a note on the markov property of stochastic processes described by nonlinear fokker planck equations abstract we study the markov property of processes described by generalized fokker planck equations that are nonlinear with respect to probability densities such as mean field fokker planck equations and fokker planck equations related to generalized thermostatistics we show that their transient solutions describe non markov processes in contrast stationary solutions can describe markov processes as a result nonlinear fokker planck equations can be used to model transient non markov processes that converge to stationary markov processes

derivations vanishing on s lets be the group of finite permutations on countably many symbols we exhibit an embedding ofs into a uhf algebra mathfrak a of glimm typen such that if is a derivation vanishing ons and satisfying 0 where is the unique trace on mathfrak a then admits an extension which is the generator of ac dynamics

synchronization of globally coupled phase oscillators singularities and scaling for general couplings abstract the onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling the effect of a stochastic temporal variation in the frequencies is also included the fokker planck equation for the coupled langevin system is reduced to a kinetic equation for the oscillator distribution function instabilities of the phase incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes depending on the coupling the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected a detailed analysis of these singularities to all orders in the normal form expansion is presented physically the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization these predictions are verified by numerically solving the kinetic equation for various couplings and frequency distributions

approximate controllability of impulsive differential equations with state dependent delay in order to describe various real world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process impulsive differential equations have been used to describe the system model in this article the problem of approximate controllability for nonlinear impulsive differential equations with state dependent delay is investigated we study the approximate controllability for nonlinear impulsive differential system under the assumption that the corresponding linear control system is approximately controllable using methods of functional analysis and semigroup theory sufficient conditions are formulated and proved finally an example is provided to illustrate the proposed theory

approximate controllability of partial neutral functional differential systems of fractional order with state dependent delay we prove the approximate controllability of control systems governed by a class of partial neutral functional differential systems of fractional order with state dependent delay in an abstract space sufficient conditions for approximate controllability of the control systems are established provided the approximate controllability of the corresponding linear control systems the results are obtained by using the krasnoselskii schaefer type fixed point theorem with the fractional power of operators an example is provided to illustrate the main results

noncommutative topological dynamics and compact actions on c algebras the classical notions of topological transitivity and minimality of a topological dynamical system are extended and analyzed in the context of c dynamical systems these notions are compared with other notions naturally arising in noncommutative ergodic theory as an application a c algebra version of a theorem of araki haag kastler and takesaki comm math phys 53 1977 97 134 about the correspondence between a compact automorphism group here assumed to be abelian and its fixed point subalgebra is proved in the presence of a commuting topologically transitive action a variation of this theorem in the setting of standard w inclusions is also presented

noncommutative topological dynamics ii this part deals with almost periodic and weakly mixing c flows and with disjointness and weak disjointness of c flows flows on c algebras the main result is a generalization to c flows of keynes and robertson s characteriza tion of minimal weakly mixing flows examples are discussed exhibiting anomalous behaviour of disjointness in the c flow case

fokker planck equation for a square lattice of coupled anharmonic oscillators and the two dimensional ising model the authors calculate the lowest non zero eigenvalue of the fokker planck equation for an infinite square lattice of coupled overdamped anharmonic oscillators bistable potentials through a variational calculation it generalises previous work for only one oscillator in a non trivial way they reduce that calculation to that of a suitable ratio of partition functions for the 2d ising model for which an exact analytical expression is obtained numerical results are also given and discussed

infinite lattice systems of interacting diffusion processes existence and regularity properties generalized stochastic gradient systems for infinite lattice models are investigated the allowed strength of the interaction depends on the dimension of the lattice the semigroup of transition probabilities is constructed and its regularity properties are also discussed some results of doss and royer 2 are improved

laplace approximations for sums of independent random vectors we consider expressions of the form zn e exp nf sn n where sn is the sum of n i i d random vectors with values in a banach space and f is a smooth real valued function by results of donsker varadhan and bahadur zabell one knows that lim 1 n log zn supxf x h x where h is the so called entropy function in an earlier paper a more precise evaluation of zn is given in the case where there was a unique point maximizing f h and the curvature at the maximum was nonvanishing the present paper treats the more delicate problem where these conditions fail to hold

