linear quadratic mean field control the invariant subspace method this note studies linear quadratic lq mean field social optimization problems with non uniform agents and indefinite state weight and analyzes the resulting linear system of ordinary differential equations odes subject to a partial initial condition and a growth condition we apply a subspace decomposition method for existence analysis and computation this is accomplished by carefully choosing the unspecified initial condition in the ode system such that the modes exceeding the admissible growth rate are not activated we further extend this method to lq mean field games

sle and virasoro representations fusion we continue the study of null vector equations in relation with partition functions of systems of schramm loewner evolutions sles by considering the question of fusion starting from n commuting sles seeded at distinct points the partition function satisfies n null vector equations at level 2 we show how to obtain higher level null vector equations by coalescing the seeds one by one as an example we extend schramm s formula for the position of a marked bulk point relatively to a chordal sle trace to an arbitrary number of sle strands the argument combines input from representation theory the study of verma modules for the virasoro algebra with regularity estimates themselves based on hypoellipticity and stochastic flow arguments

special values of multiple polylogarithms historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations much the same can be said for euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics more recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm euler sums and the riemann zeta function here we provide a general framework within which previously isolated results can now be properly understood applying the theory developed herein we prove several previously conjectured evaluations including an intriguing conjecture of don zagier

mean field games for large population multiagent systems with markov jump parameters in this paper distributed games for large population multiagent systems with random time varying parameters are investigated where the agents are coupled via their individual costs and the structure parameters are a family of independent markov chains with identical generators the cost function of each agent is a long run average tracking type functional with an unknown mean field coupling nonlinear term as reference signal to reduce the computational complexity the mean field approach is applied to construct distributed strategies the population statistics effect pse is used to approximate the average effect of all the agents and the distributed strategies are given through solving a markov jump tracking problem here the pse is a deterministic quantity and can be obtained by solving the stackelberg equilibrium of an auxiliary two player game it is shown that the closed loop system is uniformly stable and the distributed strategies are asymptotically optimal in the sense of nash equilibrium

toward a conformal field theory for schramm loewner evolutions we discuss the partition function point of view for chordal schramm loewner evolutions and their relationship with correlation functions in conformal field theory both are closely related to crossing probabilities and interfaces in critical models in two dimensional statistical mechanics we gather and supplement previous results with different perspectives point out remaining difficulties and suggest directions for future studies

periodic solutions of lagrangian systems on a compact manifold abstract let m be a smooth n dimensional manifold and let tm be its tangent bundle we consider a time periodic lagrangian of period t l t tm r and we seek t periodic solutions of the lagrange equations which in local coordinates are d dt l q t q q l q t q q 0 i 1 n our main result states that if the fundamental group of m is finite then has infinitely many t periodic solutions provided that l t satisfies certain physically reasonable assumptions

mean field games with common noise a theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise the concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations and existence of weak solutions for mean field games is shown to hold under very general assumptions examples and counter examples are provided to enlighten the underpinnings of the existence theory finally an analog of the famous result of yamada and watanabe is derived and it is used to prove existence and uniqueness of a strong solution under additional assumptions

large population lqg games involving a major player the nash certainty equivalence principle we consider linear quadratic gaussian lqg games with a major player and a large number of minor players the major player has a significant influence on others the minor players individually have negligible impact but they collectively contribute mean field coupling terms in the individual dynamics and costs to overcome the dimensionality difficulty and obtain decentralized strategies the so called nash certainty equivalence methodology is applied the control synthesis is preceded by a state space augmentation via a set of aggregate quantities giving the mean field approximation subsequently within the population limit the lqg game is decomposed into a family of limiting two player games as each is locally seen by a representative minor player next when solving these limiting two player games we impose certain interaction consistency conditions such that the aggregate quantities initially assumed coincide with the ones replicated by the closed loop of a large number of minor players this procedure leads to a set of decentralized strategies for the original lqg game which is an varepsilon nash equilibrium

