random copolymers and the morita approximation polymer adsorption and polymer localization we analyse directed walk models of random copolymer adsorption and localization ideally we would like t os olve thequenched problem but it appears to be intractable even for simple directed models the annealed approximation is solvable but is inadequate in the strong interaction regime fo rt he adsorption problem and gives a qualitatively incorrect phase diagram for the localization problem in this paper we treat these directed models using an approximation suggested by morita 1964 j math phys 5 1401 5 in which the proportion of each comonomer is fixed we find that the morita approximation leads to behaviour that is closer to that of the quenched average model and this is particularly interesting in the localization problem where the phase diagram is a tl east qualitatively very similar to that of the quenched average problem we also show that the phase boundaries in the morita approximation are bounds on the locations of the phase boundaries of the quenched model

hierarchical pinning models quadratic maps and quenched disorder we consider a hierarchical model of polymer pinning in presence of quenched disorder introduced by derrida et al j stat phys 66 1189 1213 1992 which can be re interpreted as an infinite dimensional dynamical system with random initial condition the disorder it is defined through a recurrence relation for the law of a random variable r n n 1 2 which in absence of disorder i e when the initial condition is degenerate reduces to a particular case of the well known logistic map the large n limit of the sequence of random variables 2 n log r n a non random quantity which is naturally interpreted as a free energy plays a central role in our analysis the model depends on a parameter 0 1 related to the geometry of the hierarchical lattice and has a phase transition in the sense that the free energy is positive if the expectation of r 0 is larger than a certain threshold value and it is zero otherwise it was conjectured in derrida et al j stat phys 66 1189 1213 1992 that disorder is relevant respectively irrelevant or marginally relevant if 1 2 1 2 we find the correct scaling form for weak disorder of the critical point shift

levy processes with no positive jumps at an increase time we study the behaviour of a levy process with no positive jumps near its increase times specifically we construct a local time on the set of increase times then we describe the path decomposition at an increase time chosen at random according to the local time and we evaluate the rate of escape before and after this instant

the weak coupling limit of disordered copolymer models a copolymer is a chain of repetitive units monomers that are almost identical but they differ in their degree of affinity for certain solvents this difference leads to striking phenomena when the polymer fluctuates in a non homogeneous medium for example made up by two solvents separated by an interface one may observe for instance the localization of the polymer at the interface between the two solvents a discrete model of such system based on the simple symmetric random walk on z has been investigated in bolthausen and den hollander ann probab 25 1997 1334 1366 notably in the weak polymer solvent coupling limit where the convergence of the discrete model toward a continuum model based on brownian motion has been established this result is remarkable because it strongly suggests a universal feature of copolymer models in this work we prove that this is indeed the case more precisely we determine the weak coupling limit for a general class of discrete copolymer models obtaining as limits a one parameter alpha in 0 1 family of continuum models based on alpha stable regenerative sets

pinning and wetting transition for 1 1 dimensional fields with laplacian interaction we consider a random field 1 n as a model for a linear chain attracted to the defect line 0 that is the x axis the free law of the field is specified by the density exp iv i with respect to the lebesgue measure on n where is the discrete laplacian and we allow for a very large class of potentials v the interaction with the defect line is introduced by giving the field a reward 0 each time it touches the x axis we call this model the pinning model we consider a second model the wetting model in which in addition to the pinning reward the field is also constrained to stay nonnegative we show that both models undergo a phase transition as the intensity of the pinning reward varies both in the pinning a p and in the wetting a w case there exists a critical value ca such that when ca the field touches the defect line a positive fraction of times localization while this does not happen for ca delocalization the two critical values are nontrivial and distinct 0 cp cw and they are the only nonanalyticity points of the respective free energies for the pinning model the transition is of second order hence the field at cp is delocalized on the other hand the transition in the wetting model is of first order and for cw the field is localized the core of our approach is a markov renewal theory description of the field

new results on stability analysis of markovian switching singular systems this paper addresses the stability problem for linear continuous time markovian switching singular systems considering the inherent state jump behavior at the switching instants a necessary and sufficient condition of exponential stability in the mean square sense for the markovian switching singular system is established in terms of linear matrix inequalities by means of a stochastic lyapunov approach based on the obtained stability result sufficient conditions of exponential stability in the mean square sense for the markovian switching singular system with uncertain and partly unknown transition probability are presented numerical examples are presented to illustrate the effectiveness of the proposed approach

