on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

on the linear quadratic gaussian nash game with one step delay observation sharing pattern this paper studies a two player nash dynamic discrete time linear quadratic game under the so called one step delay observation sharing pattern it is shown that under very weak assumptions the solution exists and is unique and linear in the information

wellposedness of mean field games with common noise under a weak monotonicity condition in this paper we consider mean field games in the presence of common noise relaxing the usual independence assumption of individual random noise we assume a simple linear model with terminal cost satisfying a convexity and a weak monotonicity property our main result is showing existence and uniqueness of a mean field game solution using the stochastic maximum principle the uniqueness is a result of a monotonicity property similar to that of lasry and lions we use the banach fixed point theorem to establish an existence over a small time duration and show that it can be extended over an arbitrary finite time duration

generation of conditional densities in nonlinear filtering for infinite dimensional systems consider a signal process that takes values in an infinite dimensional space and assume that this process is observed through a nonlinear diffusion process in the nonlinear filtering theory literature recursive expressions for such problems in the form of conditional expectations and consequently measure valued stochastic partial differential equations on the underlying infinite dimensional space are obtained in this work state estimation problems for such stochastic systems are considered and nonlinear filtering equations in the form of conditional densities are obtained this is achieved by first deriving the filtering equations in the form of conditional expectations for a banach space valued state process whose dynamics depends on a finite dimensional brownian motion next a recursion for the conditional density which we assume to exist with respect to a sigma finite measure is obtained for such an analysis we obtain the adjoint operator for the generator of the state process which requires to use the theory of integration by parts in the infinite dimensional domains finally we discuss some potential application of the derived filtering equations in a portfolio optimization problem for portfolios which consist of bonds whose price curve is interpreted as a stochastic partial differential equation taking values in a banach space applications in mean field games are also demonstrated

efficient interactive methods for a class of multiattribute shortest path problems given an acyclic network and a preference order relation on paths when and how can bellman s principle of optimality be combined with interactive programming to efficiently locate an optimal path we show that if preferences are defined via a collection of attributes then under common conditions the principle of optimality is valid if and only if the preferences can be represented by a linear value function over the attributes consequently an interactive programming method is suggested which assesses the value function while using the principle of optimality to efficiently search for an optimal path

on the relation of zakai s and mortensen s equations the problem of optimal control for partially observable diffusion processes is studied by the dynamic programming in function space approach first proposed by r e mortensen the density of the conditional distribution of the unobservable signal given past and present observations which satisfies the zakai equation of nonlinear filtering is viewed as the new state of the system a verification result is established for the corresponding bellman hamilton jacobi equation known as mortensen s equation

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

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

nonlinear filtering theory for mckean vlasov type stochastic differential equations we consider estimation theory for partially observed stochastic dynamical systems whose state equations are given by mckean vlasov type stochastic differential equations and hence contain a measure term corresponding to the distribution of the solution of the state process nonlinear filtering equations are derived in this framework based upon the classification that the measure term is either stochastic or deterministic and that either the state or the measure term is estimated when the measure term is deterministic the standard theory holds further when the measure is random the induced functions in the dynamics of the state become random and a similar recursion for the optimal filter is obtained the joint estimation of state and the measure term is next considered the extended state in this setup is shown to be a banach space valued stochastic process with random functions in its state dynamics and a nonlinear filtering equation for this setup is provided the first step of such an analysis requ

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

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

discrete time mean field stochastic linear quadratic optimal control problems this paper firstly presents necessary and sufficient conditions for the solvability of discrete time mean field stochastic linear quadratic optimal control problems secondly the optimal control within a class of linear feedback controls is investigated using a matrix dynamical optimization method thirdly by introducing several sequences of bounded linear operators the problem is formulated as an operator stochastic linear quadratic optimal control problem by the kernel range decomposition representation of the expectation operator and its pseudo inverse the optimal control is derived using solutions to two algebraic riccati difference equations finally by completing the square the two riccati equations and the optimal control are also obtained

