uniform positivity improving property sobolev inequalities and spectral gaps abstract we introduce a certain kind of strong ergodicity condition to study the existence of spectral gap for markov generator we can estimate the spectral gap using the ergodicity condition and a sobolev type inequality we apply our results to the dirichlet form on wiener spaces riemannian manifolds and loop spaces

essential spectral radius for markov semigroups i discrete time case using two new measures of non compactness p and w p for a positive kernel p on a polish space e we obtain a new formula of nussbaum gelfand type for the essential spectral radius ress p on b using that formula we show that different known sufficient conditions for geometric ergodicity such as doeblin s condition drift condition by means of lyapunov function geometric recurrence etc lead to variational formulas of the essential spectral radius all those can be easily transported on the weighted space bu some related results on l2 are also obtained especially in the symmetric case moreover we prove that for a strongly feller and topologically transitive markov kernel the large deviation principle of donsker varadhan for occupation measures of the associated markov process holds if and only if the essential spectral radius is zero this result allows us to show that the sufficient condition of donsker varadhan for the large deviation principle is in fact necessary the knowledge of ress p allows us to estimate eigenvalues of p in l2 in the symmetric case and to estimate the geometric convergence rate by means of that in the metric of wasserstein applications to different concrete models are provided for illustrating those general results

moment lyapunov exponent and stability index for linear conservative system with small random perturbation asymptotic expansion series for the moment lyapunov exponent and stability index are constructed and justified for the two dimensional linear stochastic system close to a harmonic oscillator as an example a one degree of freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained

lyapunov exponents and rotation number of two dimensional systems with telegraphic noise exact expressions for the exponential growth rates lyapunov exponents and rotation numbers of two dimensional systems perturbed by telegraphic noise are derived asymptotic expansions for small large slow and fast noise and the white noise limit are constructed then results are applied to the random oscillator stochastic stability in particular stabilization by noise and stochastic bifurcation

exact results for the lyapunov exponents of certain linear ito systems the lyapunov exponents of certain linear ito systems are considered the approach exploits the relations between the lyapunov exponents and the p moment lyapunov exponents as has been observed and studied by arnold siam j appl math 44 1984 pp 793 802 using the method that was developed in a leizarowitz explicit expressions for the lyapunov exponents of certain markov processes preprint the differential equations that are satisfied by the function t x 0 to e x t x 0 p and its laplace transform are derived in the two dimensional case this study yields explicit expressions for the lyapunov exponents

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

almost sure and pth moment stability and stabilization of regime switching jump diffusion systems this work focuses on regime switching jump diffusions which include three classes of random processes brownian motions poisson processes and markov chains first a scalar linear system is treated as a benchmark model then stabilization of systems with one sided linear growth is considered next nonlinear systems that have a finite explosion time are treated in which regularization explosion suppression and stabilization are achieved by introducing appropriate diffusions together with poisson and markov chain perturbations this work reveals the impact of various random effects on the underlying systems for almost sure and p th moment stability and provides insight on stability and stabilization of switching jump diffusion systems

feynman kac semigroups ground state diffusions and large deviations we study the generalized schrodinger operator l v where l is the generator of a symmetric markov semigroup pt on l2 e m and the corresponding dirichlet form ev by means of the cramer functional v we give necessary and sufficient conditions for ev to be lower bounded and for the feynman kac semigroup pvt to be bounded some sufficient conditions for the essential self adjointness of l v are also given by means of large deviations we find a new condition which ensures the existence of ground state of l v and we construct the ground state process q t whose generator is given in the diffusion case by l l 1 where is the square field operator associated to l the self adjointness of l is discussed as applications we consider perturbation of the semigroups of second quantization on an abstract wiener space the time evolution of euclidean quantum fields and stochastic quantization

lp uniqueness of schrodinger operators and the capacitary positive improving property abstract we prove several lp uniqueness results for schrodinger operators l v by means of the feynman kac formula using the m p capacity theory for general markov semigroups we show that the associated feynman kac semigroup is positive improving in the sense of m p capacity improving the well known one in the sense of measure using that capacitary positive improving property and two new inequalities for generalized ornstein uhlenbeck generators we show the essential self adjointness of the ground state diffusion generator l l 2 associated with two dimensional euclidean quantum fields