intersection of independent regenerative sets we investigate the nature of the intersection of two independent regenerative sets the approach combines bochners subordination and potential theory for a pair of markov processes in duality

a local limit theorem for random walks conditioned to stay positive we consider a real random walk s n x1 x n attracted without centering to the normal law this means that for a suitable norming sequence a n we have the weak convergence s n a n x dx x being the standard normal density a local refinement of this convergence is provided by gnedenko s and stone s local limit theorems in the lattice and nonlattice case respectively now let open image in new window denote the event s1 0 s n 0 and let s n denote the random variable s n conditioned on open image in new window it is known that s n a n x dx where x x exp x2 2 1 x 0 what we establish in this paper is an equivalent of gnedenko s and stone s local limit theorems for this weak convergence we also consider the particular case when x1 has an absolutely continuous law in this case the uniform convergence of the density of s n a n towards x holds under a standard additional hypothesis in analogy to the classical case we finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process unlike the classical proofs of the llt we make no use of characteristic functions our techniques are rather taken from the so called fluctuation theory for random walks

variational characterization of the critical curve for pinning of random polymers in this paper we look at the pinning of a directed polymer by a one dimensional linear interface carrying random charges there are two phases localized and delocalized depending on the inverse temperature and on the disorder bias using quenched and annealed large deviation principles for the empirical process of words drawn from a random letter sequence according to a random renewal process birkner greven and den hollander 6 we derive variational formulas for the quenched respectively annealed critical curve separating the two phases these variational formulas are used to obtain a necessary and sufficient criterion stated in terms of relative entropies for the two critical curves to be different at a given inverse temperature a property referred to as relevance of the disorder this criterion in turn is used to show that the regimes of relevant and irrelevant disorder are separated by a unique inverse critical temperature subsequently upper and lower bounds are derived for the inverse critical temperature from which sufficient conditions under which it is strictly positive respectively finite are obtained the former condition is believed to be necessary as well a problem that we will address in a forthcoming paper random pinning has been studied extensively in the literature the present paper opens up a window with a variational view our variational formulas for the quenched and the annealed critical curve are new and provide valuable insight into the nature of the phase transition our results on the inverse critical temperature drawn from these variational formulas are not new but they offer an alternative approach that is flexible enough to be extended to other models of random polymers with disorder

one sided local large deviation and renewal theorems in the case of infinite mean if s n n 0 is an integer valued random walk such that s n a n converges in distribution to a stable law of index 0 1 as n then gnedenko s local limit theorem provides a useful estimate for p s n r for values of r such that r a n is bounded the main point of this paper is to show that under certain circumstances there is another estimate which is valid when r a n in other words to establish a large deviation local limit theorem we also give an asymptotic bound for p s n r which is valid under weaker assumptions this last result is then used in establishing some local versions of generalized renewal theorems

intermediate hamiltonians as a new class of effective hamiltonians the theory of effective hamiltonians is well established however limitations appear in its applicability for many problems in molecular physics and quantum chemistry the standard effective hamiltonians may become strongly non hermitian when there is a large coupling between the model space in which they are defined and the outer space moreover in the presence of intruder states discontinuities appear in the matrix elements of these effective hamiltonians as a function of the internuclear distances to solve these difficulties a new class of effective hamiltonians called intermediate hamiltonians is presented only one part of their roots are exact eigen energies of the full hamiltonian the theory of these intermediate hamiltonians is presented by means of a new wave operator r which is the analogue of the wave operator omega in the theory of effective hamiltonians solutions are obtained by a generalised degenerate perturbation theory gdpt and by iterative procedures two model systems are numerically solved which demonstrate the good convergence properties of gdpt with respect to standard degenerate perturbation theory dpt continuity of the solutions is also checked in the presence of an intruder state

on the irrelevant disorder regime of pinning models recent results have lead to substantial progress in understanding the role of disorder in the de localization transition of polymer pinning models notably there is an understanding of the crucial issue of disorder relevance and irrelevance that albeit still partial is now rigorous in this work we exploit interpolation and replica coupling methods to get sharper results on the irrelevant disorder regime of pinning models in particular we compute in this regime the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal ex pansion obtained by field theory methods we also show that the quenched and the quenched averaged correlation length exponents coincide while in general they are ex pected to be different interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals 2000 mathematics subject classification 60k35 60k05 82b41 82b44

