risk sensitive and robust escape criteria the problem of controlling a noisy process so as to prevent it from leaving a prescribed set has a number of interesting applications in this paper new approaches to this problem are considered first a risk sensitive criterion for a stochastic diffusion process model is examined and it is shown that the value is a classical solution of a related pde the qualitative properties of this criterion are favorably contrasted with those of existing criteria in the risk averse limit it is proved that in the risk averse limit the value of the risk sensitive criterion converges to a viscosity solution of a first order pde it is then demonstrated that the value function of a deterministic differential game is also a viscosity solution to the pde this game gives a robust control formulation of the escape time problem and is analogous to h infty control in particular the opposing player attempts to push the process out of the prescribed set and suffers an l2 cost for his efforts lower bounds on the escape time as a function of this cost are obtained

optimal control for partially observed diffusions stochastic control problems are considered in which a state process x t and an observation process y t are governed by ito sense stochastic differential equations driven by independent brownian motions the control u t enters linearly in the dynamics of x t a separated control problem is introduced in which the state at any time t is a measure lambda t representing an unnormalized conditional distribution for x t given y s u s for s leqq t the method depends on introducing a pathwise version of lambda t which depends continuously on observation and control trajectories y u existence of an optimal control is obtained in a suitable class larger than the usual class of controls admissible in the strict sense that u t is measurable on the sigma algebra mathcal f t y generated by observations y s s leqq t the dynamics of lambda t are studied using a method of forward and backward partial differential equations under a suitable nondegeneracy condition the m

time average control of martingale problems a linear programming formulation one studies the average cost for controlled systems given as solutions of the martingale problem for their generator the control problem is reformulated as a linear programming problem and conditions are given for the existence of an optimal solution it is further shown that the optimal control can be taken to depend only on the history of the system and that this cost remains optimal for systems with different initial distributions

some examples of optimal stochastic controls or the stochastic maximum principle at work the stochastic maximum principle is applied to solve eight stochastic control problems four have been solved previously using much more complicated methods but the last four are new the main thrust of the work is to show that some completely observable stochastic control problems with special structure can be solved quite quickly and easily using the maximum principle

robustness of nonlinear filters over the infinite time interval nonlinear filtering is one of the classical areas of stochastic control from the point of view of practical usefulness it is important that the filter not be too sensitive to the assumptions made on the initial distribution the transition function of the underlying signal process and the model for the observation this is particularly acute if the filter is of interest over a very long or potentially infinite time interval then the effects of small errors in the model which is used to construct the filter might accumulate to make the output useless for large time the problem of asymptotic sensitivity to the initial condition has been treated in several papers we are concerned with this as well as with the sensitivity to the signal model uniformly over the infinite time interval it is conceivable that the effects of even small errors in the model will accumulate so that the filter will eventually be useless the robustness is shown for three classes of problems for the first two cases the signal model is markov and the observations are taken in discrete time and the observation is the usual function of the signal plus noise the last class treated is a continuous time markov process with a point process observation

lattices and multi dimensional words in the present paper we develop a formalism to generate multi dimensional words using lattices which generalizes the construction of real numbers one dimensional words from a sequence of partial quotients using continued fractions the construction was introduced in a special case by simpson and tijdeman in order to derive a multi dimensional generalization of the theorem of fine and wilf we show that the produced multi dimensional words are intrinsically connected with k dimensional sturmian words

diffusions conditionnelles i hypoellipticite partielle this paper is devoted to the proof of certain results on conditional diffusions by using the malliavin calculus of variations it is shown that under minimal conditions on the vectors fields which define a two component diffusion xt zt and their lie brackets a s for any t 0 the conditional law of xt t 0 given b zs s t a c density