fuzzy hypothesis testing for a population proportion based on set valued information abstract in this paper we propose fuzzy hypothesis testing for a proportion with crisp data as the exact generalized one tailed hypergeometric test with fuzzy non complementary hypotheses and incorporate the conventional hypergeometric test with crisp hypotheses into its framework in particular we formulate hypotheses as ontic or epistemic fuzzy sets since we assume that presumptions regarding the true population proportion can be expressed in the conjunctive or disjunctive reading further we suggest modeling of membership functions in terms of cost or frequency due to considerations of user s priorities or incomplete knowledge in relation to hypotheses formulation considering a hypothesis as union of its crisp and fuzzy areas we illustrate via real life examples that in contrast to classical test theory fuzzy hypothesis testing provides an additional partial and gradual consideration of the indifference zone for both complementary and non complementary hypotheses the test concept is introduced for both test methods a test of significance and an alternative test and leads to a crisp test decision considering further cost aspects like the sample size the generalized error criteria are derived and interpreted in relation to both hypotheses additionally a corresponding sensitivity analysis is presented for one of the most promising test types

asymptotics of sample entropy production rate for stochastic differential equations using the dimension free harnack inequality and the integration by parts formula for the associated diffusion semigroup we prove the central limit theorem the moderate deviation principle and the logarithmic iteration law for the sample entropy production rate of a family of stochastic differential equations

moment lyapunov exponents of a two dimensional system under combined harmonic and real noise excitations abstract the moment lyapunov exponents and the lyapunov exponents of a two dimensional system under combined excitations of harmonic and real noise which is modelled as an ornstein uhlenbeck process are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a regular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the real noise excitation on the parametric resonance due to the harmonic excitation is investigated

stochastic dynamics of parametrically excited two d o f systems with symmetry in this paper we investigate the non linear dynamics of a two degree of freedom system with symmetries subject to random parametric excitation the study of this non linear near hamiltonian system is simplified by using the symmetry and separation of scales present in the problem to this end we study the equations as a random perturbation of a four dimensional weakly dissipative hamiltonian system we achieve the model reduction through stochastic averaging and the reduced process is simply a markov process on a line examination of the reduced markov process on the line yields many important results namely probability density functions and stochastic bifurcations the steady state dynamics is computed explicitly phenomenological and dynamical bifurcations are investigated the approach adopted in this paper can in principle be applied to any four dimensional integrable system

correlation free forms for nonlinear stochastic systems abstract an asymptotic method for analysis of nonlinear systems with wide band and high frequency stochastic perturbation is developed correlation free ito limiting equations are derived and a generalization of the wong zakai correction for the drift term is obtained a correction for the diffusion term is also shown to exist several illustrative examples are considered

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

set based representations of conjunctive and disjunctive knowledge abstract we look at the use of sets as a method of representing information we study the use of different set operators to implement different semantic operations careful distinction is made between conjunctive and disjunctive type information

reduction and reconstruction for symmetric ordinary differential equations we discuss the reduction and reconstruction problem for ordinary differential equations that admit a linear symmetry group the goal is to prove that modulo reduction there remain only linear differential equations and to construct these explicitly extending previous work on one parameter groups we show this for certain unipotent and solvable groups and for all semisimple groups some applications to relative equilibria are given

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

moment lyapunov exponents of a two dimensional system under both harmonic and white noise parametric excitations the moment lyapunov exponents and the lyapunov exponents of a 2d system under both harmonic and white noise excitations are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a singular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated

representing uncertainty on set valued variables using belief functions a formalism is proposed for representing uncertain information on set valued variables using the formalism of belief functions a set valued variable x on a domain w is a variable taking zero one or several values in w while defining mass functions on the frame 2 2 w is usually not feasible because of the double exponential complexity involved we propose an approach based on a definition of a restricted family of subsets of 2 w that is closed under intersection and has a lattice structure using recent results about belief functions on lattices we show that most notions from dempster shafer theory can be transposed to that particular lattice making it possible to express rich knowledge about x with only limited additional complexity as compared to the single valued case an application to multi label classification in which each learning instance can belong to several classes simultaneously is demonstrated

lyapunov exponents of hybrid stochastic heat equations in this paper we investigate a class of hybrid stochastic heat equations by explicit formulae of solutions we not only reveal the sample lyapunov exponents but also discuss the pth moment lyapnov exponents moreover several examples are established to demonstrate that unstable deterministic or stochastic dynamical systems can be stabilized by markovian switching

