strategy iteration is strongly polynomial for 2 player turn based stochastic games with a constant discount factor ye 2011 showed recently that the simplex method with dantzig s pivoting rule as well as howard s policy iteration algorithm solve discounted markov decision processes mdps with a constant discount factor in strongly polynomial time more precisely ye showed that both algorithms terminate after at most o mn1 log n1 iterations where n is the number of states m is the total number of actions in the mdp and 0

the complexity of mean payoff games on graphs abstract we study the complexity of finding the values and optimal strategies of mean payoff games on graphs a family of perfect information games introduced by ehrenfeucht and mycielski and considered by gurvich karzanov and khachiyan we describe a pseudo polynomial time algorithm for the solution of such games the decision problem for which is in np conp finally we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by condon these games are also known to be in np conp but no polynomial or pseudo polynomial time algorithm is known for them

what s decidable about weighted automata abstract weighted automata map input words to values and have numerous applications in computer science a result by krob from the 90s implies that the universality problem is decidable for weighted automata over the tropical semiring with weights in n and is undecidable when the weights are in z we continue the study of the borders of decidability in weighted automata over the tropical semiring we give a complete picture of the decidability and complexity of various decision problems for them including non emptiness universality equality and containment for the undecidability results we provide direct proofs which stay in the terrain of state machines this enables us to tighten the results and apply them to a very simple class of automata in addition we provide a toolbox of algorithms and techniques for weighted automata on top of which we establish the complexity bounds

systems under the influence of white and colored poisson noise abstract we deal in this paper with systems driven by white or colored poisson noise for a free brownian particle under the influence of white poisson noise an exact generalized master equation in position space is obtained in the gaussian and smoluchowski limits known results are recovered for a general process defined by a stochastic differential equation with colored poisson noise we find an approximate generalized master equation including first order terms in the correlation time and the first correction to the gaussianity under a more restrictive approximation the stationary distribution function is given this is used to study the phase transition in the steady state for a verhulst model corrections to the gaussianity are discussed in this case

marcus versus stratonovich for systems with jump noise the famous ito stratonovich dilemma arises when one examines a dynamical system with a multiplicative white noise in physics literature this dilemma is often resolved in favour of the stratonovich prescription because of its two characteristic properties valid for systems driven by brownian motion i it allows physicists to treat stochastic integrals in the same way as conventional integrals and ii it appears naturally as a result of a small correlation time limit procedure on the other hand the marcus prescription ieee trans inform theory 24 164 1978 stochastics 4 223 1981 should be used to retain i and ii for systems driven by a poisson process levy flights or more general jump processes in present communication we present an in depth comparison of the ito stratonovich and marcus equations for systems with multiplicative jump noise by the examples of a real valued linear system and a complex oscillator with noisy frequency the kubo anderson oscillator we compare solutions obtained with the three prescriptions

global flows with invariant gibbs measures for euler and navier stokes two dimensional fluids we construct a family of probability spaces p 0 associated with the euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow t time evolution leavingp globally invariant t is obtained as the limit of galerkin approximations associated with euler equations p is also in invariant measure for a stochastic process associated with a navier stokes equation with viscosity stochastically perturbed by a white noise force

investigation of a class of stochastic processes using path integral techniques a study is made of the langevin equation x v chi g chi zeta t eta t where the noises eta t and zeta t are gaussian with zero mean and with eta t eta t 2r delta t t zeta t zeta t d tau exp mod t t mod tau path integral representations for conditional probability distributions are given for the two cases tau 0 and tau not 0 for r and d small but of the same order the appropriate path integrals are evaluated to leading order using the method of steepest descents in order to find the stationary probability distribution pst x and the mean relaxation time t for escape from a potential well analytical expressions are given for t when tau 0 and when tau is small for general tau the authors present numerical results for the stationary probability distribution using the particular forms v x 1 2ax2 1 4ax4 r 1n x and g x x which are appropriate to the dye laser

ergodicity and invariant measures of some randomly perturbed classical fields time evolution ergodic properties and invariant measures of some classical fields described by second order differential equations are discussed the classical system is not ergodic a class of markovian random perturbations is considered it is shown that the stochastic system with a fixed stochastic perturbation is uniquely ergodic we discuss in detail the euler equation in two dimensions and the wave equation in one dimension in our models an arbitrarily small stochastic perturbation selects one of the invariant measures for the deterministic system

combinatorial structure and randomized subexponential algorithms for infinite games the complexity of solving infinite games including parity mean payoff and simple stochastic is an important open problem in verification automata and complexity theory in this paper we develop an abstract setting for studying and solving such games based on function optimization over certain discrete structures we introduce new classes of recursively local global rlg and partial recursively local global prlg functions and show that strategy evaluation functions for simple stochastic mean payoff and parity games belong to these classes in this setting we suggest randomized subexponential algorithms appropriate for rlg and prlg function optimization we show that the subexponential algorithms for combinatorial linear programming due to kalai and matousek sharir welzl can be adapted for optimizing the rlg and prlg functions

undirected edge geography abstract the game of edge geography is played by two players who alternately move a token on a graph from one vertex to an adjacent vertex erasing the edge in between the player who first has no legal move is the loser we show that the decision problem of determining whether a position in this game is a win for the first player is pspace complete further the problem remains pspace complete when restricted to planar graphs with maximum degree 3 however if the underlying graph is bipartite we provide 1 a linear algebraic characterization of the p and n positions yielding 2 a polynomial time algorithm for deciding whether any given position is p or n and also 3 a polynomial time algorithm to find winning moves

