m n i with a uniform lower bound on ricci curvature and a uniform upper bound on diameter then y has a universal cover we then show that for i sufficiently large the fundamental group of mi has a surjective homeomorphism onto the group of deck transforms of y finally in the non collapsed case where the mi have an additional uniform lower bound on volume we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality a number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the mi are only assumed to be compact length spaces with a uniform upper bound on diameter

mean field games numerical methods mean field type models describing the limiting behavior as the number of players tends to infty of stochastic differential game problems have been recently introduced by j m lasry and p l lions c r math acad sci paris 343 2006 pp 619 625 c r math acad sci paris 343 2006 pp 679 684 jpn j math 2 2007 pp 229 260 numerical methods for the approximation of the stationary and evolutive versions of such models are proposed here in particular existence and uniqueness properties as well as bounds for the solutions of the discrete schemes are investigated numerical experiments are carried out

large population cost coupled lqg problems with nonuniform agents individual mass behavior and decentralized varepsilon nash equilibria we consider linear quadratic gaussian lqg games in large population systems where the agents evolve according to nonuniform dynamics and are coupled via their individual costs a state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an epsiv nash equilibrium property a stability property of the mass behavior is established and the effect of inaccurate population statistics on an isolated agent is also analyzed by variational techniques

on a mean field optimal control problem abstract in this paper we consider a mean field optimal control problem with an aggregation diffusion constraint where agents interact through a potential in the presence of a gaussian noise term our analysis focuses on a pde system coupling a hamilton jacobi and a fokker planck equation describing the optimal control aspect of the problem and the evolution of the population of agents respectively the main contribution of the paper is a result on the existence of solutions for the aforementioned system we notice this model is in close connection with the theory of mean field games systems however a distinctive feature concerns the nonlocal character of the interaction it affects the drift term in the fokker planck equation as well as the hamiltonian of the system leading to new difficulties to be addressed