a chaotic hopf bifurcation in coupled maps abstract three dimensional maps with a one dimensional invariant subspace are considered in which the dynamics in the invariant subspace is chaotic such maps arise from three coupled one dimensional maps if the coupling is uni directional and identical then the system has a z3 symmetry and this forces the two normal lyapunov exponents to coincide it is then possible to define a linearised average rotation about the invariant subspace when the normal lyapunov exponents change sign a chaotic hopf bifurcation occurs by considering similar coupled systems but with different coupling strengths the z3 symmetry is lost but there is still an so 2 symmetry on the normal linearisation and so similar results for the lyapunov exponents hold if there is no symmetry on the linearisation either then the multiple lyapunov exponents split although it is still possible to define a linearised average rotation in many cases these three different scenarios are illustrated with numerical examples

on the polynomiality of invariant rings for codimension one ga modules abstract let k be an algebraically closed field of positive characteristic and let g a be its additive group let g a act linearly on a finite dimensional vector space v let k v be the ring of polynomial functions on v and let k v g a be the ring of g a invariants in this article we show that the ring of g a invariants k v g a is a polynomial ring under the assumption that v is a codimension one g a module

spectral gaps for hyperbounded operators abstract we consider a positive and power bounded linear operator t on l p over a finite measure space and prove that if t l p l q for some q p then the essential spectral radius of t is strictly smaller than 1 as a special case we obtain a recent result of miclo who proved this assertion for self adjoint ergodic markov operators in the case p 2 and thereby solved a long open problem of simon and hoegh krohn our methods draw a connection between spectral theory and the geometry of banach spaces they rely on a result going back to groh that encodes spectral gap properties via ultrapowers and on the fact that an infinite dimensional l p space cannot be isomorphic to an l q space for q p we also prove a number of variations of our main result i it follows from theorems of lotz and martinez that the condition t l p l q can be replaced with the weaker assumption that t maps the positive part of the l p unit ball into a uniformly p integrable set ii while it is known that the positivity assumption on t cannot in general be omitted we show that we can replace it with the assumption that t is contractive both on l p and on l q iii we prove a version of the theorem which allows us under appropriate circumstances to also consider non finite measures spaces iv our result also has a uniform version there exists an upper bound c 0 1 for the essential spectral radius of t where c depends on certain quantitative properties of t l p and l q

stabilization of linear systems by noise it is proved that the biggest lyapunov number lambda max of the system dot x a f t x where a is a fixed d times d matrix and f t is a zero mean strictly stationary matrix valued stochastic process satisfies 1 d trace a leqq lambda max on the other hand for each varepsilon 0 there is a process f t for which lambda max leqq 1 d trace a varepsilon in particular the system dot x ax can be stabilized by zero mean stationary parameter noise if and only if trace a 0 the stabilization can be accomplished by a one dimensional noise source the results carry over to the case where a is a stationary process they are also true for f t white noise

reasoning with conjunctive knowledge abstract we carefully distinguish between variables whose value is the disjunction of a set of elements and those whose values are the conjunction of a set of elements we extend the theory of approximate reasoning from its initial setting of reasoning with disjunctive variables to include conjunctive type variables

maximal lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise asymptotic expansions for the exponential growth rate known as the lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise are constructed such systems arise naturally in the investigation of the stability of steady state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems almost sure stability or instability of dynamical systems depends on the sign of the maximal lyapunov exponent stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable the results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging where approximate ito equations in amplitudes and phases are obtained in the sense of weak convergence

perturbation of dirichlet forms lower semiboundedness closability and form cores we study perturbations e e qu of dirichlet forms e on some l2 space l2 m given by quadratic forms q g g d with a signed borel measure whose positive and negative parts are smooth measures with respect to the given dirichlet form we give necessary and sufficient conditions for e to be a lower semibounded closed quadratic form with a unique associated continuous semigroup pt on l2 m we also study the associated resolvent semigroup as well as the relation between e and the largest closed quadratic form which is smaller than e in particular we exhibit form cores