looking at mean payoff and total payoff through windows we consider two player games played on weighted directed graphs with mean payoff and total payoff objectives two classical quantitative objectives while for single dimensional games the complexity and memory bounds for both objectives coincide we show that in contrast to multi dimensional mean payoff games that are known to be conp complete multi dimensional total payoff games are undecidable we introduce conservative approximations of these objectives where the payoff is considered over a local finite window sliding along a play instead of the whole play for single dimension we show that i if the window size is polynomial deciding the winner takes polynomial time and ii the existence of a bounded window can be decided in np conp and is at least as hard as solving mean payoff games for multiple dimensions we show that i the problem with fixed window size is exptime complete and ii there is no primitive recursive algorithm to decide the existence of a bounded window

the complexity of mean payoff pushdown games two player games on graphs are central in many problems in formal verification and program analysis such as synthesis and verification of open systems in this work we consider solving recursive game graphs or pushdown game graphs that model the control flow of sequential programs with recursion while pushdown games have been studied before with qualitative objectives such as reachability and regular objectives in this work we study for the first time such games with the most well studied quantitative objective the mean payoff objective in pushdown games two types of strategies are relevant 1 global strategies which depend on the entire global history and 2 modular strategies which have only local memory and thus do not depend on the context of invocation but rather only on the history of the current invocation of the module our main results are as follows 1 one player pushdown games with mean payoff objectives under global strategies are decidable in polynomial time 2 two player pushdown games with mean payoff objectives under global strategies are undecidable 3 one player pushdown games with mean payoff objectives under modular strategies are np hard 4 two player pushdown games with mean payoff objectives under modular strategies can be solved in np i e both one player and two player pushdown games with mean payoff objectives under modular strategies are np complete we also establish the optimal strategy complexity by showing that global strategies for mean payoff objectives require infinite memory even in one player pushdown games and memoryless modular strategies are sufficient in two player pushdown games finally we also show that all the problems have the same complexity if the stack boundedness condition is added where along with the mean payoff objective the player must also ensure that the stack height is bounded

synthesis of optimal insertion functions for opacity enforcement our prior work has studied the enforcement of opacity security properties using insertion functions given a system that is not opaque the so called all insertion structure ais is a game structure played by the system and the insertion function that embeds all valid insertion functions in this paper we first propose a more compact ais that can be constructed with lower computational complexity we then introduce the maximum total cost and the maximum mean cost and use them as quantitative objectives to solve for optimal insertion functions specifically we first determine if an insertion function with a finite total cost exists if such an insertion function exists we synthesize an optimal total cost insertion function otherwise we construct an optimal mean cost insertion function in either case we find an optimal insertion strategy on the ais with respect to the corresponding cost objective the algorithmic procedures are adapted from results developed for minimax games and mean payoff games the resulting optimal strategy is represented as a subgraph of the ais that consists of all the system actions and the optimal insertion actions finally we use this subgraph to synthesize an optimal insertion function that is encoded as an i o automaton

nonequilibrium phenomena outlines and bibliographies of a workshop the following is a set of outlines and bibliographies for lectures presented at a summer workshop on nonequilibrium phenomena held from june 22 to july 3 1981 at the institute for theoretical physics in santa barbara these outlines were distributed to the participants in lieu of formal proceedings and they are being presented for publication in the same form in the belief that the information they contain will be useful to a wider audience it should be clearly stated however that the compilation is an informal one which does not claim to be a complete survey of the subject

travelling waves for reaction diffusion equations forced by translation invariant noise abstract inspired by applications we consider reaction diffusion equations on r that are stochastically forced by a small multiplicative noise term that is white in time coloured in space and invariant under translations we show how these equations can be understood as a stochastic partial differential equation spde forced by a cylindrical q wiener process and subsequently explain how to study stochastic travelling waves in this setting in particular we generalize the phase tracking framework that was developed in hamster and hupkes 2018 2019 for noise processes driven by a single brownian motion the main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed we illustrate our approach by two fully worked out examples which highlight the predictive power of our expansions

recursive markov decision processes and recursive stochastic games we introduce recursive markov decision processes rmdps and recursive simple stochastic games rssgs which are classes of finitely presented countable state mdps and zero sum turn based perfect information stochastic games they extend standard finite state mdps and stochastic games with a recursion feature we study the decidability and computational complexity of these games under termination objectives for the two players one player s goal is to maximize the probability of termination at a given exit while the other player s goal is to minimize this probability in the quantitative termination problems given an rmdp or rssg and probability p we wish to decide whether the value of such a termination game is at least p or at most p in the qualitative termination problem we wish to decide whether the value is 1 the important 1 exit subclasses of these models 1 rmdps and 1 rssgs correspond in a precise sense to controlled and game versions of classic stochastic models including multitype branching processes and stochastic context free grammars where the objective of the players is to maximize or minimize the probability of termination extinction we provide a number of upper and lower bounds for qualitative and quantitative termination problems for rmdps and rssgs we show both problems are undecidable for multi exit rmdps but are decidable for 1 rmdps and 1 rssgs specifically the quantitative termination problem is decidable in pspace for both 1 rmdps and 1 rssgs and is at least as hard as the square root sum problem a well known open problem in numerical computation we show that the qualitative termination problem for 1 rmdps i e a controlled version of branching processes can be solved in polynomial time both for maximizing and minimizing 1 rmdps the qualitative problem for 1 rssgs is in np conp and is at least as hard as the quantitative termination problem for condon s finite state simple stochastic games whose complexity remains a well known open problem finally we show that even for 1 rmdps more general qualitative and quantitative model checking problems with respect to linear time temporal properties are undecidable even for a fixed property