numerical approximations for stochastic differential games the markov chain approximation method is a widely used robust relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high versions of these methods have been used in applications to various two player differential and stochastic dynamic games for a long time and proofs of convergence are available for some cases mainly using pde type techniques in this paper purely probabilistic proofs of convergence are given for a broad class of such problems where the controls for the two players are separated in the dynamics and cost function and which cover a substantial class not dealt with in previous works discounted and stopping time cost functions are considered finite horizon problems and problems where the process is stopped on first hitting an a priori given boundary can be dealt with by adapting the methods of h j kushner and p dupuis numerical methods for stochastic control problems in continuous time 2nd ed springer verlag berlin new york 2001 as done in this paper for the treated problems the essential conditions are the weak sense existence and uniqueness of solutions an almost everywhere continuity condition and that a weak local consistency condition holds almost everywhere for the numerical approximations just as for the control problem there are extensions to problems with controlled variance and jumps

asymptotic stability of the optimal filter with respect toits initial condition consider the problem of estimation of a diffusion signal observed in additive white noise if the solution to the filtering equations initialized with an incorrect prior distribution approaches the true conditional distribution asymptotically in time then the filter is said to be asymptotically stable with respect to perturbations of the initial condition this paper presents asymptotic stability results for linear filtering problems and for signals with limiting ergodic behavior for the linear case stability of the riccati equation of kalman filtering is used to derive almost sure asymptotic stability of linear filters for possibly non gaussian initial conditions in the nonlinear case asymptotic stability in a weak convergence sense is shown for filters of signal diffusions which converge in law to an invariant distribution

approximation of upper lyapunov exponents of bilinear stochastic differential systems a bilinear stochastic differential system is considered whose solution starting from x is denoted by x t x it is supposed that its upper lyapunov exponent lambda exists the purpose of the paper is to propose an efficient algorithm to approximate lambda it is shown that for markov chains bar x p h x p in mathbb n defined by several approximation schemes the lyapunov exponents bar lambda h are well defined and can be computed estimates are given in terms of the discretization step h of the theoretical error lambda bar lambda h then these results are applied to an engineering problem the stability of the motion of helicopter rotor blades in a turbulent wind

balanced sequences and optimal routing the objective pursued in this paper is two fold the first part addresses the following combinatorial problem is it possible to construct an infinite sequence over n letters where each letter is distributed as evenly as possible and appears with a given rate the second objective of the paper is to use this construction in the framework of optimal routing in queuing networks we show under rather general assumptions that the optimal deterministic routing in stochastic event graphs is such a sequence

balance properties of multi dimensional words a word u is called 1 balanced if for any two factors v and w of u of equal length we have 1 v i w i1 for each letter i where v i denotes the number of occurrences of i in the factor v the aim of this paper is to extend the notion of balance to multi dimensional words we first characterize all 1 balanced words on zn in particular we prove they are fully periodic for n 1 we then give a quantitative measure of non balancedness for some words on z2 with irrational density including two dimensional sturmian words

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

on the number of factors of sturmian words abstract we prove that for m 1 card a m 1 m i 1 m i 1 i where a m is the set of factors of length m of all the sturmian words and is the euler function this result was conjectured by dulucq and gouyou beauchamps 1987 who proved that this result implies that the language m 0 a m c is inherently ambiguous we also give a combinatorial version of the riemann hypothesis

combinatorics of patterns of a bidimensional sturmian sequence resume nous etudions une generalisation des suites sturmiennes en construisant une surface plissee donnee par l approximation d un plan par trois sortes de faces carrees orientees suivant les trois plans de coordonnees a cette surface on associe par projection un pavage du plan par trois sortes de losanges on definit sur ce pavage une fonction de complexite en comptant le nombre de motifs distincts d une fenetre de taille donnee nous donnons en etudiant les prolongements des motifs la forme explicite de cette fonction dans le cas d une fenetre triangulaire et d une fenetre en forme de parallelogramme

a perturbation theory for ergodic markov chains and application to numerical approximations perturbations to markov chains and markov processes are considered the unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of foster lyapunov drift conditions the perturbations are assumed to be uniform in a weak sense on bounded time intervals the long time behavior of the perturbed chain is studied applications are given to numerical approximations of a randomly impulsed ode an ito stochastic differential equation sde and a parabolic stochastic partial differential equation spde subject to space time brownian noise existing perturbation theories for geometrically ergodic markov chains are not readily applicable to these situations since they require very stringent hypotheses on the perturbations

