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In this paper, we consider the stochastic nutrient–phytoplankton–zooplankton model with nutrient cycle. In order to take stochastic fluctuations into account, we add the stochastic increments to the variations of biomass of nutrition, phytoplankton and zooplankton during time interval Δ t , thus we obtain the corresponding stochastic model. Subsequently, we explore the existence, uniqueness and stochastically ultimate boundness of global positive solution. By constructing suitable Lyapunov function, we also obtain V-geometric ergodicity of this model. In addition, the sufficient conditions of exponential extinction and persistence in the mean of plankton are established. At last, we present some numerical simulations to validate theoretical results and analyze the impacts of some important parameters. [ABSTRACT FROM AUTHOR]

In order to obtain functional limit theorems for heavy-tailed stationary processes arising from dynamical systems, one needs to understand the clustering patterns of the tail observations of the process. These patterns are well described by means of a structure called the pilling process introduced recently in the context of dynamical systems. So far, the pilling process has been computed only for observable functions maximised at a single repelling fixed point. Here, we study richer clustering behaviours by considering correlated maximal sets, in the sense that the observable is maximised in multiple points belonging to the same orbit, and we work out explicit expressions for the pilling process when the dynamics is piecewise linear and expanding (1-dimensional and 2-dimensional). [ABSTRACT FROM AUTHOR]

Infectious illnesses like hepatitis place a heavy cost on global health, and precise mathematical models must be created in order to understand and manage them. The Adomian decomposition method (ADM) and an optimal control strategy are utilized to solve a fractional-order hepatitis model in this research. By adding fractional derivatives to account for memory effects and non-integer order dynamics, the fractional-order model expands the conventional compartmental model to take into account the complexity of hepatitis dynamics. The fractional-order hepatitis model is resolved using the ADM, a powerful and effective analytical approach. This approach offers a series solution that converges quickly, enabling the model’s precise analytical solution to be derived. To identify crucial criteria and enhance control mechanisms for the management of hepatitis, an optimal solution strategy is also introduced. The optimization procedure tries to lessen the disease’s spread and its negative effects on public health. We can find the best interventions, immunization schedules, and treatment regimens to effectively reduce the hepatitis pandemic by integrating the ADM solution with an optimization framework. The findings of this study show that the suggested method may be used to solve the fractional-order hepatitis model and optimize control measures. The analytical solution produced by ADM offers important insights into the underlying dynamics of hepatitis transmission, and the optimization process produces suggestions that public health professionals and politicians may put into practice. In the end, this research presents a promising direction for improving disease control efforts in a fractional-order context and contributes to a deeper understanding of hepatitis epidemiology. The importance of this method is that it gives solutions that coincide with that obtained using the numerical approach.

In this paper, the problem of identifying Multiple-Input-Single-Output (MISO) systems with fractional models from noisy input-output available data is studied. The proposed idea is to use Higher-Order-Statistics (HOS), like fourth-order cumulants (foc), instead of noisy measurements. Thus, a fractional fourth-order cumulants based-simplified and refined instrumental variable algorithm (frac-foc-sriv) is first developed. Assuming that all differentiation orders are known a priori, it consists in estimating the linear coefficients of all Single-Input-Single-Output (SISO) sub-models composing the MISO model. Then, the frac-foc-sriv algorithm is combined with a nonlinear optimization technique to estimate all the parameters: coefficients and orders. The performances of the developed algorithms are analyzed using numerical examples. Thanks to fourth-order cumulants, which are insensitive to Gaussian noise, and the iterative strategy of the instrumental variable algorithm, the parameters estimation is consistent. [ABSTRACT FROM AUTHOR]

We study the minimal distance between two orbit segments of length n, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called correlation dimension (or Rényi entropy). We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent. [ABSTRACT FROM AUTHOR]

In this work, we obtain weak order-2 conditions of Runge–Kutta method for the optimal control of stochastic differential equations which occurs in many areas of economics and finance and recently in cognitive sciences and neuroscience. We get the order conditions that a stochastic Runge–Kutta technique must meet to have weak order two by comparing the stochastic expansion of the approximation with the associated Taylor scheme. Moreover, we present numerical examples which verify the theoretical results. We conclude our paper by a summary and an outlook to future research and application. [ABSTRACT FROM AUTHOR]