simple and prime crossed products of c algebras by compact quantum group coactions let g a be a compact quantum group and a coaction of g on a c algebra b we give necessary and sufficient conditions for the simplicity and primeness of the crossed product b g in terms of certain fixed point algebras

unbounded derivations tangential to compact groups of automorphisms ii let g be a compact abelian group and an action of g on a c algebra u such that u u u 0 u for all g where u is the spectral subspace of u corresponding to the character on g derivations which are defined on the algebra uf of g finite elements are considered in the special case u 0 these derivations are characterized by a cocycle on g with values in the relative commutant of u in the multiplier algebra of u and these derivations are inner if and only if the cocycles are coboundaries and bounded if and only if the cocycles are bounded under various restrictions on g and properties of the cocycle are deduced which again give characterizations of in terms of decompositions into generators of one parameter subgroups of g and approximately inner derivations finally a perturbation technique is devised to reduce the case uf uf to the case uf uf and u 0 this is used to show that any derivation with d uf is wellbehaved and if furthermore g t1 and uf uf the closure of generates a one parameter group of automorphisms of u in the case g td d 2 3 finite and uf uf it is shown that extends to a generator of a group of automorphisms of the weak closure of u in any g covariant representation

minimax or feared value l 1 l control abstract this is the third paper after bernhard expected value feared value and partial information optimal control in g j olsder ed new trends in dynamic games and applications annals of the international society of dynamic games vol 3 birkhauser boston usa 1995 pp 95 100 a separation theorem for expected value and feared value discrete time control cocv vol 1 pp 191 206 smai www emath fr cocv 1996 where we attempt to develop and exploit the parallel between stochastic control and min max control induced by the use of the max plus algebra using the concept of feared value as the parallel to expected value the present paper builds on the main formula of bernhard 1996 the results of which are a subset of those given here its new contribution is twofold on the one hand we clarify the role of the integral part of the cost in that parallel this leads to a more extensive theory of the so called imperfect information minimax l control problem than apparently available in the literature including a certainty equivalence theorem on the other hand we extend the parallel to the continuous time case as much as we can in that direction the present paper slightly extends the classical framework of the variational inequality of stopping time games and gives a formal treatment of the partial information case altogether this might be the first new results obtained with the tool of mathematical fear

locally compact group actions on c algebras and compact subgroups be a compact subgroup in this paper we shall give necessary and sufficient conditions in order that the crossed product g x a be simple or prime our conditions are in terms of the dual 2 of the com pact subgroup k the paper is organized as follows in the second section we summarize some preliminary definitions and results about continuous banach representations of compact groups the same section contains the definitions of the subspaces of spherical functions sn nl l 7 n e e those of the algebras s i and the relations between these subspaces section 3 contains a study of saturated actions of g on a in terms of the ideals s s we mention that for compact groups the notion of saturated action has been defined by rieffel see 19 chap 73 the same section contains conditions in order that the crossed product be simple for compact abelian groups g our result reduces to the following g x

controllability of non linear impulsive stochastic systems in this article we consider the finite dimensional dynamical control systems described by non linear impulsive stochastic differential equations sufficient conditions for the complete and approximate controllability of non linear impulsive stochastic systems are formulated and proved under the natural assumption that the corresponding linear system is appropriately controllable

spectra for compact quantum group coactions and crossed products we present definitions of both connes spectrum and strong connes spectrum for actions of compact quantum groups on c algebras and obtain necessary and sufficient conditions for a crossed product to be a prime or a simple c algebra our results extend to the case of compact quantum ac tions the results in 8 which in turn generalize results by connes olesen and pedersen and kishimoto for abelian group actions we prove in addition that the connes spectra are closed under tensor products these results are new for compact nonabelian groups as well

a strong connes spectrum for finite group actions of simple rings abstract in this paper we give necessary and sufficient conditions for a skew group ring to be simple our conditions are in terms of the irreducible representations of the group g we also introduce a version of the strong connes spectrum for finite group actions of simple rings