on truth gaps bipolar belief and the assertability of vague propositions this paper proposes an integrated approach to indeterminacy and epistemic uncertainty in order to model an intelligent agent s decision making about the assertability of vague statements initially valuation pairs are introduced as a model of truth gaps for propositional logic sentences these take the form of lower and upper truth valuations representing absolutely true and not absolutely false respectively in particular we consider valuation pairs based on supervaluationist principles and also on kleene s three valued logic the relationship between kleene valuation pairs and supervaluation pairs is then explored in some detail with particular reference to a natural ordering on semantic precision in the second part of the paper we extend this approach by proposing bipolar belief pairs as an integrated model combining epistemic uncertainty and indeterminacy these comprise of lower and upper belief measures on propositional sentences defined by a probability distribution on a finite set of possible valuation pairs the properties of these measures are investigated together with their relationship to different types of uncertainty measure finally we apply bipolar belief measures in a preliminary decision theoretic study so as to begin to understand how the use of vague expressions can help to mitigate the risk associated with making forecasts or promises this then has potential applications to natural language generation systems

stabilization by noise for a class of stochastic reaction diffusion equations we prove uniqueness ergodicity and strongly mixing property of the invariant measure for a class of stochastic reaction diffusion equations with multiplicative noise in which the diffusion term in front of the noise may vanish and the deterministic part of the equation is not necessary asymptotically stable to this purpose we show that the l1 norm of the difference of two solutions starting from any two different initial data converges a s to zero as time goes to infinity

harnack and functional inequalities for generalized mehler semigroups abstract harnack inequalities are established for a class of generalized mehler semigroups which in particular imply upper bound estimates for the transition density moreover poincare and log sobolev inequalities are proved in terms of estimates for the square field operators furthermore under a condition well known in the gaussian case we prove that generalized mehler semigroups are strong feller the results are illustrated by concrete examples in particular we show that a generalized mehler semigroup with an stable part is not hyperbounded but exponentially ergodic and that the log sobolev constant obtained by our method in the special gaussian case can be sharper than the one following from the usual curvature condition moreover a harnack inequality is established for the generalized mehler semigroup associated with the dirichlet heat semigroup on 0 1 we also prove that this semigroup is not hyperbounded

the logarithmic sobolev inequality for continuous spin systems on a lattice abstract in this article we give cf theorem 1 6 a sufficient condition on a local specification for the associated gibbs state to satisfy a logarithmic sobolev inequality in the final section cf theorem 3 10 we relate our condition to the type of mixing condition introduced by dobrushin and shlosman in connection with their work on complete analyticity

harnack inequality and applications for stochastic generalized porous media equations by using coupling and girsanov transformations the dimension free harnack inequality and the strong feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations as applications explicit upper bounds of the l p norm of the density as well as hypercontractivity ultracontractivity and compactness of the corresponding semigroup are derived

dynamic stability analysis of non linear structures with geometrical imperfections under random loading this article presents selected research results in the field of stochastic dynamic stability problems both the geometrical properties and the loading conditions are supposed to be random in nature the stability behavior of structures excited by time dependent loads can be described by the maximum lyapunov exponent this exponent turns positive for unstable systems and can be computed by a non linear time integration with simultaneous stability analysis alternatively an approximation can be obtained by investigating a linearized version of the structural model the non linear time integration of large structures requires a huge numerical effort thus this method is limited by available computer capacities in this article both methods are applied and the respective results are compared for geometrically perfect and imperfect systems of different sizes

markov additive processes i eigenvalue properties and limit theorems on etudie d un point de vue general les proprietes d existence et de regularite des valeurs propres et des fonctions propres de l operateur de feynman kac

large deviations for vector valued functionals of a markov chain lower bounds on obtient des bornes inferieures que l on exprime en termes des parametres de convergence de certains noyaux on donne une application aux mesures empiriques de chaines de markov

almost sure asymptotic stability of a general four dimensional system driven by real noise in the first part of this paper we construct an asymptotic expansion for the maximal lyapunov exponent the exponential growth rate of solutions to a linear stochastic system and the rotation numbers for a general four dimensional dynamical system driven by a small intensity real noise process stability boundaries are obtained provided the natural frequencies are noncommensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue this work is an extension of the work of sri namachchivaya and van roessel and is general in the sense that general stochastic perturbations of nonautonomous systems with two noncommensurable natural frequencies are considered