opacity of discrete event systems and its applications in this paper we investigate opacity of discrete event systems we define two types of opacities strong opacity and weak opacity given a general observation mapping a language is strongly opaque if all strings in the language are confused with some strings in another language and it is weakly opaque if some strings in the language are confused with some strings in another language we show that security and privacy of computer systems and communication protocols can be investigated in terms of opacity in particular two important properties in security and privacy namely anonymity and secrecy can be studied as special cases of opacity we also show that by properly specifying the languages and the observation mapping three important properties of discrete event systems namely observability diagnosability and detectability can all be reformulated as opacity thus opacity has a wide range of applications also in this paper we provide algorithms for checking strong opacity and weak opacity for systems described by regular languages and having a generalized projection as the observation mapping

brownian motion with multiplicative noises revisited the langevin equation with multiplicative noise and a state dependent transport coefficient should always complemented with the proper interpretation rule of the noise such as the ito and stratonovich conventions although the mathematical relationship between the different rules and how to translate from one rule to another are well established the subject of which is a more physically natural rule still remains controversial in this communication we derive the overdamped langevin equation with multiplicative noise for brownian particles by systematically eliminating the fast degrees of freedom of the underdamped langevin equation the langevin equations obtained here vary depending on the choice of the noise conventions but they are different representations for an identical phenomenon the results apply to multi variable nonequilibrium non stationary systems and other general settings

quantifying stochastic outcomes a system consisting of two species in a fluctuating environment when the interspecies competition for resources is strong will have a stochastic outcome only one of the species will survive but there is no a priori way of knowing which one this will be it is natural in such a situation to ask what will be the probability of one or another of the species surviving this probability is calculated as a function of the average growth rates and the strengths of the interaction between the species and of the randomness this is an example of a class of stochastic problems in which multiple final states are available for occupation we refer to the choice of final states as state selection and the probabilities of final states being occupied as state selection probabilities the calculation of these probabilities is carried out in the context of a model of the system which consists of two coupled stochastic differential equations by reformulating these equations in terms of path integrals the powerful methods based on the use of optimal paths may be utilized to calculate the probability of one outcome or the other the analytical results obtained by using this technique agree well with numerical simulations when both species have the same growth rate although the method adopted rests on the assumption that the strength of the fluctuations d is small remarkably the analytic results are still found to be in good agreement with the numerical results when d is of order 1

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

cyclical games with prohibitions we consider a certain combinatorial game on a digraph for two cases of the price function for one case the game in question extends the cyclical game studied in ehrenfeucht and mycielski 1979 and gurvitch karzanov and khachiyan 1988 which in its turn is a generalization of the well known problem of finding a minimum mean cycle in an edge weighted digraph we prove the existence of optimal uniform stationary strategies for both cases and give algorithms to find such strategies

mean payoff games with partial observation mean payoff games are important quantitative models for open reactive systems they have been widely studied as games of perfect information in this paper we investigate the algorithmic properties of several subclasses of mean payoff games where the players have asymmetric information about the state of the game these games are in general undecidable and not determined according to the classical definition we show that such games are determined under a more general notion of winning strategy we also consider mean payoff games where the winner can be determined by the winner of a finite cycle forming game this yields several decidable classes of mean payoff games of asymmetric information that require only finite memory strategies including a generalization of perfect information games where positional strategies are sufficient we give an exponential time algorithm for determining the winner of the latter

nonlinear random wave equations nonlinear wave equations describing forced vibrations of a string subject to random perturbations of intensity e are studied herein two kinds of random perturbations are considered one is gaussian in the form of initial white noise and the other is a non gaussian random forcing which involves the formal derivative of a brownian sheet the solutions ue t x of the complete forced equation are a stochastic process in space and time which has a unique stochastic equilibrium the time variable t varies in a finite interval i 0 t where t is a fixed positive time and the space variable x varies in an interval j assuming that the initial conditions are of sufficient regularity the fluctuations ue t x from u0 t x the unperturbed solution are analyzed as the intensity of perturbation e 0

stochastic differential equations an approach to the generation of continuous non gaussian processes the generation of continuous random processes with jointly specified probability density and covariation functions is considered the proposed approach is based on the interpretation of the simulated process as a stationary output of a nonlinear dynamic system excited by white gaussian noise and described by a system of a first order stochastic differential equations sde the authors explore how the statistical characteristics of the equation s solution depends on the form of its operator and on the intensity of the input noise some aspects of the approximate synthesis of stochastic differential equations and examples of their application to the generation of non gaussian continuous processes are considered the approach should be useful in signal processing when it is necessary to translate the available a priori information on the real random process into the language of its markov model as well as in simulation of continuous correlated processes with the known probability density function

nonlinear fokker planck equations whose stationary solutions make entropy like functionals stationary abstract the present study extends the correspondence principle of martinez et al that establishes a link between nonlinear fokker planck equations nlfpes and the variational principle approach of the theory of canonical ensembles by virtue of the extended correspondence principle we reobtain results of kaniadakis and quarati for bose and fermi systems and find relations similar to those derived by plastino and plastino and recently by borland concerning the entropy of nonextensive thermostatistics introduced in physics by tsallis moreover we propose nlfpes related to the renyi entropy and to the entropy proposed by sharma and mittal the latter comprises tsallis entropy and the renyi entropy

directed transport induced by spatially modulated levy flights in this paper we study the dynamics of a particle in a ratchet potential subject to multiplicative stable levy noise in the limit of a noise amplitude we compare the dynamics for ito and marcus multiplicative noises and obtain the explicit asymptotics of the escape time in the wells and transition probabilities between the wells a detailed analysis of the noise induced current is performed for the seebeck ratchet with a weak multiplicative noise for

a subexponential randomized algorithm for the simple stochastic game problem we describe a randomized algorithm for the simple stochastic game problem that requires 2o n expected operations for games with n vertices this is the first subexponential time algorithm for this problem