periodic behavior of the stochastic brusselator in the mean field limit we prove a propagation of chaos result for the mean field limit of a model for a trimolecular chemical reaction called brusselator then we show that the pair of nonlinear i e law dependent stochastic differential equations describing the evolution of the concentration of the molecules at a given site in the mean field limit has a solution with a periodic law in t finally we identify the limit and establish a central limit theorem for the periodic law in the case where the noise tends to zero

discontinuous differential games and control systems with supremum cost abstract this paper concerns with the existence of a value for a zero sum two player differential game with supremum cost of the form c t 0 x 0 u v sup t 0 t h x t 0 x 0 u v under isaacs condition we characterize the value function as the unique solution in a suitable sense to a pde namely the hamilton jacobi isaacs equation as a byproduct we obtain a pde characterization of the value function for control system

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

the pontry agin maximum principle for minimax problems of optimal control the subject of this paper is the pontryagin maximum principle for the minimax problem of optimal control that is we will derive the necessary conditions which a control must satisfy in order to be optimal for the problem in which the cost is measured pointwise in time by the essential supremum over t of some function of time the state and the control it is desired to use controls to minimize this cost

stochastic analyses of the dynamics of generalized little hopfield hemmen type neural networks stochastic analyses are conducted of model neural networks of the generalized little hopfield hemmen type in which the synaptic connections with linearly embeddedp sets of patterns are free of symmetric ones and a glauber dynamics of a markovian type is assumed two kinds of approaches are taken to study the stochastic dynamical behavior of the network system first by developing the method of the nonlinear master equation in the thermodynamic limitn an exact self consistent equation is derived for the time evolultion of the pattern overlaps which play the role of the order parameters of the system the self consistent equation is shown to describe almost completely the macroscopic dynamical behavior of the network system second conducting the system size expansion of the master equation for then body probability distribution of the glauber dynamics makes it possible to analyze the fluctuations in the course of the analysis the self consistent equation for the pattern overlaps is derived again the main result of the rigorous fluctuation analysis is that as far as the fluctuations are concerned the time course of the pattern overlap fluctuations behaves independently of the fluctuations in the remaining modes of the system s macrovariables in accordance with the self determining property of the macroscopic motion of the pattern overlaps for neural networks with linear synaptic couplings

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

coexistence of stable limit cycles in a generalized curie weiss model with dissipation in this paper we modify the langevin dynamics associated to the generalized curie weiss model by introducing noisy and dissipative evolution in the interaction potential we show that when a zero mean gaussian is taken as single site distribution the dynamics in the thermodynamic limit can be described by a finite set of odes depending on the form of the interaction function the system can have several phase transitions at different critical temperatures because of the dissipation effect not only the magnetization of the systems displays a self sustained periodic behavior at sufficiently low temperature but in certain regimes any finite number of stable limit cycles can exist we explore some of these peculiarities with explicit examples

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

free joinings of c dynamical systems joinings of c dynamical systems are defined in terms of free products of c algebras as an analogue of joinings of classical dynamical systems we then consider disjointness in this context in particular for ergodic versus identity systems lastly we show how multi time correlation functions appearing in quantum statistical mechanics naturally fit into this joining framework

discontinuous control problems for non convex dynamics and near viability for singularly perturbed control systems abstract the aim of this paper is to study two classes of discontinuous control problems without any convexity assumption on the dynamics in the first part we characterize the value function for the mayer problem and the supremum cost problem using viscosity tools and the notion of e viability near viability these value functions are given with respect to discontinuous cost functionals in the second part we obtain results describing the e viability near viability of singularly perturbed control systems

on reachability and minimum cost optimal control questions of reachability for continuous and hybrid systems can be formulated as optimal control or game theory problems whose solution can be characterized using variants of the hamilton jacobi bellman or isaacs partial differential equations the formal link between the solution to the partial differential equation and the reachability problem is usually established in the framework of viscosity solutions this paper establishes such a link between reachability viability and invariance problems and viscosity solutions of a special form of the hamilton jacobi equation this equation is developed to address optimal control problems where the cost function is the minimum of a function of the state over a specified horizon the main advantage of the proposed approach is that the properties of the value function uniform continuity and the form of the partial differential equation standard hamilton jacobi form continuity of the hamiltonian and simple boundary conditions make the numerical solution of the problem much simpler than other approaches proposed in the literature this fact is demonstrated by applying our approach to a reachability problem that arises in flight control and using numerical tools to compute the solution