the novel sufficient conditions of almost sure exponential stability for semi markov jump linear systems abstract this paper discusses the almost sure exponential stability problem for semi markov jump linear systems by comprehensively utilizing the coupled lyapunov matrices and the ergodic property of semi markov switching process the novel sufficient stability conditions for the considered systems are obtained which are expressed in terms of linear matrix inequalities and the probability structure of semi markov switching process finally an example is given to illustrate the effectiveness of our results

the equivalence of the logarithmic sobolev inequality and the dobrushin shlosman mixing condition given a finite range lattice gas with a compact continuous spin space it is shown cf theorem 1 2 that a uniform logarithmic sobolev inequality cf 1 4 holds if and only if the dobrushin shlosman mixing condition cf 1 5 holds as a consequence of our considerations we also show cf theorems 3 2 and 3 6 that these conditions are equivalent to a statement about the uniform rate at which the associated glauber dynamics tends to equilibrium in this same direction we show cf theorem 3 19 that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the glauber dynamics

stochastic stabilization of differential systems with general decay rate some sufficient conditions concerning stability of solutions of stochastic differential evolution equations with general decay rate are first proved then these results are interpreted as suitable stabilization ones for deterministic and stochastic systems also they permit us to construct appropriate linear stabilizers in some particular situations

poincare inequality for weighted first order sobolev spaces on loop spaces abstract let e be the loop space over a compact connected riemannian manifold with a torsion skew symmetric connection let l d be the ornstein uhlenbeck operator on a nonempty connected component d of the loop space e and let v d r be the restriction on d of the potential in the logarithmic sobolev inequality found by l gross on the loop group by s aida and by f z gong and z m ma on the loop space respectively we prove that the schrodinger operator l v l d v always has a spectral gap at the bottom 0 v of its spectrum and thus has its ground state transformed operator 1 l v 0 v where is the unique ground state of l v in particular our result proves l gross s conjecture about the existence of a spectral gap for the ground state transform of the schrodinger operator studied by him on the loop group in addition in all the above cases we identify the domain of the dirichlet forms associated with the ground state transforms as weighted first order sobolev spaces with weight given by 2 thus establishing a poincare inequality for them all these results are consequences from some new results in this paper on dirichlet forms characterizing certain classes with spectral gaps and from results by s aida and m hino

identifying a large deviation rate function assume a sequence of probabilities pn has a large deviation rate function i it is proved that i takes a form analogous to a convex conjugate if i is also assumed convex then i is a convex conjugate of an explicitly defined function ft the results are applied to the empirical law of a markov chain yielding universal bounds on i examples are given of markov chains in which the empirical law has a large deviation rate strictly between the given bounds

quasi compactness and absolutely continuous kernels we show how the essential spectral radius re q of a bounded positive kernel q acting on bounded functions is linked to the lower approximation of q by certain absolutely continuous kernels the standart doeblin s condition can be interpreted in this context and when suitably reformulated it leads to a formula for re q this results may be used to characterize the markov kernels having a quasi compact action on a space of measurable functions bounded with respect to some test function when no irreducibilty and aperiodicity are assumed

turbulence effects on fluidelastic instability of a cylinder in a shear flow abstract fluidelastic instability is the main mechanism that causes severe damages in flow induced vibration experiments show that grid generated turbulence can affect the stability of the system which is modeled as an ornstein uhlenbeck process the dynamic stability of a four dimensional system under real noise excitation is studied through the determination of the p th moment lyapunov exponent and the lyapunov exponent the partial differential eigenvalue equation governing the moment lyapunov exponent is established for small noise amplitudes a method of regular perturbation is used to determine analytical expansions of the moment lyapunov exponents and lyapunov exponents which are shown to be in good agreement with those obtained using numerical approaches the analytical method is applied to study the stability of a cylinder in a shear flow where fluidelastic instability is known to occur the results demonstrate that the unstable system can be stabilized under certain conditions and the effect of stabilization is proportional to the turbulence intensity which agrees with experimental observations

criteria of spectral gap for markov operators let e f be a probability space and let p be a markov operator on l2 with 1 a simple eigenvalue such that p i e is an invariant probability measure of p then p 12 p p has a spectral gap i e 1 is isolated in the spectrum of p if and only if p limr sup f2 1 f pf r 1 this strengthens a conjecture of simon and hoegh krohn on the spectral gap for hyperbounded operators solved recently by l miclo in 10 consequently for a symmetric conservative irreducible dirichlet form on l2 a poincare log sobolev type inequality holds if and only if so does the corresponding defective inequality extensions to sub markov operators and non conservative dirichlet forms are also presented