marginal relevance of disorder for pinning models the effect of disorder on pinning and wetting models has attracted much attention in theoretical physics in particular it has been predicted on the basis of the harris criterion that disorder is relevant annealed and quenched model have different critical points and critical exponents if the return probability exponent alpha a positive number that characterizes the model is larger than 1 2 weak disorder has been predicted to be irrelevant i e coinciding critical points and exponents if alpha 1 2 recent mathematical work has put these predictions on firm grounds in renormalization group terms the case alpha 1 2 is a marginal case and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality the question is particularly intriguing also because the case alpha 1 2 includes the classical models of two dimensional wetting of a rough substrate of pinning of directed polymers on a defect line in dimension 3 1 or 1 1 and of pinning of an heteropolymer by a point potential in three dimensional space here we prove disorder relevance both for the general alpha 1 2 pinning model and for the hierarchical version of the model proposed by b derrida v hakim and j vannimenus jsp 1992 in the sense that we prove a shift of the quenched critical point with respect to the annealed one in both cases we work with gaussian disorder and we show that the shift is at least of order exp 1 beta 4 for beta small if beta is the standard deviation of the disorder

the effect of disorder on polymer depinning transitions we consider a polymer with monomer locations modeled by the trajectory of a markov chain in the presence of a potential that interacts with the polymer when it visits a particular site 0 we assume that probability of an excursion of length n is given by n c varphi n for some 1 3 2 at high temperature the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point the size of this neighborhood scales as beta 1 2c 3 where is the inverse temperature for c 0 for sufficiently high temperature the quenched and annealed curves are within a factor of 1 epsilon for all u near the critical point in particular the quenched and annealed critical points are equal for c 3 2 the regime depends on the slowly varying function varphi

a note on the diffusion of directed polymers in a random environment a simple martingale argument is presented which proves that directed polymers in random environments satisfy a central limit theorem ford 3 and if the disorder is small enough this simplifies and extends an approach by j imbrie and t spencer

stability analysis for stochastic hybrid systems this survey addresses stability analysis for stochastic hybrid systems shs which are dynamical systems that combine continuous change and instantaneous change and that also include random effects we re emphasize the common features found in most of the models that have appeared in the literature which include stochastic switched systems markov jump systems impulsive stochastic systems switching diffusions stochastic impulsive systems driven by renewal processes diffusions driven by levy processes piecewise deterministic markov processes general stochastic hybrid systems and stochastic hybrid inclusions then we review many of the stability concepts that have been studied including lyapunov stability lagrange stability asymptotic stability and recurrence next we detail lyapunov based sufficient conditions for these properties and additional relaxations of lyapunov conditions many other aspects of stability theory for shs like converse lyapunov theorems and robustness theory are not fully developed hence we also formulate some open problems to serve as a partial roadmap for the development of the underdeveloped pieces

the continuum disordered pinning model any renewal processes on mathbb n 0 with a polynomial tail with exponent alpha in 0 1 has a non trivial scaling limit known as the alpha stable regenerative set in this paper we consider gibbs transformations of such renewal processes in an i i d random environment called disordered pinning models we show that for alpha in left frac 1 2 1 right these models have a universal scaling limit which we call the continuum disordered pinning model cdpm this is a random closed subset of mathbb r in a white noise random environment with subtle features any fixed a s property of the alpha stable regenerative set e g its hausdorff dimension is also an a s property of the cdpm for almost every realization of the environment nonetheless the law of the cdpm is singular with respect to the law of the alpha stable regenerative set for almost every realization of the environment the existence of a disordered continuum model such as the cdpm is a manifestation of disorder relevance for pinning models with alpha in left frac 1 2 1 right