approximations and optimal control for the pathwise average cost per unit time and discounted problems for wideband noise driven systems the average cost per unit time problem for wide bandwidth noise driven control systems is considered where the average cost is in the pathwise sense no expectations are used let t time of control and text bw bandwidth for the class of processes considered here various unformity properties are proven for the convergence of the pathwise average costs as t to infty text bw to infty let u delta cdot be a smooth delta optimal control for the limit controlled diffusion the limit as text bw to infty for the mean average cost per unit time problem it is shown that for large enough t and text bw u delta cdot is 2 delta optimal with a probability arbitrarily close to one for the pathwise wide bandwidth problem this uniformity is important in applications for often there is only one long sequence to control and the expectation is inappropriate also otherwise as text bw to infty it might take longer and longer to approxim

numerical approximations for stochastic differential games the ergodic case the markov chain approximation method is a widely used relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high we consider a class of stochastic differential games with a reflected diffusion system model and ergodic cost criterion and where the controls for the two players are separated in the dynamics and cost function it is shown that the value of the game exists and that the numerical method converges to this value as the discretization parameter goes to zero the actual numerical method solves a stochastic game for a finite state markov chain and ergodic cost criterion the essential conditions are nondegeneracy and that a weak local consistency condition hold almost everywhere for the numerical approximations just as for the control problem

occupation measures for controlled markov processes characterization and optimality for controlled markov processes taking values in a polish space control problems with ergodic cost infinite horizon discounted cost and finite horizon cost are studied each is posed as a convex optimization problem wherein one tries to minimize a linear functional on a closed convex set of appropriately defined occupation measures for the problem these are characterized as solutions of a linear equation asssociated with the problem this characterization is used to establish the existence of optimal markov controls the dual convex optimization problem is also studied

full counting statistics of chaotic cavities from classical action correlations we present a trajectory based semiclassical calculation of the full counting statistics of quantum transport through chaotic cavities in the regime of many open channels our method to obtain the mth moment of the density of transmission eigenvalues requires two correlated sets of m classical trajectories therefore generalizing previous works on conductance and shot noise the semiclassical results agree for all values of m with the corresponding predictions from random matrix theory

approximation and limit results for nonlinear filters over an infinite time interval this paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval since the optimal filter can rarely be constructed one needs to compute with numerically feasible approximations the signal model can be a jump diffusion or just a process that is approximated by a jump diffusion the observation noise can be either white or of wide bandwidth the observations can be taken in either discrete or continuous time the cost of interest is the pathwise error per unit time over a long time interval it is shown under quite reasonable conditions on the approximating filter and the signal and noise processes that as time bandwidth process and filter approximation etc go to their limit in any way at all the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter were used analogous results are obtained with appropriate scaling if the observations are taken in discrete time and the sampling interval also goes to zero for these cases the approximating filter is a numerical approximation to the optimal filter for the presumed limit signal observation noise problem

markov property and ergodicity of the nonlinear filter in this paper we first prove under quite general conditions that the nonlinear filter and the pair signal filter are feller markov processes the state space of the signal is allowed to be nonlocally compact and the observation function h can be unbounded our proofs in contrast to those of kunita j multivariate anal 1 1971 pp 365 393 spatial stochastic processes birkhauser 1991 pp 233 256 and stettner stochastic differential equations springer verlag 1989 pp 279 292 do not depend upon the uniqueness of the solutions to the filtering equations we then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process these results extend those of kunita and stettner which hold for locally compact state space and bounded h to our general framework finally we show that the recent results of ocone and pardoux siam j control optim 34 1996 pp 226 243 on asymptotic stability of the nonlinear filter which use the kunita stettner setup hold for the general situation considered in this paper

conservative stochastic differential equations mathematical and numerical analysis we consider stochastic differential equations on the whole euclidean space possessing a scalar invariant along their solutions the stochastic dynamics therefore evolves on a hypersurface of the ambient space using orthogonal coordinate systems we show the existence and uniqueness of smooth solutions of the kolmogorov equation under some ellipticity conditions over the invariant hypersurfaces if we assume moreover the existence of an invariant measure we show the exponential convergence of the solution towards its average in the second part we consider numerical approximation of the stochastic differential equation and show the convergence and numerical ergodicity of a class of projected schemes in the context of molecular dynamics this yields numerical schemes that are ergodic with respect to the microcanonical measure over isoenergy surfaces