an empirical investigation of some bicriterion shortest path algorithms abstract several applications involve finding shortest paths in bicriterion networks those in which each arc possesses two characteristics such as time and cost in these circumstances it is appropriate to find efficient pareto optimal paths between various points in the network that is paths are sought for which no alternative path has a smaller value for the first characteristic without a larger value for the second in this paper several implementations of a general label correcting algorithm are discussed for bicriterion networks these implementations are then compared on networks of varying size and having varying degrees of correlation between the two criteria the empirical findings shed some light on the ability to solve bicriterion problems in practice and on those factors that significantly influence the computational effort

electric load model synthesis by diffusion approximation of a high order hybrid state stochastic system a statistical approach is used to model the dynamics of the electric demand of large aggregates of electric space heaters or air conditioners the importance of such loads is twofold first they account for a significant portion of power system dynamics following a power outage second because they are associated with energy storage they are often selected for load shedding within a load management program the derivation of the aggregate electrical dynamics is considered first for a homogeneous group of devices subsequently a perturbation analysis yields the dynamics for a nonhomogeneous group the homogeneons group aggregrate load model is a system of coupled ordinary and partial differential equations fokker planck equations it is obtained by writing evolution equations for the probability density of a hybrid state markov system used to model the switching dynamics of individual devices this result is new and could give a clue to the analysis of a broad class of hybrid state stochastic systems in turn this could provide a new impetus not only in the area of electric load modeling but other areas such as power system reliability and the design of relay control systems where stochastic hybrid state models occur frequently simulation results which illustrate the dynamical properties of the model are presented

hierarchically accelerated dynamic programming for finite state machines a procedure called hierarchically accelerated dynamic programming hadp is presented which at the cost of a degree of suboptimality can significantly accelerate dynamic programming algorithms for discrete event systems modeled by finite state machines fsms the methodology is based upon the possibly iterated dynamical abstraction of a given fsm by state aggregation in order to generate a so called partition machine hierarchy necessary and sufficient conditions for the hadp procedure to generate globally optimal solutions are given as well as bounds on the degree of suboptimality of the method a group of examples called the broken manhattan grid problems is used to illustrate an implementation of hadp with two and three level hierarchies a set of open problems is described concerning the construction and selection of the partition machine abstractions and the improvement of the estimation of hadp suboptimality

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

ito s formula in umd banach spaces and regularity of solutions of the zakai equation using the theory of stochastic integration for processes with values in a umd banach space developed recently by the authors an ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in umd banach spaces the abstract results are applied to prove regularity in space and time of the solutions of the zakai equation

a general decomposition approach for multi criteria decision trees in this paper we address the dynamic and multi criteria decision making problems under uncertainty generally represented by multi criteria decision trees decision making consists of choosing at each period a decision that maximizes the decision maker outcomes these outcomes should often be measured against a set of heterogeneous and conflicting criteria generating the set of non dominated solutions is a common approach considered in the literature to solve the multi criteria decision trees but it becomes very challenging for large problems we propose a new approach to solve multi criteria decision trees without generating the set of all non dominated solutions which should reduce the computation time and the cardinality of the solution set in particular the proposed approach combines the advantages of decomposition with the application of multi criteria decision aid mcda methods at each decision node a generalization of the bellman s principle of decomposition to the multi criteria context is put forward a decomposition theorem is therefore proposed under the sufficient conditions stated by the theorem the principle of decomposition will generate the set of best compromise strategies seven mcda methods are then characterized lexicographic weighted sum multi attribute value theory topsis electre iii and promethee ii against the conditions of the theorem of decomposition and against other properties neutrality anonymous fidelity dominance independency in order to confirm or infirm their applicability with the proposed decomposition principle moreover the relationship between independency and temporal consistence is discussed as well as the effects of incomparableness rank reversals and use of thresholds two conjectures resulted from this characterization

extremal metrics for spectral functions of dirac operators in even and odd dimensions abstract let m n g be a closed smooth riemannian spin manifold and denote by its atiyah singer dirac operator we study the variation of riemannian metrics for the zeta function and functional determinant of 2 and prove finiteness of the morse index at stationary metrics and local extremality at such metrics under general i e not only conformal change of metrics in even dimensions which is also a new case for the conformal laplacian the relevant stability operator is of log polyhomogeneous pseudodifferential type and we prove new results of independent interest on the spectrum for such operators we use this to prove local extremality under variation of the riemannian metric which in the important example when m n g is the round n sphere gives a partial verification of branson s conjecture on the pattern of extremals thus det 2 has a local max max min min when the dimension is 4 k 4 k 1 4 k 2 4 k 3 respectively