In this article, a SIRS epidemic model with a general incidence rate is proposed and investigated. We briefly verify the global existence of a unique positive solution for the proposed system. Moreover, and unlike other works, we were able to find the stochastic threshold R s of the proposed model which was used for the discussion of the persistence in mean and extinction of the disease. Moreover, we utilize stochastic Lyapunov functions to show under sufficient conditions the existence and uniqueness of stationary distributions of the solution. Lastly, numerical simulation is executed to conform our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the optimal timing to convert the risk of business for an insurance company in order to improve its solvency. The cash flow of company evolves according to a jump-diffusion process. Business conversion option offers the company an opportunity to transfer the jump risk business out. In exchange for this option, the company needs to pay both fixed and proportional transaction costs. The proportional cost can also be seen as the profit loading of the jump risk business. We formulated this problem as an optimal stopping problem. By solving this stopping problem, we find that the optimal timing of business conversion mainly depends on the profit loading of the jump risk business. A larger profit loading would make the conversion option valueless. The fixed cost, however, only delays the optimal timing of business conversion. In the end, numerical results are provided to illustrate the impacts of transaction costs and environmental parameters to the optimal strategies. [ABSTRACT FROM AUTHOR]

A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in R n with a quadratic, conservative nonlinearity B(x, x) and a linear damping term—Ax which is degenerate in the sense that ker A ≠ ∅ . We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforward if A is full rank, but otherwise, energy could potentially accumulate in ker A and lead to almost-surely unbounded trajectories, making the existence of stationary measures impossible. We give a relatively simple and general sufficient condition based on time-averaged coercivity estimates along trajectories in neighborhoods of ker A and many examples where such estimates can be made. [ABSTRACT FROM AUTHOR]

We discuss the qualitatively new properties of random walks on groups that arise in the situation when the entropy of the step distribution is infinite. [ABSTRACT FROM AUTHOR]

We study the set of invariant idempotent probabilities for place-dependent idempotent iterated function systems defined in compact metric spaces. Using well-known ideas from dynamical systems, such as the Mañé potential and the Aubry set, we provide a complete characterization of the densities of such idempotent probabilities. As an application, we provide an alternative formula for the attractor of a class of fuzzy iterated function systems. [ABSTRACT FROM AUTHOR]

Releasing Wolbachia-infected male mosquitoes into the wild to sterilize wild female mosquitoes is a bio-safe and effective strategy to block the transmission of the dengue virus. Since the spread of the dengue virus is often subject to some randomness, stochastic differential equations can be used to describe the transmission difference of dengue affected by environmental changes. Considering the latent period of virus transmission, we established a delay stochastic differential system model based on the release of Wolbachia-infected males, which has received relatively little attention. By constructing suitable Lyapunov functions, we obtain sufficient conditions for the extinction of the dengue virus and the existence of a unique stationary distribution. To study the efficacy of the releases Wolbachia-infected males, we find two threshold values r 1 ∗ and r 2 ∗ of the release ratio r , where r 1 ∗ > r 2 ∗ . If r > r 1 ∗ , then the dengue virus is extinct. If r < r 2 ∗ , a unique stationary distribution exists, which indicates that the virus is transmitted between humans and mosquitoes. We present some numerical examples to illustrate the impact of environmental noise on the virus dynamics, and we also simulate the dynamics of viruses under different release ratios. [ABSTRACT FROM AUTHOR]

This paper discusses a variational principle on subsets for topological pressure of non-autonomous dynamical systems. Let (X , f 1 , ∞) be a non-autonomous dynamical system and ψ be a continuous potential on X, where (X, d) is a compact metric space and f 1 , ∞ = (f n) n = 1 ∞ is a sequence of continuous maps f n : X → X . We define the Pesin–Pitskel topological pressure P f 1 , ∞ B (Z , ψ) and weighted topological pressure P f 1 , ∞ W (Z , ψ) for any subset Z of X. Also, we define the measure-theoretic pressure P μ , f 1 , ∞ (X , ψ) for any μ ∈ M (X) , where M (X) denotes the set of all Borel probability measures on X. Then, for any nonempty compact subset Z of X, we show the following variational principle for topological pressure P f 1 , ∞ B (Z , ψ) = P f 1 , ∞ W (Z , ψ) = sup { P μ , f 1 , ∞ (X , ψ) : μ ∈ M (X) , μ (Z) = 1 }. Moreover, we show that the Pesin–Pitskel topological pressure and weighted topological pressure can be determined by the measure-theoretic pressure of Borel probability measures. In particular, we have the same results for topological entropy. [ABSTRACT FROM AUTHOR]