mean field linear quadratic control uniform stabilization and social optimality abstract this paper is concerned with uniform stabilization and social optimality for general mean field linear quadratic control systems where subsystems are coupled via individual dynamics and costs and the state weight is not assumed with the definiteness condition for the finite horizon problem we first obtain a set of forward backward stochastic differential equations fbsdes from variational analysis and construct a feedback type control by decoupling the fbsdes for the infinite horizon problem by using solutions to two riccati equations we design a set of decentralized control laws which is further proved to be asymptotically social optimal some equivalent conditions are given for uniform stabilization of the systems in different cases respectively finally the proposed decentralized controls are compared to the asymptotic optimal strategies in previous works

symplectic manifolds with contact type boundaries an example of a 4 dimensional symplectic manifold with disconnected boundary of contact type is constructed a collection of other results about symplectic manifolds with contact type boundaries are derived using the theory ofj holomorphic spheres in particular the following theorem of eliashberg floer mcduff is proved if a neighbourhood of the boundary of v is symplectomorphic to a neighbourhood ofs2n 1 in standard euclidean space and if vanishes on all 2 spheres inv thenv is diffeomorphic to the ballb2n

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

periodic orbits in magnetic fields and ricci curvature of lagrangian systems where da f on the space of closed curves on the manifold mn here a is a 1 form i e f is an exact 2 form this functional is a natural generalization of the usual functional of length and its closed extremals correspond to periodic trajectories of the motion of particles on the riemannian manifold mn when the kinetic energy is defined by the metric tensor and the form f defines a magnetic field also this functional corresponds to the periodic orbits for other problems of classical mechanics and mathematical physics as it was shown in n2 n3 ns when the lagrangian function

asymptotically optimal decentralized control for large population stochastic multiagent systems the interaction of interest coupled decision makers and the uncertainty of individual behavior are prominent characteristics of multiagent systems mas how to break through the framework of conventional control theory which aims at single decision maker and single decision objective and to extend the methodology and tools in the stochastic adaptive control theory to analyze mas are of great significance in this paper a preliminary exploration is made in this direction and the decentralized control problem is considered for large population stochastic mas with coupled cost functions different from the deterministic discounted costs in the existing differential game models a time averaged stochastic cost function is adopted for each agent the decentralized control law is constructed based on the state aggregation method and tracking like quadratic optimal control by using probability limit theory the stability and optimality of the closed loop system are analyzed the main contributions of this paper include the following points 1 the concepts of asymptotic nash equilibrium in probability and almost surely respectively are introduced and the relationship between these concepts is illuminated which provide necessary tools for analyzing the optimality of the decentralized control laws 2 the closed loop system is shown to be almost surely uniformly stable and bounded independently of the number of agents n 3 the population state average psa is shown to converge to the infinite population mean ipm trajectory in the sense of both l2 norm and time average almost surely as n increases to infinity 4 the decentralized control law is designed and shown to be almost surely asymptotically optimal the cost of each agent based on local measurements converges to that based on global measurements almost surely as n increases to infinity

a c 2 smooth counterexample to the hamiltonian seifert conjecture in r 4 we construct a proper c 2 smooth function on r 4 such that its hamiltonian flow has no periodic orbits on at least one regular level set this result can be viewed as a c 2 smooth counterexample to the hamiltonian seifert conjecture in dimension four

mean field production output control with sticky prices nash and social solutions this paper presents an application of mean field control to dynamic production optimization with sticky prices and adjustment costs both noncooperative and cooperative solutions are considered by solving auxiliary limiting optimal control problems subject to consistent mean field approximations two sets of decentralized strategies are obtained and further shown to asymptotically attain nash equilibria and social optima respectively a numerical example is given to compare market prices firms outputs and costs under the two solution frameworks

periodic solutions on hypersurfaces and a result by c viterbo on considere un champ vectoriel hamiltonien x j h x x h x sur x r 2n h etant une fonction lisse dont le gradient h est defini par rapport a la metrique euclidienne on cherche des solutions periodiques sur une surface d energie donnee s x r 2n h x const supposee reguliere