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

maximal pattern complexity of two dimensional words the maximal pattern complexity of one dimensional words has been studied in several papers t kamae l zamboni sequence entropy and the maximal pattern complexity of infinite words ergodic theory dynam systems 22 4 2002 1191 1199 t kamae l zamboni maximal pattern complexity for discrete systems ergodic theory dynam systems 22 4 2002 1201 1214 t kamae h rao pattern complexity over l letters e comb j to appear t kamae y m xue two dimensional word with 2k maximal pattern complexity osaka j math 41 2 2004 257 265 we study the maximal pattern complexity p k of two dimensional words a two dimensional version of the notion of strong recurrence is introduced it is shown that if is strongly recurrent then either is doubly periodic or p k 2k k 1 2 accordingly we define a two dimensional pattern sturmian word as a strongly recurrent word with p k 2k examples of pattern sturmian words are given

abelian properties of parry words abelian complexity of a word u is a function that counts the number of pairwise non abelian equivalent factors of u of length n we prove that for any c balanced parry word u the values of the abelian complexity function can be computed by a finite state automaton the proof is based on the notion of relative parikh vectors the approach works for any function f n that can be expressed in terms of the set of relative parikh vectors corresponding to the length n for example we show that the balance function of a c balanced parry word is computable by a finite state automaton as well

on the functional central limit theorem and the law of the iterated logarithm for markov processes let xt t 0 be an ergodic stationary markov process on a state space s if is its infinitesimal generator on l2 s dm where m is the invariant probability measure then it is shown that for all f in the range of hat a n 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 int limits 0 nt f x s text ds text t underline underline 0 converges in distribution to the wiener measure with zero drift and variance parameter 2 2 f g 2 g g where g is some element in the domain of such that g f theorem 2 1 positivity of 2 is proved for nonconstant f under fairly general conditions and the range of is shown to be dense in 1 a functional law of the iterated logarithm is proved when the 2 th moment of f in the range of is finite for some 0 theorem 2 7 a under the additional condition of convergence in norm of the transition probability p t x d y to m dy as t for each x the above results hold when the process starts away from equilibrium theorems 2 6 2 7 b applications to diffusions are discussed in some detail

on the existence of optimal controls the optimal control problem where the state is governed by an ito stochastic differential equation possibly just an ordinary differential equation is formulated in martingale terms under a coercivity condition which is weaker than compactness of the control set a convexity condition and mild continuity hypotheses on the data it is shown by the direct method that optimal controls exist hard and soft constraints are allowed in the absence of soft constraints it is shown that there exists an optimal control that is a function only of the present time and state i e the synthesis problem has a solution the main tool here is krylov s markovian selection theorem

t stability of the semi implicit euler method for delay differential equations with multiplicative noise the paper deals with the t stability of the semi implicit euler method for delay differential equations with multiplicative noise a difference equation is obtained by applying the numerical method to a linear test equation in which the wiener increment is approximated by a discrete random variable with two point distribution the conditions under which the method is t stable are considered and the numerical experiments are given

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

statistical distribution of the wigner smith time delay matrix moments for chaotic cavities we derive the joint distribution of the moments trq 1 of the wigner smith matrix for a chaotic cavity supporting a large number of scattering channels n this distribution turns out to be asymptotically gaussian and we compute explicitly averages and covariances the results are in a compact form and have been verified numerically the general methodology of proof and computations has a wide range of applications

episturmian words and some constructions of de luca and rauzy abstract in this paper we study infinite episturmian words which are a natural generalization of sturmian words to an arbitrary alphabet a characteristic property is they are closed under reversal and have at most one right special factor of each length they are first obtained by a construction due to de luca which utilizes the palindrome closure they can also be obtained by the way of extended rauzy rules