generalized dynamic programming for multicriteria optimization abstract applications of dynamic programming dp to multicriteria sequential decision problems involving the optimization of a multicriteria preference function have been rare this is due to the ease with which dp s monotonicity assumption can be violated in such situations generalized dp avoids the potential pitfalls created by this absence of monotonicity thereby guaranteeing optimality the methodology is applied to a prototypical multicriteria dp problem namely a multicriteria version of the shortest path problem

short time solution to the master equation of a first order mean field game abstract the goal of this paper is to show existence of short time classical solutions to the so called master equation of first order mean field games which can be thought of as the limit of the corresponding master equation of a stochastic mean field game as the individual noises approach zero despite being the equation of an idealistic model its study is justified as a way of understanding mean field games in which the individual players randomness is negligible in this sense it can be compared to the study of ideal fluids we restrict ourselves to mean field games with smooth coefficients but do not impose any monotonicity conditions on the running and initial costs and we do not require convexity of the hamiltonian thus extending the result of gangbo and swiech to a considerably broader class of hamiltonians

stochastic integration in umd banach spaces in this paper we construct a theory of stochastic integration of processes with values in l h e where h is a separable hilbert space and e is a umd banach space i e a space in which martingale differences are unconditional the integrator is an h cylindrical brownian motion our approach is based on a two sided l p decoupling inequality for umd spaces due to garling which is combined with the theory of stochastic integration of l h e valued functions introduced recently by two of the authors we obtain various characterizations of the stochastic integral and prove versions of the ito isometry the burkholder davis gundy inequalities and the representation theorem for brownian martingales

asymptotic solutions to weakly coupled stochastic teams with nonclassical information the authors develop a new iterative approach toward the solution of a class of two agent dynamic stochastic teams with nonclassical information when the coupling between the agents is weak either through the state dynamics or through the information channel in each case the weak coupling is characterized in terms of a small perturbation parameter when this parameter value say in is set equal to zero the original fairly complex dynamic team with a nonclassical information pattern is decomposed into or converted to relatively simple stochastic control or team problems the solution of which makes up the zeroth order approximation in a function space to the team optical solution of the original problem the fact that the zeroth order solution approximates the optimal cost up to at least o in is shown it is also shown that approximations of all orders can be obtained by solving a sequence of stochastic control and or simpler team problems

epsilon nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents this paper studies large population dynamic games involving nonlinear stochastic dynamical systems with agents of the following mixed types i a major agent and ii a population of n minor agents where n is very large the major and minor agents are coupled via both i their individual nonlinear stochastic dynamics and ii their individual finite time horizon nonlinear cost functions this problem is analyzed by the so called epsilon nash mean field game theory a distinct feature of the mixed agent mean field game problem is that even asymptotically as the population size n approaches infinity the noise process of the major agent causes random fluctuation of the mean field behavior of the minor agents to deal with this the overall asymptotic n rightarrow infty mean field game problem is decomposed into i two nonstandard stochastic optimal control problems with random coefficient processes which yield forward adapted stochastic best response control processes determined from the

continuum and finite player noncooperative models of competition the anonymous interaction of large numbers of economic agents is a kind of noncooperative situation which is markedly different from small numbers strategic conflict the mathematical model of a nonatomic game or a game with a continuum of players has been introduced as a model for these many agent situations on the basis that its equilibria should closely approximate those of games with large finite numbers of players this paper contains a precise definition of what it means for a nonatomic game to be the limit of a sequence of finite player games and a theorem which states when the limit of equilibria of finite player games will be an equilibrium of the nonatomic limit game this is analogous to theorems prompted by edgeworth s conjecture in core theory it is derived from a general set of sufficient conditions for the graph of a noncooperative equilibrium correspondence to be closed