Let X 1 , X 2 ,... be the digits in the base-q expansion of a random variable X defined on [0, 1) where q ≥ 2 is an integer. For n = 1 , 2 ,... , we study the probability distribution P n of the (scaled) remainder T n (X) = ∑ k = n + 1 ∞ X k q n - k : If X has an absolutely continuous CDF then P n converges in the total variation metric to the Lebesgue measure μ on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that T N (X) follows μ and is independent of (X 1 ,... , X N) , and second exponentially fast convergence of P n and its PDF f n . We discuss how many digits are needed and show examples of our results. [ABSTRACT FROM AUTHOR]

This article uses hospital capacity to determine the treatment rate for an infectious disease. To examine the impact of random jamming and hospital capacity on the spread of the disease, we propose a stochastic SIR model with nonlinear treatment rate and degenerate diffusion. Our findings demonstrate that the disease’s persistence or eradication depends on the basic reproduction number R 0 s . If R 0 s < 1 , the disease is eradicated with a probability of 1, while R 0 s > 1 results in the disease being almost surely strongly stochastically permanent. We also demonstrate that if R 0 s > 1 , the Markov process has a unique stationary distribution and is exponentially ergodic. Additionally, we identify a critical capacity which determines the minimum hospital capacity required. [ABSTRACT FROM AUTHOR]

In order to detect radionuclides and dangerous substances in a fractured medium with the least amount of time and maximum probability, this paper provides a revolutionary tracking model. It relies on a coordinated tracking strategy using set of programmed nano-sensors. The radionuclide particle has a linear flow from d-directions to spread in a single direction with Random Walk process. We assume that this particle has been spread over a short period of time along phantomic cylindrical path inside the fluid. Two nano-sensors will track or search along each cylinder's route, starting from any location in the fragmented medium (space). A vector of random variables with a known probability distribution makes up the particle's initial position. We add the discounted effort-reward track as a reduction parameter in the distance function. The expected value of the first meeting time between one of the nano-sensors and a Random walk target will be reduced as a result. We extract various supplementary results regarding the optimal tracking strategy in addition to presenting the analytical solution that demonstrates the existence of a finite tracking technique. [ABSTRACT FROM AUTHOR]

This paper concerns with a stochastic version of the discrete-time Mitra–Wan forestry model defined as follows. Consider a system composed by a large number of N trees of the same species, classified according to their ages ranging from 1 to s. At each stage, all trees have a common nonnegative probability of dying (known as the mortality rate). Further, there is a central controller who must decide how many trees to harvest in order to maximize a given reward function. Considering the empirical distribution of the trees over the ages, we introduce a suitable stochastic control model M N to analyze the system. However, due N is too large and the complexity involved in defining an optimal steady policy for long-term behavior, as is typically done in deterministic cases, we appeal to the mean field theory. That is we study the limit as N → ∞ of the model M N . Then, under a suitable law of large numbers we obtain a control model M , the mean field control model, that is deterministic and independent of N, and over which we can obtain a stationary optimal control policy π ∗ under the long-run average criterion. It turns out that π ∗ is one of the so-called normal forest policy, which is completely determined by the mortality rate. Consequently, our goal is to measure the deviation from optimality of π ∗ when it is used to control the original process in M N . [ABSTRACT FROM AUTHOR]

From the mesoscopic point of view, a definition of soft point is introduced by considering the attributes of geometric profile and mass distribution. After that, this concept is used to develop the soft matching technique to simulate the chaotic behaviors of the equations. Especially, a tennis model with deformation factor a(t) is proposed to derive a generalized Newton-Stokes equation v′(t) = λ(vT − a(t)v(t)). Furthermore, a concept of duality of deformation factor a(t) and velocity v(t) with respect to the generalized Newton-Stokes equation is established. To solve this equation, two data-driven models of a(t) are provided, one is based on the concept of soft matching, while the other is by using the amplitude modulation. Finally, the related iterative algorithm is developed to simulate the motion of the falling body via the duality of a(t) and v(t). Numerical examples successfully demonstrate the phenomenon of chaos, which consists of the continual random oscillations and sudden accelerations. Moreover, the algorithm is tested by using larger coefficients corresponding to the terminal velocity and shows more satisfactory results. It may enable us to characterize the total energy of the dynamical system more accurately. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic periodic behavior of a chemostat model with periodic nutrient input. We first prove the existence of global unique positive solution with any initial value for stochastic non-autonomous periodic chemostat system. After that, the sufficient conditions are established for the existence of nontrivial positive T - periodic solution. Moreover, we also analyze the conditions for extinction exponentially of microorganism, and we find that there exists a unique boundary periodic solution for stochastic chemostat model, which is globally attractive. At the same time, in the end of this paper, we also give some numerical simulations to illustrate our main conclusions. [ABSTRACT FROM AUTHOR]