anomalous diffusion in systems driven by the stable levy noise with a finite noise relaxation time and inertia dynamical systems driven by a general levy stable noise are considered the inertia is included and the noise represented by a generalized ornstein uhlenbeck process has a finite relaxation time a general linear problem the additive noise is solved the resulting distribution converges with time to the distribution for the white noise massless case moreover a multiplicative noise is discussed it can make the distribution steeper and the variance which is finite depends sublinearly on time subdiffusion for a small mass a white noise limit corresponds to the stratonovich interpretation on the other hand the distribution tails agree with the ito interpretation if the inertia is very large an escape time from the potential well is calculated

complexity of path forming games abstract for a number of two player games where players alternately choose the next vertex of a simple or an elementary path in a graph we consider the problem to determine whether for a given game instance there is a winning strategy for the first player we show several of these problems to be pspace complete in some special cases we obtain polynomial time algorithms based on graph rewriting or an intricate form of dynamic programming e g we show generalized geography and some other pspace complete problems to be linear time solvable on graphs with constant bounded treewidth

the complexity of stochastic games abstract we consider the complexity of stochastic games simple games of chance played by two players we show that the problem of deciding which player has the greatest chance of winning the game is in the class np co np

qualitative analysis of concurrent mean payoff games we consider concurrent games played by two players on a finite state graph where in every round the players simultaneously choose a move and the current state along with the joint moves determine the successor state we study the most fundamental objective for concurrent games namely mean payoff or limit average objective where a reward is associated to each transition and the goal of player 1 is to maximize the long run average of the rewards and the objective of player 2 is strictly the opposite i e the games are zero sum the path constraint for player 1 could be qualitative i e the mean payoff is the maximal reward or arbitrarily close to it or quantitative i e a given threshold between the minimal and maximal reward we consider the computation of the almost sure resp positive winning sets where player 1 can ensure that the path constraint is satisfied with probability 1 resp positive probability almost sure winning with qualitative constraint exactly corresponds to the question of whether there exists a strategy to ensure that the payoff is the maximal reward of the game our main results for qualitative path constraints are as follows 1 we establish qualitative determinacy results that show that for every state either player 1 has a strategy to ensure almost sure resp positive winning against all player 2 strategies or player 2 has a spoiling strategy to falsify almost sure resp positive winning against all player 1 strategies 2 we present optimal strategy complexity results that precisely characterize the classes of strategies required for almost sure and positive winning for both players and 3 we present quadratic time algorithms to compute the almost sure and the positive winning sets matching the best known bound of the algorithms for much simpler problems such as reachability objectives for quantitative constraints we show that a polynomial time solution for the almost sure or the positive winning set would imply a solution to a long standing open problem of solving the value problem of turn based deterministic mean payoff games that is not known to be solvable in polynomial time

ito vs stratonovich in the presence of absorbing states it is widely assumed that there exists a simple transformation from the ito interpretation to the one by stratonovich and back for any stochastic differential equation of applied interest while this transformation exists under suitable conditions and transforms one interpretation into the other at the price of modifying the drift of the equation it cannot be considered universal we show that a class of stochastic differential equations characterized by the presence of absorbing states and of interest in applications does not admit such a transformation in particular formally applying this transformation may lead to the disappearance of some absorbing states in turn this modifies the long time and even the intermediate time behavior of the solutions the number of solutions can also be modified by the unjustified application of the mentioned transformation as well as by a change in the interpretation of the noise we discuss how these facts affect the classical debate on the ito vs stratonovich dilemma it is widely assumed that there exists a simple transformation from the ito interpretation to the one by stratonovich and back for any stochastic differential equation of applied interest while this transformation exists under suitable conditions and transforms one interpretation into the other at the price of modifying the drift of the equation it cannot be considered universal we show that a class of stochastic differential equations characterized by the presence of absorbing states and of interest in applications does not admit such a transformation in particular formally applying this transformation may lead to the disappearance of some absorbing states in turn this modifies the long time and even the intermediate time behavior of the solutions the number of solutions can also be modified by the unjustified application of the mentioned transformation as well as by a change in the interpret

state dependent fire models and related renewal processes we introduce a general class of stochastic processes forced by instantaneous random fires i e jumps that reset the state variable x to a given value since in many physical systems the fire activity is often dependent on the actual value of the state variable as in the case of natural fires in ecosystems and firing dynamics in neuronal activity the frequency of fire occurrence is assumed to be state dependent such dynamics leads to independent interfire statistics i e to renewal point processes various functions relating the frequency of fire occurrence to x t are analyzed and compared the relation between the probabilistic dynamics of x t and the interfire statistics is derived and some exact probability distribution of both x t and the interfire times are obtained for systems with different degrees of complexity after studying processes in which the fire activity is coupled only to a deterministic drift we also analyze processes forced by either additive or multiplicative gaussian white noise

adiabatic elimination for systems of brownian particles with nonconstant damping coefficients we discuss the problem of eliminating the momentum variable in the phase space langevin equations for a system of brownian particles in two related situations i position dependent damping and ii existence of hydrodynamic interactions we discuss the problems associated with the conventional elimination and we develop an alternative elimination procedure in the lagevin framework which leads to the correct smoluchowski equation we give a heuristic argument on the basis of stochastic differential equations for the smoluchowski limit and establish rigorously the limit for the general case of position dependent friction and diffusion coefficents