combinatorial theory of the semiclassical evaluation of transport moments i equivalence with the random matrix approach to study electronic transport through chaotic quantum dots there are two main theoretical approachs one involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging the other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories there are established evaluation procedures within the semiclassical evaluation that for several linear and non linear transport moments to which they were applied have always resulted in the agreement with random matrix predictions we prove that this agreement is universal any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory the equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs maps with certain properties and exhibiting systematic cancellations among their contributions remaining trajectory sets can be identified with primitive palindromic factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices the equivalence is proved for systems with and without time reversal symmetry

ergodic control of multidimensional diffusions 1 the existence results the existence of optimal stable markov relaxed controls for the ergodic control of multidimensional diffusions is established by direct probabilistic methods based on a characterization of a s limit sets of empirical measures the optimality of the above is established in the strong i e almost sure sense among all admissible controls under very general conditions

average cost markov decision processes optimality conditions we are concerned in this paper with discrete time markov decision processes mdps with borel state and action spaces x and a respectively and the long run expected average cost criterion when x is a denumerable set man y necessary and or sufficient conditions for the existence of optimal control policies are known however when x is a borel space i e a borel subset of a complete separable metric space most of the available results impose on the mdp very restrictive topological conditions e g compactness and or strong recurrence assumptions such as doeblin s condition see e g 4 9 12 and their references another related work is 7 where we have studied mdps from the viewpoint of the recurrence or ergodicity properties of the state process ln the present paper however we are concerned with the existence of average optimal policies by looking at static optimization problems see condition cs in section 4 related in sorne cases equivalent to the existence of a bounded solution to the so called optimality equation see c4 in section 4 these optimization problems are dual in the sense that under appropriate conditions the existence of an optimal solution to one of the problems implies existence of an optimal

survey paper a partial history of the early development of continuous time nonlinear stochastic systems theory this article is a survey of the early development of selected areas in nonlinear continuous time stochastic control key developments in optimal control and the dynamic programming principle existence of optimal controls under complete and partial observations nonlinear filtering stochastic stability the stochastic maximum principle and ergodic control are discussed issues concerning wide bandwidth noise for stability modeling filtering and ergodic control are dealt with the focus is on the earlier work but many important topics are omitted for lack of space

stability indices for randomly perturbed power systems this paper considers the stability of moments of stochastic systems such as stability of the mean or mean square stability the exponential growth behavior of moments is compared to almost sure exponential growth via lyapunov exponents we develop a series of indices that are useful to describe system performance under random perturbations the theory is applied to two examples including an electric power system

balancedness and coboundaries in symbolic systems abstract this paper studies balancedness for infinite words and subshifts both for letters and factors balancedness is a measure of disorder that amounts to strong convergence properties for frequencies it measures the difference between the numbers of occurrences of a given word in factors of the same length we focus on two families of words namely dendric words and words generated by substitutions the family of dendric words includes sturmian and arnoux rauzy words as well as codings of regular interval exchanges we prove that dendric words are balanced on letters if and only if they are balanced on words in the substitutive case we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness

on positive recurrence of constrained diffusion processes let g c r k be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces g i i 1 n where n i and d i denote the inward normal and direction of constraint associated with g i respectively stability properties of a class of diffusion processes constrained to take values in g are studied under the assumption that the skorokhod problem defined by the data n i d i i 1 n is well posed and the skorokhod map is lipschitz continuous explicit conditions on the drift coefficient b of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure define c i d i i 0 i 1 n then the key condition for stability is that there exists 0 and a bounded subset a of g such that for all x g a b x c and dist b x c where c denotes the boundary of c

a criterion for invariant measures of markov processes using the martingale formulation for markov processes introduced by stroock and varadhan we develop a criterion for checking if a measure happens to be invariant