approximate controllability of second order damped mckean vlasov stochastic evolution equations in this paper sufficient conditions for the approximate controllability of a class of second order nonlinear stochastic functional differential equations of mckean vlasov type are derived the nonlinearities at a given time t considered depend not only on the state of the solution at time t but also on the corresponding probability distribution at time t an example is given to illustrate the theory

maximal subalgebras of c algebras associated with periodic flows we find necessary and sufficient conditions for the subalgebra of analytic elements associated with a periodic c dynamical system to be a maximal norm closed subalgebra our conditions are in terms of the arveson spectrum of the action we also describe equivalent properties of the system in terms of the strong connes spectrum and the simplicity of the crossed product c 2012 elsevier inc all rights reserved

worst case analysis of nonlinear systems the authors work out a framework for evaluating the performance of a continuous time nonlinear system when this is quantified as the maximal value at an output port under bounded disturbances the disturbance problem this is useful in computing gain functions and l sub spl infin induced norms which are often used to characterize performance and robustness of feedback systems the approach is variational and relies on the theory of viscosity solutions of hamilton jacobi equations convergence of euler approximation schemes via discrete dynamic programming is established the authors also provide an algorithm to compute upper bounds for value functions differences between the disturbance problem and the optimal control problem are noted and a proof of convergence of approximation schemes for the control problem is given case studies are presented which assess the robustness of a feedback system and the quality of trajectory tracking in the presence of structured uncertainty

topological automorphism groups of compact quantum groups we study the topological structure of the automorphism groups of compact quantum groups showing that in parallel to a classical result due to iwasawa the connected component of identity of the automorphism group and of the inner automorphism group coincide for compact matrix quantum groups which can be thought of as quantum analogues of compact lie groups we prove that the inner automorphism group is a compact lie group and the outer automorphism group is discrete applications of this to the study of group actions on compact quantum groups are highlighted we end by providing examples of compact matrix quantum groups with infinitely generated fusion rings in stark contrast with the classical situation along the way we study the invariant theory of finite group actions on free laurent rings and show that the rings of invariants are in general not finitely generated

a note on noncommutative unique ergodicity and weighted means abstract in this paper we study unique ergodicity of c dynamical system a t consisting of a unital c algebra a and a markov operator t a a relative to its fixed point subspace in terms of riesz summation which is weaker than cesaro one namely it is proven that a t is uniquely ergodic relative to its fixed point subspace if and only if its riesz means 1 p 1 p n k 1 n p k t k x converge to e t x in a for any x a as n here e t is an projection of a to the fixed point subspace of t it is also constructed a uniquely ergodic entangled markov operator relative to its fixed point subspace which is not ergodic

mixing properties of quantum systems we generalize the classical notion of topological mixing for automorphisms ofc algebras in two ways we show that for galilean invariant fermi systems the weaker form of mixing is satisfied with some additional requirement on the range of the interaction we can also demonstrate the stronger mixing property

the existence of value and saddle point in games of fixed duration differential games of fixed duration are defined the definition of strategy follows that of friedman while the definition of payoff follows that of krasovskii and subbotin it is shown by relatively elementary methods that games of fixed duration which satisfy the isaacs condition have values and saddle points it is also shown under appropriate hypotheses on the data of the problem that if the isaacs condition holds then the value is uniformly lipschitz continuous on bounded sets and satisfies the isaacs equation at all points of differentiability the relationship of the value as defined here to other values is studied