langevin dynamics for vector variables driven by multiplicative white noise a functional formalism we discuss general multidimensional stochastic processes driven by a system of langevin equations with multiplicative white noise in particular we address the problem of how time reversal diffusion processes are affected by the variety of conventions available to deal with stochastic integrals we present a functional formalism to build up the generating functional of correlation functions without any type of discretization of the langevin equations at any intermediate step the generating functional is characterized by a functional integration over two sets of commuting variables as well as grassmann variables in this representation time reversal transformation became a linear transformation in the extended variables simplifying in this way the complexity introduced by the mixture of prescriptions and the associated calculus rules the stochastic calculus is codified in our formalism in the structure of the grassmann algebra we study some examples such as higher order derivative langevin equations and the functional representation of the micromagnetic stochastic landau lifshitz gilbert equation

symmetric powers and modular invariants of elementary abelian p groups abstract let e be an elementary abelian p group of order q p n let w be a faithful indecomposable representation of e with dimension 2 over a field k of characteristic p and let v s m w with m q we prove that the rings of invariants k v e are generated by elements of degree q and relative transfers this extends recent work of wehlau 18 on modular invariants of cyclic groups of order p if m p we prove that k v e is generated by invariants of degree 2 q 3 extending a result of fleischmann sezer shank and woodcock 6 for cyclic groups of order p our methods are primarily representation theoretic and along the way we prove that for any d q with d m q s d v is projective relative to the set of subgroups of e with order m and that the sequence s d v d 0 is periodic with period q modulo summands which are projective relative to the same set of subgroups these results extend results of almkvist and fossum 1 on cyclic groups of prime order

transformation invariant stochastic catastrophe theory catastrophe theory is a mathematical formalism for modeling nonlinear systems whose discontinuous behavior is determined by smooth changes in a small number of driving parameters fitting a catastrophe model to noisy data constitutes a serious challenge however because catastrophe theory was formulated specifically for deterministic systems loren cobb addressed this challenge by developing a stochastic counterpart of catastrophe theory sct based on ito stochastic differential equations in sct the stable and unstable equilibrium states of the system correspond to the modes and the antimodes of the empirical probability density function respectively unfortunately sct is not invariant under smooth and invertible transformations of variables this is an important limitation since invariance to diffeomorphic transformations is essential in deterministic catastrophe theory from the ito transformation rules we derive a generalized version of sct that does remain invariant under transformation and can include cobb s sct as a special case we show that an invariant function is obtained by multiplying the probability density function with the diffusion function of the stochastic process this invariant function can be estimated by a straightforward time series analysis based on level crossings we illustrate the invariance problem and its solution with two applications

concurrent reachability games we consider concurrent two player games with reachability objectives in such games at each round player 1 and player 2 independently and simultaneously choose moves and the two choices determine the next state of the game the objective of player 1 is to reach a set of target states the objective of player 2 is to prevent this these are zero sum games and the reachability objective is one of the most basic objectives determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification there are three types of winning states according to the degree of certainty with which player 1 can reach the target from type 1 states player 1 has a deterministic strategy to always reach the target from type 2 states player 1 has a randomized strategy to reach the target with probability 1 from type 3 states player 1 has for every real e 0 a randomized strategy to reach the target with probability greater than email protected we show that for finite state spaces all three sets of winning states can be computed in polynomial time type 1 states in linear time and type 2 and type 3 states in quadratic time the algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies

ito versus stratonovich a survey is given of the facts and fancies concerning the nonlinear langevin or ito equation actually it is merely a pre equation which becomes an equation when an interpretation rule is added the rules of ito and stratonovich differ but both are mathematically consistent and therefore equally admissible conventions the reason why they seem to lead to physical differences is that the langevin approach used to arrive at the equation involves a tacit assumption for systems with external noise this assumption can be justified and it is then clear that the stratonovich rule applies systems with internal noise however can only be properly described by a master equation and the ito stratonovich controversy never enters afterward one is free to model the resulting fluctuations either with an ito or a stratonovich scheme but that does not lead to any new information

a test particle approach in the stochastic stability analysis of charged particle beams abstract the stochastic stability of a charged particle beam propagating in a randomly unhomogeneous medium is analyzed using an internal test particle approach the effect of the random medium is included by adding a stochastic fluctuating term in the neutralization factors the effects of weakly colored noise and of fluctuations called telegraphic noise i e two valued markov process are studied the exact solution of the telegraphist noise is compared with the approximate solution corresponding to the weakly colored noise which was obtained using the stratonovich approach by means of an expansion technique it is concluded that the beam is always probabilistically unstable whenever the power spectral density of the fluctuations contains a component at the frequency of the radial oscillations of the test particles

state dependent diffusion thermodynamic consistency and its path integral formulation the friction coefficient of a particle can depend on its position as it does when the particle is near a wall we formulate the dynamics of particles with such state dependent friction coefficients in terms of a general langevin equation with multiplicative noise whose evaluation requires the introduction of specific rules two common conventions the ito and the stratonovich provide alternative rules for evaluation of the noise but other conventions are possible we show that the requirement that a particle s distribution function approach the boltzmann distribution at long times dictates that a drift term must be added to the langevin equation this drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise we explore the consequences of this result in a number of examples with spatially varying diffusion coefficients we also derive a path integral representation for arbitrary interpretation of the noise and use it in a perturbative study of correlations in a simple system