lyapunov exponent and rotation number of two dimensional linear stochastic systems with small diffusion by means of a singular perturbation scheme and large deviation estimates we expand the lyapunov exponent and rotation number in terms of the small noise intensity varepsilon for all 2 times 2 systems dx t varepsilon ax t varepsilon dt sqrt varepsilon sum nolimits k 1 r b k x t varepsilon circ dw t k for which a is complex diagonalizable examples are given

functional inequalities and spectrum estimates the infinite measure case abstract as a continuation of 13 where a poincare type inequality was introduced to study the essential spectrum on the l 2 space of a probability measure this paper provides a modification of this inequality so that the infimum of the essential spectrum is well described even if the reference measure is infinite high order eigenvalues as well as the corresponding semigroup are estimated by using this new inequality criteria of the inequality and estimates of the inequality constants are presented finally some concrete examples are considered to illustrate the main results in particular estimates of high order eigenvalues obtained in this paper are sharp as checked by two examples on the euclidean space

a constructive approach to a class of ergodic hjb equations with unbounded and nonsmooth cost we consider a class of ergodic hamilton jacobi bellman hjb equations related to long time asymptotics of nonsmooth multiplicative functional of diffusion processes under suitable ergodicity assumptions on the underlying diffusion we show existence of these asymptotics and that they solve the related hjb equation in the viscosity sense

on the almost sure stability condition for a co dimension two bifurcation system under the parametric excitation of a real noise abstract in this paper the almost sure stability condition for a co dimension two bifurcation system on a three dimensional central manifold which is parametrically excited by a real noise is investigated a model of enhanced generality is developed by assuming the real noise as the first component of an output of a linear filter system a zero mean stationary gaussian diffusion vectoral process which conforms to the detailed balance condition the strong mixing condition which is the essential theoretic basis for the stochastic averaging method is removed in the present study to solve the complicated problem encountered in this work the asymptotic analysis approach and the eigenfunction expansion of the solutions to the relevant fokker planck equations are employed in the construction of the asymptotic expansions of the invariant measures and the maximal lyapunov exponents for the relevant system

brief paper almost sure exponential stabilisation of stochastic systems by state feedback control so far a major part of the literature on the stabilisation issues of stochastic systems has been dedicated to mean square stability this paper develops a new class of criteria for designing a controller to stabilise a stochastic system almost surely which is unable to be stabilised in mean square sense the results are expressed in terms of linear matrix inequalities lmis which are easy to be checked in practice by using matlab toolbox moreover the control structure in this paper appears not only in the drift part but also in the diffusion part of the underlying stochastic system

lyapunov exponent and rotation number of two dimensional linear stochastic systems with telegraphic noise in this paper the lyapunov exponent and rotation number of a family of two dimensional systems with general telegraphic noise where the unperturbed system has oscillatory behavior are studied using a martingale approach an iterative computational procedure is derived and analytic expansions in the noise strength are obtained for the lyapunov exponent and the rotation number these expansions are convergent in a nonmarginal way the positivity of the exponent is also established for the random harmonic oscillator with zero mean telegraphic noise and lower and upper bounds are derived for the exponent when the noise is the random telegraph process

instability of the harmonic oscillator with small noise we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is small and is defined by a temporally homogeneous markov process with a finite number of states in the case of two states the telegraph process we obtain additional terms in the expansions affording comparison between the exact values and the asymptotic formulas

on hyperboundedness and spectrum of markov operators consider an ergodic markov operator m reversible with respect to a probability measure mu on a general measurable space it is shown that if m is bounded from mathbb l 2 mu to mathbb l p mu where p 2 then it admits a spectral gap this result answers positively a conjecture raised by hoegh krohn and simon j funct anal 9 121 80 1972 in the more restricted semi group context the proof is based on isoperimetric considerations and especially on cheeger inequalities of higher order for weighted finite graphs recently obtained by lee et al proceedings of the 2012 acm symposium on theory of computing 1131 1140 acm new york 2012 it provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general in addition the usual cheeger inequality is extended to the higher eigenvalues in the compact riemannian setting and the exponential behaviors of the small eigenvalues of witten laplacians at small temperature are recovered