the bloch wave operator generalizations and applications part i the time independent case this is part 1 of a two part review on wave operator theory and methods the basic theory of the time independent wave operator is presented in terms of partitioned matrix theory for the benefit of general readers with a discussion of the links between the matrix and projection operator approaches the matrix approach is shown to lead to simple derivations of the wave operators and effective hamiltonians of lowdin bloch des cloizeaux and kato as well as to some associated variational forms the principal approach used throughout stresses the solution of the nonlinear equation for the reduced wave operator leading to the construction of the effective hamiltonians of bloch and of des cloizeaux several mathematical techniques which are useful in implementing this approach are explained some of them being relatively little known in the area of wave operator calculations the theoretical discussion is accompanied by several specimen numerical calculations which apply the described techniques to a selection of test matrices taken from the previous literature on wave operator methods the main emphasis throughout is on the use of numerical methods which use iterative or perturbation algorithms with simple pade approximant methods being found sufficient to deal with most of the cases of divergence which are encountered the use of damping factors and relaxation parameters is found to be effective in stabilizing calculations which use the energy dependent effective hamiltonian of lowdin in general the computations suggest that the numerical applications of the nonlinear equation for the reduced wave operator are best carried out with the equation split into a pair of equations in which the bloch effective hamiltonian appears as a separate entity the presentation of the theoretical and computational details throughout is accompanied by references to and discussion of many works which have used wave operator methods in physics chemistry and engineering some of the techniques described in this part 1 will be further extended and applied in part 2 of the review which deals with the changes which are required to extend wave operator theory to the case of a time dependent hamiltonian such as that which describes the interaction of a laser pulse with an atom or molecule

local limit theorem for the maximum of asymptotically stable random walks let s n n 0 be an asymptotically stable random walk and let m n denote it s maximum in the first n steps we show that the asymptotic behaviour of local probabilities for m n can be approximated by the density of the maximum of the corresponding stable process if and only if the renewal mass function based on ascending ladder heights is regularly varying at infinity we also give some conditions on the random walk which guarantee the desired regularity of the renewal mass function finally we give an example of a random walk for which the local limit theorem for m n does not hold

intersections and limits of regenerative sets a regenerative set m is a random set of real numbers that has a type of markov property not necessarily at arbitrary times like the set of instances following independent exponentially distributed waiting times but only at those times belonging to m early uses of the term regenerative set appear in meyer 1970 and maisonneuve and morando 1970 and earlier tackas 1956 used regeneration point for a member of a certain type of regenerative set as clarification for some readers but possibly prematurely for others i mention that feller 1968 in chapter xiii used the term recurrent event but this term has been largely discarded because a regenerative set is not what is usually called an event hoffman jorgensen 1969 uses the term markov set whereas krylov and yushkevich 1965 use markov random set for a stochastic process related to a regenerative set breiman 1968 and kingman 1972 focus on the indicator function of m using the terms renewal process and regenerative phenomenon respectively we will stick to regenerative set

a numerical approach to copolymers at selective interfaces we consider a model of a random copolymer at a selective interface which undergoes a localization delocalization transition in spite of the several rigorous results available for this model the theoretical characterization of the phase transition has re mained elusive and there is still no agreement about several important issues for example the behavior of the polymer near the phase transition line from a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known in this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram our main results include various numerical observations that suggest that the critical line lies strictly in between the two bounds a rigorous statistical test based on concentration inequalities and super additivity for determining whether a given point of the phase diagram is in the localized phase this is applied in particular to show that with a very low level of error the lower bound does not coincide with the critical line an analysis of the precise asymptotic behavior of the partition function in the de localized phase with particular attention to the effect of rare atypical stretches in the disorder sequence and on whether or not in the delocalized regime the polymer path has a brownian scaling a new proof of the lower bound on the critical line this proof relies on a characteri zation of the localized regime which is more appealing for interpreting the numerical data

stability analysis for semi markovian switched singular stochastic systems abstract the stability problem is investigated for switched singular stochastic systems with semi markovian switching signals considering the inherent mode dependent state jumps at the switching instants sufficient conditions of stochastic stability exponential stability in the mean square and almost surely exponential stability are obtained by using the stochastic analysis theory and lyapunov function method finally an example is provided to illustrate the effectiveness of the proposed results

local behaviour of first passage probabilities suppose that s is an asymptotically stable random walk with norming sequence cn and that tx is the time that s first enters x where x 0 the asymptotic behaviour of p t0 n has been described in a recent paper of vatutin and wachtel probab theory relat fields 143 177 217 2009 and here we build on that result to give three estimates for p tx n which hold uniformly as n in the regions x o cn x o cn and x cn respectively