construction of non gaussian self similar random fields with hierarchical structure in the present work we construct non gaussian self similar random fields with hierarchical structure the construction is based on non gaussian solutions of the main nonlinear equation of the hierarchical models theory the existence of such solutions was proved originally by sinai and the author and later by another method by collet and eckmann next we establish the uniqueness of a gibbs state for the constructed self similar field finally for a class of hierarchical models we prove the convergence of renormalization transformations of a random field at the critical point to the self similar field

a mini tutorial on measure valued markov processes and nonlinear martingale problems as a reply to mccauley s comment abstract one goal of this mini tutorial is to provide an introduction into the theory of measure valued markov processes and nonlinear martingales defined by strongly nonlinear fokker planck equations and to discuss the physical relevance of the associated processes another goal is to reply to mccauley s comment on t d frank physica a 331 391 2004 the tutorial addresses in detail two approaches found in physics and mathematics the first approach exploits a mapping between linear and nonlinear fokker planck equations the second approach exploits martingale theory several examples of markov processes and martingales in quantum mechanical nonextensive and self organizing systems defined by nonlinear fokker planck equations are discussed

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

compact actions commuting with ergodic actions and applications to crossed products let a k be a c dynamical system with k compact in this paper we prove a duality result for saturated actions theorem 3 3 the proof of this result can also be considered as an alternate proof of the corresponding result for von neumann algebras due to araki haag kastler and takesaki 14 we also obtain results concerning the simplicity and the primeness of the crossed product a k in terms of the ergodicity of the commutant of propositions 5 3 and 5 4

derivations commuting with abelian gauge actions on lattice systems let be an action of a compact abelian groupg on ac algebraa and assume that the fixed point subalgebraa is an af algebra we show that if is a closed derivation ona commuting with and the restriction of toa generates a one parameter group of automorphisms then itself is a generator in particular the result applies if is an infinite product action ofg on a uhf algebra furthermore if in this situation 1 and 2 are two derivations both satisfying the hypotheses on and 1 and 2 have the same restriction toa then there exists a one parameter subgroup of the action with generator 0 such thatd 1 d 2 d 0 is a joint core for the three derivations and 2 1 0 on this core

diffusion with interactions and collisions between coloured particles and the propagation of chaos one considers a system ofn particles on the real line which are of two different types colours their dynamics is given by a stochastic differential equation with constant diffusion part the drift felt by a particle of either type depends on the empirical measures of type 1 and 2 particles at every instant further a reflection condition is imposed so that particles of different type are not allowed to cross each other the article studies the vlasov mckean limit of the system asn propagation of chaos and an evolution equation for the limiting empirical measures is established from where in particular an equation for the separating front between the two types follos

approximate controllability for a class of semilinear abstract equations in this paper a class of control systems governed by the semilinear abstract equation dot y ay f y bv is considered a sufficient condition for the approximate controllability is obtained which improves j henry s results on a nonlinear parabolic control systems with some more serious restrictions it is suitable not only to the infinite dimensional case but also to the finite dimensional case two examples are given to explain the applications of the theory

viscosity solutions of isaacs equations and differential games with lipschitz controls abstract it is demonstrated that the upper and lower values of a two person zero sum differential game solve the respective upper and lower isaacs equations in the viscosity sense introduced by crandall and lions trans amer math soc 277 1983 1 42 since such solutions are unique this yields a fairly simple proof that the game has value should the minimax condition hold as a further application of viscosity techniques a new and simpler proof that the upper and lower values can be approximated by the values of certain games with lipschitz controls is given

symmetries of quantum spaces subgroups and quotient spaces of quantum su 2 and so 3 groups we prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group we give the formula for the corresponding multiplicities in the case of the quotient quantum spaces we describe the subgroups and the quotient spaces of quantumsu 2 andso 3 groups

approximate controllability for abstract measure differential systems abstract in this paper we investigate approximate controllability for abstract measure differential systems based on generalizing knowledge for ordinary differential systems we first introduce new concepts of the reachable set and approximate controllability for abstract measure differential systems then based on the nonlinear alternative for condensing mapping in banach space we present sufficient conditions of approximate controllability for a class of abstract measure differential systems our results in dealing with approximate controllability are less conservative than those in the previous literature finally an example is given to illustrate the availability of our results for approximate controllability