This paper concerns a class of stochastic recursive systems which we treat as stochastic decision models. Problems for optimal control are considered via the limiting path of the system. The existence of such a limiting path is provided by a law of large numbers type theorem. This limiting path is shown to be the integral path of a differential equation closely related to the dynamics of the recursive system. Examples of simulation for convergence are provided. Applications to a simple model of data transmission and a simple model of high frequency trading with optimal strategies are discussed. [ABSTRACT FROM AUTHOR]

Let X(t) be an ergodic stationary random process or an ergodic homogeneous random field on R m , m ≥ 2 , and let M(B) be a mixing homogeneous locally finite random Borel measure with mean density γ on R m , m ≥ 1 . We assume that X and M are independent and possess finite expectations. If { T n } is an increasing sequence of bounded convex sets, containing balls of radii r n → ∞ , then lim n → ∞ 1 λ (T n) ∫ T n X (t) M (d t , w) = γ E [ X (0) ] a.s. and in L 1. Special cases are ergodic theorems with averages over finite random sets. Example: If S is an independent-of-X Poisson random set in R m with mean density γ , then lim n → ∞ 1 λ (T n) ∑ t ∈ S ∩ T n X (t) = γ E [ X (0) ] a.s. and in L 1 (card (S ∩ T n) < ∞ a.s.). These theorems offer a universal way of constructing consistent estimators using observations on finite sets. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of extremal points of C , the family of all bivariate coherent distributions on [ 0 , 1 ] 2 . It is well-known that the set C is convex and weak ∗ compact, and all extreme points of C must be supported on sets of Lebesgue measure zero. Conversely, examples of extreme coherent measures with a finite or countably infinite number of atoms have been successfully constructed in the literature. The main purpose of this article is to bridge the natural gap between those two results: We provide an example of extreme coherent distribution with an uncountable support and with no atoms. Our argument is based on classical tools and ideas from dynamical systems theory. This unexpected connection can be regarded as an independent contribution of the paper. [ABSTRACT FROM AUTHOR]

We consider random products of SL(2,R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense Gδ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrödinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

We consider random products of SL(2,R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense Gδ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrödinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

This paper presents new consistent methods for continuous-time Multiple-Input-Single-Output (MISO) Errors-In-Variables (EIV) systems by fractional models. The proposed idea is to use Higher-Order Statistics (HOS), such as fourth-order cumulants (foc), instead of noisy input and output measurements to obtain unbiased estimates. Firstly, all differentiation orders are assumed to be known a priori and linear coefficients are estimated. The developed estimator is based on minimizing the equation error and it is called fractional fourth-order based-least squares estimator ( f r a c - f o c - l s ). Secondly, the global commensurability of the fractional MISO system is considered. The f r a c - f o c - l s is combined with a non linear technique to estimate the global commensurate order along with linear coefficients. The developed algorithm is based on minimizing the output error and called fractional fourth-order cumulants based-least squares combined with global commensurate order optimization ( f r a c - f o c - g c o o l s ). The consistency of the developed estimators, in presence of high levels of noise corrupting both the input and output measurements, is assessed through a numerical example with the help of Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

We describe the mathematical modelling of the group behaviour of workers in the labor market. The worker receives the salary and seeks to improve his qualifications in order to receive higher wages. The worker enlarges his qualification by the investments in human capital. At a random moment of time, a vacancy appears that provides a jump in the worker's salary. The mathematical model of the worker's behaviour in the labor market is presented as an optimal control problem on an infinite time horizon. The paper presents the derivation of the Kolmogorov–Fokker–Planck equation for the Lévy process, which describes the behaviour of a large amount of workers within a social layer. The numerical solution of the Kolmogorov–Fokker–Planck equation and the calculation results are presented. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish limit laws for volume preserving almost Anosov maps of T 2 flows on 3-three manifolds having a transversally neutral periodic orbit of cubic saddle type. In the process, we derive estimates for the Dulac maps for neutral saddles in planar vector fields. [ABSTRACT FROM AUTHOR]