existence of markov controls and characterization of optimal markov controls given a solution of a controlled martingale problem it is shown under general conditions that there exists a solution having markov controls which has the same cost as the original solution this result is then used to show that the original stochastic control problem is equivalent to a linear program over a space of measures under a variety of optimality criteria existence and characterization of optimal markov controls then follows an extension of echeverria s theorem characterizing stationary distributions for uncontrolled markov processes is obtained as a corollary in particular this extension covers diffusion processes with discontinuous drift and diffusion coefficients

filtering and control for wide bandwidth noise driven systems much of modern stochastic control theory uses ideal white noise driven models ito equations if the observed data are corrupted by noise then the noise is usually assumed to be white gaussian typically if the underlying models are linear one uses a kalman bucy filter to get an estimate of the state and then bases the control on this estimate in practice the noises are rarely white and the reference signals and the systems are only approximations in some sense to a diffusion nevertheless owing to lack of viable alternatives one still uses the kalman bucy filter etc then the estimates are not optimal and indeed might be quite far from being optimal the same is true for the corresponding control examples are given to illustrate this the sense in which the estimates and or control is useful needs to be examined in order to justify the use of the commonly used procedure the issue is much deeper than mere robustness in the usual sense since basic questions of interpretation of the results are involved this paper deals with these questions for the filtering problem where the signal is a near gauss markov process and the observation noise wide band it is shown that the usual method is nearly optimal with respect to a class of alternative data processors this alternative class is rather natural and includes the data processors which one would normally want to use it is unlikely that the class can be enlarged very much in general the asymptotic in time and bandwidth problem is treated as is the much harder conditional gaussian case and a case where the observations are nonlinear the basic techniques are those of weak convergence theory similar results are obtained for the combined filtering and control problem where it is shown that good controls for the ideal model are also good for the actual physical model with respect to a natural case of alternative controls the control problem over a finite interval as well as the average cost per unit time problem are considered

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

nearly optimal singular controls for wideband noise driven systems singular stochastic control problems arise in many applications for example in storage inventory finite fuel consumption and investment and limits of impulsive control problems here the increment of the control effort is not of the usual form u t dt but is the differential of a nondecreasing and suitably adapted process the diffusion process models used are only approximations to some physical process which might be a wideband noise driven system or a suitably scaled discrete parameter process since the optimal controls for the physical processes are usually impossible to obtain it is of interest to know whether nearly optimal controls for the diffusion model are nearly optimum when applied to the physical problem it is shown that this is true under broad conditions for discounted and average cost per unit time problems the usual weak convergence analysis via the skorokhod topology on d 0 infty is not appropriate here due to the nature of the singular controls and it is ne

balances for fixed points of primitive substitutions an infinite word defined over a finite alphabet a is balanced if for any pair of factors of the same length and for any letter a in the alphabet a a 1 where a denotes the number of occurrences of the letter a in the word in this paper we generalize this notion and introduce a measure of balance for an infinite sequence in the case of fixed points of primitive substitutions we show that the asymptotic behaviour of this measure is in part ruled by the spectrum of the incidence matrix associated with the substitution connections with frequencies of letters and other balance properties are also discussed

moments of the wigner delay times the wigner time delay is a measure of the time spent by a particle inside the scattering region of an open system for chaotic systems the statistics of the individual delay times whose average is the wigner time delay are thought to be well described by random matrix theory here we present a semiclassical derivation showing the validity of random matrix results in order to simplify the semiclassical treatment we express the moments of the delay times in terms of correlation functions of scattering matrices at different energies in the semiclassical approximation the elements of the scattering matrix are given in terms of the classical scattering trajectories requiring one to study correlations between sets of such trajectories we describe the structure of correlated sets of trajectories and formulate the rules for their evaluation to the leading order in inverse channel number this allows us to derive a polynomial equation satisfied by the generating function of the moments along with showing the agreement of our semiclassical results with the moments predicted by random matrix theory we infer that the scattering matrix is unitary to all orders in the semiclassical approximation

on the factors of the sturmian sequences resume cet article a pour objet l etude d une construction associant a toute droite de pente p q p et q premiers entre eux et q n un mot de longueur n sur l alphabet 0 1 nous montrons que nous obtenons par cette construction le langage constitue de tous les facteurs des suites de sturm nous formulons apres avoir obtenu une equation fonctionelle dont la solution est la serie generatrice de ce langage une conjecture reliant cette serie generatrice a la fonction d euler