quantitative fair simulation games simulation is an attractive alternative for language inclusion for automata as it is an under approximation of language inclusion but usually has much lower complexity for non deterministic automata while language inclusion is pspace complete simulation can be computed in polynomial time simulation has also been extended in two orthogonal directions namely 1 fair simulation for simulation over specified set of infinite runs and 2 quantitative simulation for simulation between weighted automata again while fair trace inclusion is pspace complete fair simulation can be computed in polynomial time for weighted automata the quantitative language inclusion problem is undecidable for mean payoff automata and the decidability is open for discounted sum automata whereas the quantitative simulation reduce to mean payoff games and discounted sum games which admit pseudo polynomial time algorithms in this work we study quantitative simulation for weighted automata with buchi acceptance conditions i e we generalize fair simulation from non weighted automata to weighted automata we show that imposing buchi acceptance conditions on weighted automata changes many fundamental properties of the simulation games for example whereas for mean payoff and discounted sum games the players do not need memory to play optimally we show in contrast that for simulation games with buchi acceptance conditions i for mean payoff objectives optimal strategies for both players require infinite memory in general and ii for discounted sum objectives optimal strategies need not exist for both players while the simulation games with buchi acceptance conditions are more complicated e g due to infinite memory requirements for mean payoff objectives as compared to their counterpart without buchi acceptance conditions we still present pseudo polynomial time algorithms to solve simulation games with buchi acceptance conditions for both weighted mean payoff and weighted discounted sum automata

elementary divisors of tensor products and p ranks of binomial matrices abstract let f be a field of characteristic p and let a and b be two jordan blocks with nonzero eigenvalues the reduction formulas of c w norman enable us to compute the elementary divisors of a b we investigate connections between the elementary divisors of a b and binomial matrices in particular we show that norman s reduction formulas can be used to compute the p rank of certain binomial matrices

mean payoff games and propositional proofs we associate a cnf formula to every instance of the mean payoff game problem in such a way that if the value of the game is non negative the formula is satisfiable and if the value of the game is negative the formula has a polynomial size refutation in s 2 frege i e dnf resolution this reduces mean payoff games to the weak automatizability of s 2 frege and to the interpolation problem for s 2 2 frege since the interpolation problem for s 1 frege i e resolution is solvable in polynomial time our result is close to optimal up to the computational complexity of solving mean payoff games the proof of the main result requires building low depth formulas that compute the bits of the sum of a constant number of integers in binary notation and low complexity proofs of the required arithmetic properties

exterior and symmetric powers of modules for cyclic 2 groups abstract we prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2 group this makes computation straightforward previously a complete description was only known for cyclic groups of prime order

numerical method for solving stochastic differential equations with dichotomous noise we propose a numerical method for solving stochastic differential equations with dichotomous markov noise the numerical scheme is formulated such that i the stochastic formula used follows the stratonovichtaylor form over the entire range of noise correlation times including the gaussian white noise limit and ii the method is readily applicable to dynamical systems driven by arbitrary types of noise provided there exists a way to describe the random increment of the stochastic process expressed in the stratonovich taylor form we further propose a simplified taylor scheme that significantly reduces the computation time while still satisfying the moment properties up to the required order the accuracies and efficiencies of the proposed algorithms are validated by applying the schemes to two prototypical model systems that possess analytical solutions

randomness for free we consider two player zero sum games on finite state graphs these games can be classified on the basis of the information of the players and on the mode of interaction between them on the basis of information the classification is as follows a partial observation both players have partial view of the game b one sided complete observation one player has complete observation and c complete observation both players have complete view of the game on the basis of mode of interaction we have the following classification a concurrent players interact simultaneously and b turn based players interact in turn the two sources of randomness in these games are randomness in the transition function and randomness in the strategies in general randomized strategies are more powerful than deterministic strategies and probabilistic transitions give more general classes of games we present a complete characterization for the classes of games where randomness is not helpful in a the transition function probabilistic transitions can be simulated by deterministic transitions and b strategies pure strategies are as powerful as randomized strategies as a consequence of our characterization we obtain new undecidability results for these games

contingent planning under uncertainty via stochastic satisfiability we describe a new planning technique that efficiently solves probabilistic propositional contingent planning problems by converting them into instances of stochastic satisfiability ssat and solving these problems instead we make fundamental contributions in two areas the solution of ssat problems and the solution of stochastic planning problems this is the first work extending the planning as satisfiability paradigm to stochastic domains our planner zander can solve arbitrary goal oriented finite horizon partially observable markov decision processes pomdps an empirical study comparing zander to seven other leading planners shows that its performance is competitive on a range of problems

multiplicative levy processes ito versus stratonovich interpretation langevin equation with a multiplicative stochastic force is considered that force is uncorrelated it has the levy distribution and the power law intensity the fokker planck equations which correspond both to the ito and stratonovich interpretation are presented they are solved for the case without drift and for the harmonic oscillator potential the variance is evaluated it is always infinite for the ito case whereas for the stratonovich one it can be finite and rise with time slower that linearly which indicates subdiffusion analytical results are compared with numerical simulations

prescription induced jump distributions in multiplicative poisson processes generalized langevin equations gle with multiplicative white poisson noise pose the usual prescription dilemma leading to different evolution equations master equations for the probability distribution contrary to the case of multiplicative gaussian white noise the stratonovich prescription does not correspond to the well known mid point or any other intermediate prescription by introducing an inertial term in the gle we show that the ito and stratonovich prescriptions naturally arise depending on two time scales the one induced by the inertial term and the other determined by the jump event we also show that when the multiplicative noise is linear in the random variable one prescription can be made equivalent to the other by a suitable transformation in the jump probability distribution we apply these results to a recently proposed stochastic model describing the dynamics of primary soil salinization in which the salt mass balance within the soil root zone requires the analysis of different prescriptions arising from the resulting stochastic differential equation forced by multiplicative white poisson noise whose features are tailored to the characters of the daily precipitation a method is finally suggested to infer the most appropriate prescription from the data