large deviations for random walks under subexponentiality the big jump domain for a given one dimensional random walk sn with a subexponential step size distribution we present a unifying theory to study the sequences xn for which as n uniformly for x xn we also investigate the stronger local analogue our theory is self contained and fits well within classical results on domains of partial attraction and local limit theory when specialized to the most important subclasses of subexponential distributions that have been studied in the literature we reproduce known theorems and we supplement them with new results

directed polymers in random environment are diffusive at weak disorder in this paper we consider directed polymers in random environment with discrete space and time for transverse dimension at least equal to 3 we prove that diffusivity holds for the path in the full weak disorder region i e where the partition function differs from its annealed value only by a non vanishing factor deep inside this region we also show that the quenched averaged energy has fluctuations of order 1 in complete generality arbitrary dimension and temperature we prove monotonicity of the phase diagram in the temperature

stochastic stability properties of jump linear systems jump linear systems are defined as a family of linear systems with randomly jumping parameters usually governed by a markov jump process and are used to model systems subject to failures or changes in structure the authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties it is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability and a testable necessary and sufficient condition for second moment stability is derived the lyapunov exponent method for the study of almost sure sample stability is discussed and a theorem which characterizes the lyapunov exponents of jump linear systems is presented finally for one dimensional jump linear system it is proved that the region for delta moment stability is monotonically converging to the region for almost sure stability at delta down arrow 0 sup

notes on random walks in the cauchy domain of attraction the goal of these notes is to fill some gaps in the literature about random walks in the cauchy domain of attraction which has been in many cases left aside because of its additional technical difficulties we prove here several results in that case a fuk nagaev inequality and a local version of it a large deviation theorem two types of local large deviation theorems we also derive two important applications of these results a sharp estimate of the tail of the first ladder epochs and renewal theorems extending standard renewal theorems to the case of random walks most of our techniques carry through to the case of random walks in the domain of attraction of an alpha stable law with alpha in 0 2 so we also present results in that case since many of them seem to be missing in the literature

stability of singular stochastic systems with markovian switching in recent years singular hybrid systems have received considerable attention however few results are concerned with stability of singular hybrid stochastic systems based on a stability result for singular hybrid systems this note studies mean square stability of singular hybrid stochastic systems it is found that the diffusions and the markov chain may play important roles in system analysis in mean square sense as a special case a criterion for mean square stability of hybrid stochastic systems is developed by the frequency domain approach

stability analysis of switched linear singular systems this paper addresses the stability analysis problem for switched linear continuous time singular systems first based on the equivalent dynamics decomposition form a refined description for state jumps of the switched singular system is presented which indicates that overall state jumps are resulted by two sequential state jumps second sufficient conditions for exponential stability of the switched singular system with stable subsystems are presented it is shown that the stability property of the system is completely determined by the switched reduced order dynamic subsystem and the switching law induced state jumps then a sufficient stability condition for the switched singular system with both stable and unstable subsystems is obtained finally numerical examples are presented to illustrate the effectiveness of the proposed approach

stochastic stabilization of hybrid differential equations this paper aims to determine whether or not a stochastic state feedback control can stabilize a given linear or nonlinear hybrid system new methods are developed and sufficient conditions on the stability for hybrid stochastic differential equations are provided these results are then used to examine stochastic stabilization by stochastic feedback controls two types of structure controls namely state feedback and output injection are discussed our stabilization criteria are in terms of linear matrix inequalities lmis whence the feedback controls can be designed more easily in practice a couple of examples and computer simulations are worked out to illustrate our theory