We consider level-2 large deviations for the one-sided countable full shift without assuming the existence of Bowen’s Gibbs state. To deal with non-compact closed sets, we provide a sufficient condition in terms of inducing which ensures the exponential tightness of a sequence of Borel probability measures constructed from periodic configurations. Under this condition we establish the level-2 Large Deviation Principle. We apply our results to the continued fraction expansion of real numbers in [0, 1) generated by the Rényi map, and obtain the level-2 Large Deviation Principle, as well as a weighted equidistribution of a set of quadratic irrationals to equilibrium states of the Rényi map. [ABSTRACT FROM AUTHOR]

This work concerns with the numerical approximation for the stochastic Lotka–Volterra model originally studied by Mao et al. (Stoch Process Appl 97(1):95–110, 2002). The natures of the model including multi-dimension, super-linearity of both the drift and diffusion coefficients and the positivity of the solution make most of the existing numerical methods fail. In particular, the super-linearity of the diffusion coefficient results in the explosion of the 1st moment of the analytical solution at a finite time. This becomes one of our main technical challenges. As a result, the convergence framework is to be set up under the θ th moment with 0 < θ < 1 . The idea developed in this paper will not only be able to cope with the stochastic Lotka–Volterra model but also work for a large class of multi-dimensional super-linear SDE models. [ABSTRACT FROM AUTHOR]

For a class of irrational numbers, depending on their Diophantine properties, we construct explicit rank-one transformations that are totally ergodic and not weakly mixing and classify when the measure is finite or infinite. In the finite case they are isomorphic to irrational rotations, giving explicit rank-one cutting and spacer parameters for these irrational rotations. In the infinite case we use the constructions to provide examples of non-weakly mixing infinite measure-preserving ergodic transformations which do not have any nontrivial probability preserving factors with discrete spectrum, thereby contributing to the program of Aaronson and Nadkarni and a question of Glasner and Weiss. We also obtain nonsingular versions of these examples for each Krieger ration set. [ABSTRACT FROM AUTHOR]

We show an elementary way to calculate the exact Lyapunov exponent of affine Boole transformations using the interchangeability theorem for differentiation and integration due to Leibniz. [ABSTRACT FROM AUTHOR]

This article consists of a detailed and novel stochastic optimal control analysis of a coupled non-linear dynamical system. The state equations are modelled as an additional food-provided prey–predator system with Holling type III functional response for predator and intra-specific competition among predators. We first discuss the optimal control problem as a Lagrangian problem with a linear quadratic control. Second, we consider an optimal control problem in the time-optimal control setting. We initially establish the existence of optimal controls for both these problems and later characterize these optimal controls using the Stochastic maximum principle. Further numerical simulations are performed based on stochastic forward-backward sweep methods for realizing the theoretical findings. The results obtained in these optimal control problems are discussed in the context of biological conservation and pest management.

In statistical seismology, the Epidemic Type Aftershocks Sequence (ETAS) model is a branching process used world-wide to forecast earthquake intensity rates and reproduce many statistical features observed in seismicity catalogs. In this paper, we describe a fractional differential equation that governs the earthquake intensity rate of the pure temporal ETAS model by using the Caputo fractional derivative and we solve it analytically. We highlight that the tools and special functions of fractional calculus simplify the classical methods employed to obtain the intensity rate and let us describe the change of solution decay for large times. We also apply and discuss the theoretical results to the Japanese catalog in the period 1965-2003. [ABSTRACT FROM AUTHOR]

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein–Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled. [ABSTRACT FROM AUTHOR]

Subshifts of deterministic substitutions are ubiquitous objects in dynamical systems and aperiodic order (the mathematical theory of quasicrystals). Two of their most striking features are that they have low complexity (zero topological entropy) and are uniquely ergodic. Random substitutions are a generalisation of deterministic substitutions where the substituted image of a letter is determined by a Markov process. In stark contrast to their deterministic counterparts, subshifts of random substitutions often have positive topological entropy, and support uncountably many ergodic measures. The underlying Markov process singles out one of the ergodic measures, called the frequency measure. Here, we develop new techniques for computing and studying the entropy of these frequency measures. As an application of our results, we obtain closed form formulas for the entropy of frequency measures for a wide range of random substitution subshifts and show that in many cases there exists a frequency measure of maximal entropy. Further, for a class of random substitution subshifts, we prove that this measure is the unique measure of maximal entropy. These subshifts do not satisfy Bowen's specification property or the weaker specification property of Climenhaga and Thompson and hence provide an interesting new class of intrinsically ergodic subshifts. [ABSTRACT FROM AUTHOR]