new scaling algorithms for the assignment and minimum mean cycle problems in this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems our assignment algorithm is based on applying scaling to a hybrid version of the recentauction algorithm of bertsekas and the successive shortest path algorithm the algorithm proceeds by relaxing the optimality conditions and the amount of relaxation is successively reduced to zero on a network with 2n nodes m arcs and integer arc costs bounded byc the algorithm runs in o sqrt n m log nc time and uses very simple data structures this time bound is comparable to the time taken by gabow and tarjan s scaling algorithm and is better than all other time bounds under thesimilarity assumption i e c o nk for somek we next consider the minimum mean cycle problem themean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains theminimum mean cycle problem is to identify a cycle whose mean cost is minimum we show that by using ideas of the assignment algorithm in an approximate binary search procedure the minimum mean cycle problem can also be solved in o sqrt n m lognc time under the similarity assumption this is the best available time bound to solve the minimum mean cycle problem

statistical behavior and coherent structures in two dimensional inviscid turbulence the ergodic and mixing questions are studied for the dynamical systems obtained by truncation of the fourier transformed two dimensional euler equations with periodic boundary conditions such statistical properties are important because they bear a close relationship to the existence of coherent structures in turbulent flow fields these theoretical relationships are derived in particular for the case of macroscopic vortex structures the main emphasis is on the numerical analysis of these statistical properties several different systems with various a priori statistical properties ergodic nonergodic unknown have been computed in this manner comparison standards are established for systems with unknown statistical properties a careful error analysis is presented for the results in particular it is found that final integration times for the modal truncations must be kept small relative to a relaxation time in order to insure numerical accuracy the consequences of this fact are important for th

adams operations on the green ring of a cyclic group of prime power order we consider the green ring r kc for a cyclic p group c over a field k of prime characteristic p and determine the adams operations psi n in the case where n is not divisible by p this gives information on the decomposition into indecomposables of exterior powers and symmetric powers of kc modules

coupled map lattices as models of deterministic and stochastic differential delay equations we discuss the probabilistic properties of a class of differential delay equations dde s by first reducing the equations to coupled map lattices and then considering the spectral properties of the associated transfer operators the analysis is carried out for the deterministic case and a stochastic case perturbed by additive or multiplicative white noise this scheme provides an explicit description of the evolution of phase space densities in dde s and yields an evolution equation that approximates the analog for delay equations of the generalized liouville and fokker planck equations it is shown that in many cases of interest for both stochastic and deterministic delay equations the phase space densities reach a limit cycle in the asymptotic regime this statistical cycling is observed numerically in continuous time systems with delay and discussed in light of our analytical description of the transfer operators 1995 the american physical society

wealth condensation in a simple model of economy abstract we introduce a simple model of economy where the time evolution is described by an equation capturing both exchange between individuals and random speculative trading in such a way that the fundamental symmetry of the economy under an arbitrary change of monetary units is insured we investigate a mean field limit of this equation and show that the distribution of wealth is of the pareto power law type the pareto behaviour of the tails of this distribution appears to be robust for finite range models as shown using both a mapping to the random directed polymer problem as well as numerical simulations in this context a phase transition between an economy dominated by a few individuals and a situation where the wealth is more evenly spread out is found an interesting outcome is that the distribution of wealth tends to be very broadly distributed when exchanges are limited either in amplitude or topologically favouring exchanges and less surprisingly increasing taxes seems to be an efficient way to reduce inequalities

an optimal on line algorithm for metrical task system in practice almost all dynamic systems require decisions to be made on line without full knowledge of their future impact on the system a general model for the processing of sequences of tasks is introduced and a general on line decision algorithm is developed it is shown that for an important class of special cases this algorithm is optimal among all on line algorithms specifically a task system s d for processing sequences of tasks consists of a set s of states and a cost matrix d where d i j is the cost of changing from state i to state j we assume that d satisfies the triangle inequality and all diagonal entries are 0 the cost of processing a given task depends on the state of the system a schedule for a sequence t 1 t 2 t k of tasks is a sequence s 1 s 2 s k of states where s i is the state in which t i is processed the cost of a schedule is the sum of all task processing costs and the state transition costs incurred an on line scheduling algorithm is one that chooses s i only knowing t 1 t 2 t i such an algorithm is w competitive if on any input task sequence its cost is within an additive constant of w times the optimal offline schedule cost the competitive ratio w s d is the infimum w for which there is a w competitive on line scheduling algorithm for s d it is shown that w s d 2 s 1 for every task system in which d is symmetric and w s d o s 2 for every task system finally randomized on line scheduling algorithms are introduced it is shown that for the uniform task system in which d i j 1 for all i j the expected competitive ratio w s d o log s

nonlinear stochastic equations with multiplicative levy noise the langevin equation with a multiplicative levy white noise is solved the noise amplitude and the drift coefficient have a power law form a validity of ordinary rules of the calculus for the stratonovich interpretation is discussed the solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the stratonovich interpretation the problem of escaping from a potential well is analyzed numerically predictions of different interpretations of the stochastic integral are compared

black scholes option pricing within ito and stratonovich conventions abstract options are financial instruments designed to protect investors from the stock market randomness in 1973 black scholes and merton proposed a very popular option pricing method using stochastic differential equations within the ito interpretation herein we derive the black scholes equation for the option price using the stratonovich calculus along with a comprehensive review aimed to physicists of the classical option pricing method based on the ito calculus we show as can be expected that the black scholes equation is independent of the interpretation chosen we nonetheless point out the many subtleties underlying black scholes option pricing method

stationary solutions of linear stochastic delay differential equations applications to biological systems recently k uchler and mensch stochastics stochastics rep 40 23 1992 derived exact stationary probability densities for linear stochastic delay differential equations this paper presents an alternative derivation of these solutions by means of the fokker planck approach introduced by guillouzic phys rev e 59 3970 1999 61 4906 2000 applications of this approach which is argued to have greater generality are discussed in the context of stochastic models for population growth and tracking movements