a replica coupling approach to disordered pinning models we consider a renewal process 0 1 on the integers where the law of i i 1 has a power like tail p i i 1 n n 1 l n with 0 and l slowly varying we then assign a random n dependent reward penalty to the occurrence of the event that the site n belongs to in such generality this class of problems includes among others 1 d dimensional models of pinning of directed polymers on a one dimensional random defect 1 1 dimensional models of wetting of disordered substrates and the poland scheraga model of dna denaturation by varying the average of the reward the system undergoes a transition from a localized phase where occupies a finite fraction of mathbb n to a delocalized phase where the density of vanishes in absence of disorder i e if the reward is independent of n the transition is of first order for 1 and of higher order for 1 2 11 in physical terms disorder is relevant in this situation in agreement with the heuristic harris criterion on the other hand for 0 1 2 it has been proven recently by k alexander 2 that if disorder is sufficiently weak critical exponents are not modified by randomness disorder is irrelevant in this work generalizing techniques which in the framework of spin glasses are known as replica coupling and interpolation we give a new simpler proof of the main results of 2 moreover we partially justify a small disorder expansion worked out in 9 for 1 2 showing that it provides a free energy upper bound which improves the annealed one

adsorption of a random heteropolymer at a potential well revisited location of transition point and design of sequences the adsorption of an ideal heteropolymer loop at a potential point well is investigated within the framework of standard random matrix theory on the basis of a semi analytical semi numerical approach a histogram of transition points for the ensemble of quenched heteropolymer structures with bimodal symmetric distribution of types of chain links is constructed it is shown that the sequences having transition points in the tail of the histogram display correlations between nearest neighbour monomers

polymer dynamics in the depinned phase metastability with logarithmic barriers we consider the stochastic evolution of a 1 1 dimensional polymer in the depinned regime at equilibrium the system exhibits a double well structure the polymer lies essentially either above or below the repulsive line as a consequence one expects a metastable behavior with rare jumps between the two phases combined with a fast thermalization inside each phase however the energy barrier between these two phases is only logarithmic in the system size l and therefore the two relevant time scales are only polynomial in l with no clear cut separation between them the whole evolution is governed by a subtle competition between the diffusive behavior inside one phase and the jumps across the energy barriers our main results are i a proof that the mixing time of the system lies between l frac 5 2 and l frac 5 2 2 ii the identification of two regions associated with the positive and negative phase of the polymer together with the proof of the asymptotic exponentiality of the tunneling time between them with rate equal to a half of the spectral gap

the set of real numbers left uncovered by random covering intervals random covering intervals are placed on the real line in a poisson manner lebesgue measure governs their random locations and an arbitrary measure governs their random lengths the uncovered set is a regenerative set in the sense of hoffmann jorgensen s generalization of regenerative phenomena introduced by kingman thus as has previously been obtained by mandelbrot it is the closure of the image of a subordinator one that is identified explicitly well known facts about subordinators give shepp s necessary and sufficient condition on for complete coverage and when the coverage is not complete a formula for the hausdorff dimension of the uncovered set the method does not seem to be applicable when the covering is not done in a poisson manner or if the line is replaced by the plane or higher dimensional space

stability analysis of switched singular stochastic linear systems this paper is devoted to study the stability of switched singular stochastic linear systems with both stable and unstable subsystems by using the method of multiple lyapunov functions and the noti

on the localization transition of random copolymers near selective interfaces in this note we consider the de localization transition for random directed 1 1 dimensional copolymers in the proximity of an interface separating selective solvents we derive a rigorous lower bound on the free energy this yields a substantial improvement on the bounds from below on the critical line that were known so far our result implies that the critical curve does not lie below the critical curve set forth by monthus eur phys j b 13 111 130 2000 on the base of a renormalization group analysis we discuss this result in the light of the rigorous and non rigorous approaches present in the literature and by making an analogy with a particular asymptotics of a disordered wetting model we propose a simplified framework in which the question of identifying the critical curve as well as understanding the nature of the transition may be approached

exponential stability analysis for delayed semi markovian recurrent neural networks a homogeneous polynomial approach this paper investigates the exponential stability analysis issue for a class of delayed recurrent neural networks rnns with semi markovian parameters by constructing a stochastic lyapunov functional and using some zoom techniques to estimate its weak infinitesimal operator the exponential mean square stability criteria have been proposed for the markovian neural networks with certain transition probabilities we then generalize the homogeneous polynomial approach for the delayed markovian rnns with uncertain transition probabilities during the stability analysis theoretical results have obtained by introducing an appropriate technique for dealing with a large number of complex homogeneous polynomial matrix inequalities finally numerical examples are provided to demonstrate the effectiveness of the proposed technique