transportation information inequalities for markov processes in this paper one investigates the transportation information tci inequalities tc i for all probability measures on a metric space mathcal x d where is a given probability measure tc is the transportation cost from to with respect to the cost function c x y on mathcal x 2 i is the fisher donsker varadhan information of with respect to and 0 0 is a left continuous increasing function using large deviation techniques it is shown that tci is equivalent to some concentration inequality for the occupation measure of a reversible ergodic markov process related to i the tensorization property of tci and comparisons of tci with poincare and log sobolev inequalities are investigated several easy to check sufficient conditions are provided for special important cases of tci and several examples are worked out

a stochastic model for the evolution of bone metastasis persistence and recovery abstract bone metastasis is a complex disease that modifies the natural dynamics of the bone remodeling process which is vital for maintaining a healthy bone structure nowadays there are deterministic differential models that describe the bone metastasis evolution by considering interaction mechanisms between cancer cells and basic bone cells such as osteoclasts and osteoblasts based on these deterministic models a stochastic differential model is proposed to describe the bone metastasis evolution in order to analyze the effect of environmental fluctuations inclusions conditions for the existence and uniqueness of a nonnegative solution of the proposed stochastic model are given to characterize whether the invasion of cancer persists or is eradicated finally we illustrate different dynamical behaviors of the stochastic solution by means of numerical simulations

functional inequalities on abstract hilbert spaces and applications we study the essential spectrum and the semigroup property for self adjoint operators on abstract hilbert spaces by using functional inequalities some known results obtained on the l 2 space w r t a measure space are generalized the functional inequality is also used to study non symmetric semigroups

upper and lower images of a fuzzy set induced by a fuzzy relation applications to fuzzy inference and diagnosis in fuzzy set theory the image of a fuzzy set induced by a fuzzy relation is usually obtained by a sup t norm composition this corresponds to the upper image which gathers the elements in relation with at least one element of the fuzzy set however the dual point of view leading to the definition of the lower image as the fuzzy set of elements in relation with all the elements of the fuzzy set whose image is computed is not often considered fuzzy sets and fuzzy relations depending on the situations can be interpreted either in a conjunctive manner or as subsets of mutually exclusive possible values for variables whose precise values are ill known disjunctive view the applications of upper and lower images are investigated in both interpretations the generalized modus ponens used in fuzzy rule based systems corresponds to the disjunctive view the interest of upper and lower images is also emphasized for diagnosis problems where the conjunctive interpretation is encountered

hypercontractivity and applications for stochastic hamiltonian systems abstract the hypercontractivity is proved for the markov semigroup associated with a class of stochastic hamiltonian systems on hilbert spaces consequently the markov semigroup converges exponentially to the invariant probability measure in entropy and is compact for large time these strengthen the hypocoercivity results derived in the literature since the log sobolev inequality is invalid we introduce a new argument to prove the hypercontractivity using coupling and dimension free harnack inequality the main results are illustrated by concrete examples of the kinetic fokker planck equation and highly degenerate diffusion processes

an estimate of the gap of spectrum of schrodinger operators which generate hyperbounded semigroups abstract logarithmic sobolev inequalities with potential functions on loop spaces were proved by l gross 1991 j funct anal 102 265 313 s aida 1996 in proceedings of the fifth gregynog symposium stochastic analysis and applications pp 1 19 world scientific singapore and f z gong and z m ma 1998 j funct anal 599 623 the generators of the dirichlet forms are schrodinger operators on loop spaces and they generate hyperbounded semigroups recently f z gong m rockner and l m wu 2000 poincare inequality for weighted first order sobolev spaces on loop spaces submitted proved the existence of the gap of spectrum at the lowest eigenvalue of the schrodinger operators by using m hino s exponential decay estimates of the semigroup 2000 osaka j math 37 603 624 and the author s strong ergodicity property of the semi group s aida 1998 j funct anal 158 152 185 in this paper we will give a lower bound on the gap by using a weak poincare inequality which was introduced by m rockner and f y wang 2000 weak poincare inequalities and l 2 convergence rates of markov semigroups preprint also we will give estimates on the distribution function of ground states using the weak poincare inequality