diffusion approximations for nonlinear phase locked loop type systems with wide band inputs abstract communication systems often involve differential equation models whose inputs are noises and signals with wide bandwidths it is frequently of interest to approximate them by some markov diffusion process since then many analytical and numerical methods can be used here recent results on getting diffusion approximations to systems with such inputs are applied to three classes of detection systems which are very important in applications 1 a phase locked loop with a limiter 2 a quadricorrelator with and without a limiter the function is to track changes in phase and frequency 3 a squaring loop whose purpose is the tracking of the carrier frequency despite the carrier modulation in 3 a type of pulse phase modulation is used the method is natural systematic and relatively straightforward under natural scaling of the signals and noises the appropriate diffusion approximations for band pass but wide band noise are obtained the approximation is in the sense of weak convergence the first two problems have been hard to analyze owing to the nature of the non linearity and the results clearly indicate the advantage and disadvantages of the use of the limiter the third problem has been difficult to analyze partly due to the periodicities which occur naturally in such problems all three classes represent widely used and important systems and much information can be obtained from the limit process for example the results show that the use of a limiter can actually improve the tracking ability of the systems when the noise is small the system signal and noise models to which the methods can be applied are much broader than those used here but the results together with the results in h j kushner ieee trans inform theory it 26 1980 715 725 for different classes of problems illustrate the great potential of the approximation methods for problems in control and communication theory in certain cases the limit processes are of the type which have been obtained via more formal arguments

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

simple necessary and sufficient conditions for the stability of constrained processes in recent years a new approach has emerged for analyzing the stability properties of constrained stochastic processes in this approach one associates with the stochastic model a deterministic model or a family of deterministic models and under appropriate conditions stability of the stochastic model follows if all solutions of the deterministic model are attracted to the origin in the present work we show that a rather sharpcharacterization for the stability of the deterministic model is possible when it can be represented in terms of what we call a regular skorokhod map let g be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces gi i 1 n where ni and di denote the inward normal and direction of constraint associated with gi suppose that the skorokhod problem defined by the data ni di i 1 n is regular under these conditions the deterministic model mentioned above will correspond to a law of large numbers limit of the original stochas

stability of linear stochastic systems via lyapunov exponents and applications to power systems this paper studies linear systems under sustained random perturbations the stochastic perturbation model is given by a bounded markov diffusion process as it appears e g in the description of load or generation uncertainties of power systems for such systems the lyapunov exponents describe necessary and sufficient conditions for almost sure asymptotic stability the paper presents several numerical methods for the computation of lyapunov exponents and applies this methodology to linear oscillators in dimension 2 and 3 and to a one machine infinite bus electric power system

approximation and limit results for nonlinear filters over an infinite time interval part ii random sampling algorithms the paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval since the optimal filter can rarely be constructed one needs to compute with numerically feasible approximations the signal model can be a jump diffusion reflected or not the observations can be taken either in discrete or continuous time the cost of interest is the pathwise error per unit time over a long time interval in a previous paper of the authors a budhiraja and h j kushner siam j control optim 37 1999 pp 1946 1979 it was shown under quite reasonable conditions on the approximating filter and on the signal and noise processes that as time bandwidth process and filter approximation etc go to their limit in any way at all the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter was used when suitable approximating filters cannot be readily constructed due to excessive computational requirements or to problems associated with a high signal dimension approximations based on random sampling methods or perhaps combinations of sampling and analytical methods become attractive and are the subject of a great deal of attention the work of the previous paper is extended to a wide class of such algorithms under quite broad conditions covering virtually all the cases considered to date it is shown that the pathwise average errors converge to the same limit that would be obtained if the optimal filter was used as time goes to infinity and the approximation parameter goes to its limit in any way at all all the extensions e g wide bandwidth observation or system driving noise in a budhiraja and h j kushner siam j control optim 37 1999 pp 1946 1979 hold for our random sampling algorithms as well