scaling limits of equilibrium wetting models in 1 1 dimension we study the path properties for the pinning wetting model in 1 1 dimension in other terms we consider a random walk model with fairly general continuous increments conditioned to stay in the upper half plane and with a measure reward for touching zero that is the boundary of the forbidden region it is well known that such a model displays a localization delocalization transition according to the size of the reward our focus is on getting a precise pathwise description of the system in both the delocalized phase that includes the critical case and in the localized one from this we extract the brownian scaling limits of the model

stochastic stability and stabilization conditions of semi markovian jump systems with mode transition dependent sojourn time distributions this paper is concerned with the stochastic stability and stabilization problem for a class of continuous time semi markovian jump systems s mjss with mode transition dependent sojourn time distributions in contrast to other approaches this paper offers the possibility to explore more refined probability models on the sojourn time by considering the mode transition dependent sojourn time distributions instead of mode dependent ones here to construct a finite set of linear matrix inequalities lmis from sojourn time dependent matrix inequalities a less conservative relaxation method is proposed in such a way that all available constraints on time varying transition rates can be exploited simulation examples report the effectiveness of the proposed method

copolymers at selective interfaces new bounds on the phase diagram we investigate the phase diagram of disordered copolymers at the interface between two selective solvents and in particular its weak coupling behavior encoded in the slope mc of the critical line at the origin we focus on the directed walk case which has turned out to be in spite of the apparent simplicity extremely challenging in mathematical terms the partition function of such a model does not depend on all the details of the markov chain that models the polymer but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns this observation leads to a natural generalization of the model in terms of arbitrary laws of return times the most interesting case being the one of return times with power law tails with exponent 1 1 2 in the case of the symmetric random walk the main results we present here are

stability analysis and stabilization for nonlinear continuous time descriptor semi markov jump systems this paper investigates the stochastic stability and the state feedback control design for a class of nonlinear continuous time descriptor semi markov jump systems whose transition rates are time varying which are more general than the descriptor markov jump systems first by deriving the infinitesimal generator for stochastic lyapunov functional of descriptor semi markov jump systems a stochastic stability condition is established which guarantees this kind of systems are regular impulse free have a unique solution and are stochastically stable in order to design the state feedback controller a linear matrix inequality lmi stability condition is developed based on the lower and upper bounds of the time varying transition probability and singular value decomposition approach furthermore the state feedback controller design is developed in terms of lmi approach last numerical examples are given to demonstrate the effectiveness of the obtained methods

spitzer s condition and ladder variables in random walks spitzer s condition holds for a random walk if the probabilities n p n 0 converge in cesaro mean to where 0 1 we answer a question which was posed both by spitzer 12 and by emery 5 by showing that whenever this happens it is actually true that n converges to this also enables us to give an improved version of a result in doney and greenwood 4 and show that the random walk is in a domain of attraction without centering if and only if the first ladder epoch and height are in a bivariate domain of attraction

admissibility and stabilization of stochastic singular markovian jump systems with time delays abstract this paper deals with the admissibility analysis and stabilization of stochastic singular markovian jump systems ssmjss with time delays more general conditions for the existence and uniqueness of the impulse free solution to delayed ssmjss are presented consequently by constructing stochastic lyapunov krasovskii functional and applying generalized ito formula new sufficient conditions for the stability of ssmjss and delayed ssmjss are obtained in terms of strict linear matrix inequalities state feedback controller is designed to ensure stabilization of the delayed ssmjss three illustrative examples including an r l c circuit network and an oil catalytic cracking process are employed to verify the effectiveness and usefulness of the obtained results

splitting at backward times in regenerative sets by a backward time is meant a random time which only depends on the future in the same sense as a stopping time only depends on the past we show that backward times taking values in a regenerative set m split m into conditionally independent subsets the conditional distributions of the past may further be identified with the palm distributions p t with respect to the local time random measure xi of m both a e e xi and wherever p t has a continuous version continuity of p t occurs essentially where e xi has a continuous density and the latter continuity set may be described rather precisely in terms of the growth rate and regularity properties of the levy measure of m