critical metrics for the determinant of the laplacian in odd dimensions let m be a closed compact n dimensional manifold with n odd we calculate the first and second variations of the zeta regularized determinants det a and det l as the metric on m varies where a denotes the laplacian on functions and l denotes the conformal laplacian we see that the behavior of these functionals depends on the dimension indeed every critical metric for 1 n 1 2det a or 1 n 1 21 det l has finite index consequently there are no local maxima if n 4rm 1 and no local minima if n 4m 3 we show that the standard 3 sphere is a local maximum for det a while the standard 4m 3 sphere with m 1 2 is a saddle point by contrast for all odd n the standard n sphere is a local extremal for det l an important tool in our work is the canonical trace on odd class operators in odd dimensions this trace is related to the determinant by the formula det q tr log q and we prove some basic results on how to calculate this trace

risk sensitive mean field games in this paper we study a class of risk sensitive mean field stochastic differential games we show that under appropriate regularity conditions the mean field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a hamilton jacobi bellman hjb equation with an additional quadratic term we provide an explicit solution of the mean field best response when the instantaneous cost functions are log quadratic and the state dynamics are affine in the control an equivalent mean field risk neutral problem is formulated and the corresponding mean field equilibria are characterized in terms of backward forward macroscopic mckean vlasov equations fokker planck kolmogorov equations and hjb equations we provide numerical examples on the mean field behavior to illustrate both linear and mckean vlasov dynamics

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

uniqueness of the functional determinant the functional determinant of the conformal laplacian and the square of the dirac operator are known to be extremized at the standard round metric of the four sphere among all conformal metrics up to gauge equivalence in this article we show that this is the unique critical point thus extending the work of onofri and osgood phillips and sarnak for the functional determinant on s 2 which characterized the constant curvature metric as the unique critical point of the determinant in addition we introduce a new symmetric two tensor field which is defined on any conformally flat four manifold and can be viewed as a fourth order generalization of the einstein gravitational tensor as a consequence we prove a pohozaev identity for manifolds with boundary which admit conformal killing vector fields

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

uplink power adjustment in wireless communication systems a stochastic control analysis this paper considers mobile to base station power control for lognormal fading channels in wireless communication systems within a centralized information stochastic optimal control framework under a bounded power rate of change constraint the stochastic control problem and its associated hamilton jacobi bellman hjb equation are analyzed by the viscosity solution method then the degenerate hjb equation is perturbed to admit a classical solution and a suboptimal control law is designed based on the perturbed hjb equation when a quadratic type cost is used without a bound constraint on the control the value function is a classical solution to the degenerate hjb equation and the feedback control is affine in the system power in addition in this case we develop approximate but highly scalable solutions to the hjb equation in terms of a local polynomial expansion of the exact solution when the channel parameters are not known a priori one can obtain on line estimates of the parameters and get adaptive versions of the control laws in numerical experiments with both of the above cost functions the following phenomenon is observed whenever the users have different initial conditions there is an initial convergence of the power levels to a common level and then subsequent approximately equal behavior which converges toward a stochastically varying optimum

non uniqueness results for critical metrics of regularized determinants in four dimensions the regularized determinant of the paneitz operator arises in quantum gravity see connes in noncommutative geometry 1994 iv 4 an explicit formula for the relative determinant of two conformally related metrics was computed by branson in commun math phys 178 301 309 1996 a similar formula holds for cheeger s half torsion which plays a role in self dual field theory see juhl in families of conformally covariant differential operators q curvature and holography progress in mathematics vol 275 2009 and is defined in terms of regularized determinants of the hodge laplacian on p forms p n 2 in this article we show that the corresponding actions are unbounded above and below on any conformal four manifold we also show that the conformal class of the round sphere admits a second solution which is not given by the pull back of the round metric by a conformal map thus violating uniqueness up to gauge equivalence these results differ from the properties of the determinant of the conformal laplacian established in commun math phys 149 241 262 1992 ann math 142 171 212 1995 commun math phys 189 655 665 1997

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

the principle of optimality in dynamic programming with returns in partially ordered sets a general sequential model is defined where returns are in a partially ordered set a distinction is made between maximal nondominated returns and greatest returns we show that bellman s principle of optimality is valid with respect to maximal returns and it leads to an algorithm to approximate these returns we argue that significantly greater effort is needed to apply this algorithm to maximal returns than to greatest returns