on the linear discrepancy model and risky shifts in group behavior a nonlinear fokker planck perspective using a nonlinear fokker planck perspective we re formulate the linear discrepancy model proposed by boster and colleagues that describes the emergence of risky shifts during group decision making analytical expressions for the stationary case are derived and risky shifts are obtained by monte carlo simulations striking similarities with the kuramoto model for group synchronization are pointed out

unbounded derivations commuting with compact group actions let be a closed derivation in ac algebra mathfrak a which commutes with an ergodic action of a compact group on mathfrak a then generates ac dynamics of mathfrak a similar results are obtained for non ergodic actions on abelianc algebras and on the algebra of compact operators

existence and controllability results for second order impulsive stochastic evolution systems with state dependent delay in this paper we discuss a kind of impulsive second order stochastic evolution systems with state dependent delay in a real separable hilbert space the results concerning the existence and controllability of mild solutions have been addressed by means of the fixed point techniques some sufficient conditions are formulated as well as an application involving partial differential equation with impulses is presented

approximate equivalence for representations of c algebras and c dynamical systems abstract approximate equivalence unitary equivalence up to arbitrarily small compact perturbations is studied for representations of c algebras and covariant representations of c dynamical systems approximate equivalence for representations of c algebras is shown to be determined by a trace condition analogous to the trace condition for unitary equivalence of representations of finite groups for c dynamical systems with a compact abelian group acting approximate equivalence is shown to be determined by certain associated representations of the fixed point algebra of the action

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

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

barriers of the mckean vlasov energy via a mountain pass theorem in the space of probability measures abstract we show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise induced metastability the result is based on an analysis of the associated mckean vlasov free energy which for suitable attractive interaction potentials has at least two distinct global minimisers at the critical parameter value c on the torus one of these states is the spatially homogeneous constant state and the other is a clustered state we show that a third critical point exists at this value as a result we obtain that the probability of transition of the empirical process from the constant state scales like exp n with the energy gap at c the proof is based on a version of the mountain pass theorem for lower semicontinuous and geodesically convex functionals on the space of probability measures p 2 m equipped with the 2 wasserstein metric where m is a complete connected and smooth riemannian manifold

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

time evolution of a one dimensional point system a note on fritz s paper in a companion paper ref 5 fritz studies the time evolution of a one dimensional point system which was introduced by spitzer as a model of traffic in the present paper we improve some of the results of ref 5 by using a different approach our results are obtained in a very simple and straightforward way but the techniques employed require conditions somewhat stronger than those assumed in ref 5

a λ convexity based proof for the propagation of chaos for weakly interacting stochastic particles abstract in this work we give a proof of the mean field limit for convex potentials using a purely variational viewpoint our approach is based on the observation that all evolution equations that we study can be written as gradient flows of functionals at different levels in the set of probability measures in the set of symmetric probability measures on n variables and in the set of probability measures on probability measures this basic fact allows us to rely on convergence tools for gradient flows to complete the proof by identifying the limits of the different terms in the evolutionary variational inequalities evis associated to each gradient flow the convexity of the confining and interaction potentials is crucial for the unique identification of the limits and for deriving the evis at each description level of the interacting particle system

approximate controllability of semilinear neutral evolution systems with delay this paper considers the approximate controllability for a class of neutral control systems governed by semilinear neutral delayed equations sufficient conditions for approximate controllability are established by constructing fundamental solutions and using resolvent condition banach contraction mapping principle and techniques on fractional power operators the obtained results improve some existing analogous ones on this topic to some extent an example is presented finally to illustrate the applications of the obtained results

a galois correspondence for compact group actions on c algebras abstract in this paper we prove a galois correspondence for compact group actions on c algebras in the presence of a commuting minimal action namely we show that there is a one to one correspondence between the c subalgebras that are globally invariant under the compact action and the commuting minimal action that in addition contain the fixed point algebra of the compact action and the closed normal subgroups of the compact group

strong asymptotic abelianness for entropic k systems we prove that in entropick systems of type ii1 the automorphism is strongly asymptotically abelian