directed polymers with random interaction an exactly solvable case we propose a model for two d 1 dimensional directed polymers subjected to a mutual function interaction with a random coupling constant and present an exact renormalization group study for this system the exact function evaluated through an e 1 d expansion for second and third moments of the partition function exhibits the marginal relevance of the disorder at d 1 and the presence of a phase transition from a weak to strong disorder regime for d 1 the length scale exponent for the critical point is 2 e 1 we give details of the renormalization we show that higher moments do not require any new interaction and hence the function remains the same for all moments the method is extended to multicritical systems involving an m chain interaction the corresponding disorder induced phase transition for d d m 1 m 1 has the critical exponent v m 2d m 1 2 1 for both the cases an essential singularity appears for the length scale right at the upper critical dimension d m we also discuss the strainge behavior of an annealed system with more than two chains with pairwise random interactions among each other

non hermitian susy hydrogen like hamiltonians with real spectra it is shown that the radial part of the hydrogen hamiltonian factorizes as the product of two not mutually adjoint first order differential operators plus a complex constant t he 1 susy approach is used to construct non hermitian operators with hydrogen spectra other non hermitian hamiltonians are shown to admit an extra complex energy at n ew self adjoint hydrogen like hamiltonians are also derived by using a 2 susy transformation with complex conjugate pairs

stabilizing randomly switched systems this article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals the switching signal is modeled as a jump stochastic not necessarily markovian process independent of the system state it selects at each instant of time the active subsystem from a family of systems sufficient conditions for stochastic stability almost sure in the mean and in probability of the switched system are established when the subsystems do not possess control inputs and not every subsystem is required to be stable these conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control the analysis is carried out with the aid of multiple lyapunov like functions and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control design

local probabilities for random walks conditioned to stay positive let s0 0 sn n 1 be a random walk generated by a sequence of i i d random variables x1 x2 and let tau rm min n geq 1 s n leq 0 and tau rm min n geq1 s n 0 assuming that the distribution of x1 belongs to the domain of attraction of an stable law we study the asymptotic behavior as n rightarrow infty of the local probabilities bf p tau pm n and prove the gnedenko and stone type conditional local limit theorems for the probabilities bf p s n in x x delta tau n with fixed and x x n in 0 infty

operator renewal theory and mixing rates for dynamical systems with infinite measure we develop a theory of operator renewal sequences in the context of innite ergodic theory for large classes of dynamical systems preserving an innite measure we determine the asymptotic behaviour of iterates l n of the transfer operator this was previously an intractable problem examples of systems covered by our results include i parabolic rational maps of the complex plane and ii not necessarily markovian nonuniformly expanding interval maps with indierent xed points in addition we give a particularly simple proof of pointwise dual ergodicity asymptotic behaviour of p n 1 l j for the class of systems under consideration in certain situations including pomeau manneville intermittency maps we obtain higher order expansions for l n and rates of mixing also we obtain error estimates in the associated dynkin lamperti arcsine laws

hierarchical pinning model with site disorder disorder is marginally relevant we study a hierarchical disordered pinning model with site disorder for which like in the bond disordered case derrida et al in j stat phys 66 1189 1213 1992 and giacomin et al in probab theor rel fields 2009 arxiv 0711 4649 math pr there exists a value of a parameter b which enters in the definition of the hierarchical lattice that separates an irrelevant disorder regime and a relevant disorder regime we show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model the proof goes beyond the technique used in giacomin et al probab theor rel fields 2009 arxiv 0711 4649 math pr and it takes explicitly advantage of the inhomogeneous character of the green function of the model

stability and stabilization for singular switching semi markovian jump systems with generally uncertain transition rates this paper studies the problems of stability and stabilization for a class of singular switching semi markovian jump systems the general transition rates in the semi markov process cover completely unknown and uncertain bounded as two special cases first sufficient conditions are developed to ensure the unforced system to be regular impulse free and exponentially mean square stable then by proposing a state feedback controller sufficient conditions in terms of strict linear matrix inequalities are derived to guarantee the closed loop system to be stochastically stabilziable finally a numerical example is provided to show the effectiveness of the obtained results

darling erdős theorems for normalized sums of i i d variables close to a stable law let 1 be i i d real valued random variables and s n 1 n in the case when the distribution of is close to a stable law for some 0 1 1 2 we investigate the asymptotic behavior in distribution of the maximum of normalized sums max k 1 k 1 s k this completes the darling erdos limit theorem for the case 2