controllability stabilizability and continuous time markovian jump linear quadratic control consideration is given to the control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes the relationship between appropriately defined controllability stabilizability properties and the solution of the infinite time jump linear quadratic jlq optimal control problems is also examined although the solution of the continuous time markov jlq problem with finite or infinite time horizons is known only sufficient conditions for the existence of finite cost constant stabilizing controls for the infinite time problem appear in the literature in this paper necessary and sufficient conditions are established these conditions are based on new definitions of controllability observability stabilizability and detectability that are appropriate for continuous time markovian jump linear systems these definitions play the same role for the jlq problem as the deterministic properties do for the linear quadratic regulator lqr problem

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

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

a class of team problems with discrete action spaces optimality conditions based on multimodularity in this paper we discuss a class of team problems with discrete action spaces we introduce multimodularity into team theory as a natural alternative to convexity in continuous spaces the main result relates coordinatewise optimal cw optimal points to the optimal team decision for a class of team problems the method is based on a characterization of coordinatewise minima of multimodular functions

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

schramm s formula and the green s function for multiple sle we construct martingale observables for systems of multiple sle curves by applying screening techniques within the cft framework recently developed by kang and makarov extended to admit multiple sles we illustrate this approach by rigorously establishing explicit expressions for the green s function and schramm s formula in the case of two curves growing towards infinity in the special case when the curves are fused and start at the same point some of the formulas we prove were predicted in the physics literature

loop erased walks and total positivity we consider matrices whose elements enumerate weights of walks in planar directed weighted graphs not necessarily acyclic these matrices are totally nonnegative more precisely all their minors are formal power series in edge weights with nonnegative coefficients a combinatorial explanation of this phenomenon involves loop erased walks applications include total positivity of hitting matrices of brownian motion in planar domains

social optima in mean field linear quadratic gaussian models with markov jump parameters this paper investigates social optima of mean field linear quadratic gaussian lqg control models with markov jump parameters the common objective of the agents is to minimize a social cost the cost average of the whole society in the cost functions there are coupled mean field terms first we consider the centralized case and get a parameterized equation of mean field effect then we design a set of distributed strategies by solving a limiting optimal control problem in an augmented state space subject to the consistency requirement for mean field approximation it is shown that the set of distributed strategies is asymptotically team optimal and the asymptotically optimal social cost value can be obtained explicitly the optimal social average cost is compared with the optimal individual cost in mean field games by virtue of the explicit expressions and the difference is further illustrated by a numerical example

stabilization of continuous time jump linear systems we investigate almost sure and moment stabilization of continuous time jump linear systems with a finite state markov jump form process we first clarify the concepts of spl delta moment stabilizability exponential spl delta moment stabilizability and stochastic spl delta moment stabilizability we then present results on the relationships among these concepts coupled riccati equations that provide necessary and sufficient conditions for mean square stabilization are given in detail and an algorithm for solving the coupled riccati equations is proposed moreover we show that individual mode controllability implies almost sure stabilizability which is not true for other types of stabilizability finally we present some testable sufficient conditions for spl delta moment stabilizability and almost sure stabilizability

some remarks on sle bubbles and schramm s two point observable simmons and cardy recently predicted a formula for the probability that the chordal sle8 3 path passes to the left of two points in the upper half plane in this paper we give a rigorous proof of their formula starting from this result we derive explicit expressions for several natural connectivity functions for sle8 3 bubbles conditioned to be of macroscopic size by passing to a limit with such a bubble we construct a certain chordal restriction measure and in this way obtain a proof of a formula for the probability that two given points are between two commuting sle8 3 paths the one point version of this result has been predicted by gamsa and cardy finally we derive an integral formula for the second moment of the area of an sle8 3 bubble conditioned to have radius 1 we evaluate the area integral numerically and relate its value to a hypothesis that the area follows the airy distribution

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