modeling heterogeneous materials via two point correlation functions basic principles heterogeneous materials abound in nature and man made situations examples include porous media biological materials and composite materials diverse and interesting properties exhibited by these materials result from their complex microstructures which also make it difficult to model the materials yeong and torquato phys rev e 57 495 1998 introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions in this first part of a series of two papers we collect the known necessary conditions on the standard two point correlation function s2 r and formulate a conjecture in particular we argue that given a complete two point correlation function space s2 r of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space we provide examples of realizable two point correlation functions and suggest a set of analytical basis functions we also discuss an exact mathematical formulation of the re construction problem and prove that s2 r cannot completely specify a two phase heterogeneous material alone moreover we devise an efficient and isotropy preserving construction algorithm namely the lattice point algorithm to generate realizations of materials from their two point correlation functions based on the yeong torquato technique subsequent analysis can be performed on the generated images to obtain desired macroscopic properties these developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two point correlation functions we will mainly focus on basic principles in this paper the algorithmic details and applications of the general scheme are given in the second part of this series of two papers

sine gordon solitons in the presence of a noisy potential abstract we study the interaction of sine gordon solitons with two kinds of impurity potentials localized in space and varying randomly in time the production of modified kink antikink pairs is observed with different features depending respectively on the additive or multiplicative nature of the noise

an invariant measure for the equation u_ tt u_ xx u 3 0 numerical studies of the initial boundary value problem of the semilinear wave equationu tt u xx u 3 0 subject to periodic boundary conditionsu t 0 u t 2 u t t 0 u t t 2 and initial conditionsu 0 x u 0 x u t 0 x v 0 x whereu 0 x andv 0 x satisfy the same periodic conditions suggest that solutions ultimately return to a neighborhood of the initial stateu 0 x v 0 x after undergoing a possibly chaotic evolution in this paper an appropriate abstract space is considered in this space a finite measure is constructed this measure is invariant under the flow generated by the hamiltonian system which corresponds to the original equation this enables one to verify the above returning property

opacity enforcing supervisory strategies via state estimator constructions state based notions of opacity such as initial state opacity and infinite step opacity emerge as key properties in numerous security applications of discrete event systems we consider systems that are modeled as partially observed nondeterministic finite automata and tackle the problem of constructing a minimally restrictive opacity enforcing supervisor moes which limits the system s behavior within some prespecified legal behavior while enforcing initial state opacity or infinite step opacity requirements we characterize the solution to moes under some mild assumptions in terms of the supremal element of certain controllable normal and opaque languages we also show that this supremal element always exists and that it can be implemented using state estimators the result is a supervisor that achieves conformance to the pre specified legal behavior while enforcing initial state opacity by disabling at any given time a subset of the controllable system events in a way that minimally restricts the range of allowable system behavior although infinite step opacity cannot be easily translated to language based opacity we show that by using a finite bank of supervisors the aforementioned approach can be extended to enforce infinite step opacity in a minimally restrictive way

random walk in a one dimensional random medium abstract the random walk of a particle in a one dimensional random medium is examined by means of the equivalent transfer rates technique in the discrete as well as in the continuous version of the model the probability distributions of the energy dependent equivalent transfer rates are found analytically either by a matching procedure in the discrete case or exactly in the continuous model both discrete and continuous models are shown to belong to the same universality class the average probability of presence of the particle at its initial point is then computed for a non zero global bias it decreases at large times according to a negative power law with an exponent depending on disorder and bias when there is no global bias sinai s model the decay at large times follows a logarithmic law

a pseudo polynomial algorithm for mean payoff stochastic games with perfect information and few random positions abstract we consider two person zero sum stochastic mean payoff games with perfect information or bwr games given by a digraph g v e with local rewards r e z and three types of positions black v b white v w and random v r forming a partition of v it is a long standing open question whether a polynomial time algorithm for bwr games exists or not even when v r 0 in fact a pseudo polynomial algorithm for bwr games would already imply their polynomial solvability in this paper 1 we show that bwr games with a constant number of random positions can be solved in pseudo polynomial time more precisely in any bwr game with v r o 1 a saddle point in uniformly optimal pure stationary strategies can be found in time polynomial in v w v b the maximum absolute local reward and the common denominator of the transition probabilities

self diffusion in a non uniform one dimensional system of point particles with collisions we generalize the results of spitzer jepsen and others 1 4 on the motion of a tagged particle in a uniform one dimensional system of point particles undergoing elastic collisions to the case where there is also an external potential u x when u x is periodic or random bounded and statistically translation invariant then the scaled trajectory of a tagged particle y a t y at mathord left vphantom y a t y at sqrt a right kern nulldelimiterspace sqrt a converges as a to a brownian motion w d t with diffusion constant d rho min left langle v right rangle mathord left vphantom d rho min left langle v right rangle hat rho 2 right kern nulldelimiterspace hat rho 2 where hat rho is the average density left langle v right rangle sqrt 2 mathord left vphantom 2 pi beta m right kern nulldelimiterspace pi beta m is the mean absolute velocity and 1 the temperature of the system when u x is itself changing on a macroscopic scale i e u a x u x mathord left vphantom x sqrt a right kern nulldelimiterspace sqrt a then the limiting process is a spatially dependent diffusion the stochastic differential equation describing this process is now non linear and is particularly simple in stratonovich form this lends weight to the belief that heuristics are best done in that form

random non linear wave equations smoothness of the solutions we show existence and uniqueness for the solution of a onedimensional wave equation with non linear random forcing then we give sufficient conditions for the solution at a given time and a given point to have a density and for this density to be smooth the proof uses the extension of the malliavin calculus to the two parameters wiener functionals