A discrete time process, with law μ , is quasi-exchangeable if for any finite permutation σ of time indices, the law μ σ of the resulting process is equivalent to μ . For a quasi-exchangeable stationary process, our main results are (1) if the process is ergodic then it is isomorphic to a Bernoulli process and (2) if the family of all Radon–Nikodym derivatives { d μ σ d μ } is uniformly integrable then the process is a mixture of Bernoulli processes, which generalizes De Finetti's Theorem. We give application of (1) to some determinantal processes. [ABSTRACT FROM AUTHOR]

We introduce the concept of mixed random-quasiperiodic linear cocycles. We characterize the ergodicity of the base dynamics and establish a large deviations type estimate for certain types of observables. For the fiber dynamics we prove the uniform upper semicontinuity of the maximal Lyapunov exponent. This paper is meant to introduce a model to be studied in depth in further projects. [ABSTRACT FROM AUTHOR]

Repairable systems are submitted to corrective maintenance and condition-based preventive maintenance actions. Condition-based preventive maintenance occurs at times which are determined according to the results of inspections and degradation or operation controls. The generalization of the models suggested makes it possible to integrate the dependence between corrective and preventive maintenances. In order to take into account this dependency and the possibility of imperfect maintenances, generalized competing risks models have been presented in Doyen and Gaudoin (2006). In this study, we revise the general case in which the potential times to next corrective and preventive maintenance are independent conditionally to the past of the maintenance process. We address the identifiability issue and we find a result similar to that of Zhou, Lu, Shi and Cheng (2018) for usual competing risks. We propose realistic models with exponential risks and derive their likelihood functions. [ABSTRACT FROM AUTHOR]

The paper relies on the practice process of entrepreneurship education in colleges and universities to carry out research. It summarizes the optimal models suitable for students’ employment and entrepreneurship based on statistical analysis using the probability theory. We built a pyramid structured model of college students’ entrepreneurial ability based on this model. The thesis uses statistical methods to carry out a factor analysis of various abilities. It provides a basis for the adjustment and implementation of employment-oriented entrepreneurship education models in colleges and universities.

In this paper, we introduce a new concept of preinvex functions which is called general h-harmonic preinvex for real-valued stochastic processes. Further, we define the multidimensional general h-harmonic preinvex stochastic processes. We prove the Hermite-Hadamard inequality and obtain some important results for these processes. Some previous results are special cases of the results obtained in this paper. [ABSTRACT FROM AUTHOR]

In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle $S^{1}$ , providing a new result on stochastic circle dynamics even for $C^1$ -diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains. [ABSTRACT FROM AUTHOR]

Let G be a real Lie group, Λ ≤ G a lattice, and Ω = G / Λ . We study the equidistribution properties of the left random walk on Ω induced by a probability measure μ on G. It is assumed that μ has a finite first moment, and that the Zariski closure of the group generated by the support of μ in the adjoint representation is semisimple without compact factors. We show that for every starting point x ∈ Ω , the μ -walk with origin x has no escape of mass, and equidistributes in Cesàro averages toward some homogeneous measure. This extends several fundamental results due to Benoist-Quint and Eskin-Margulis for walks with finite exponential moment. [ABSTRACT FROM AUTHOR]

We obtain Krickeberg mixing for a class of Z -extensions of Gibbs Markov semiflows with roof function and displacement function not in L 2 , where previous methods fail. This is done via a 'smooth tail' estimate for the isomorphic suspension flow. [ABSTRACT FROM AUTHOR]

We study the exit time of two-dimensional dynamical systems perturbed by a small noise that exhibits two peculiar behaviors: (1) The maximum of the probability density function of trajectories is not located at the point attractor. The distance between the maximum and the attractor increases with the noise amplitude σ , as shown by using WKB approximation and numerical simulations. (2) For such systems, exiting from the basin of attraction is not sufficient to guarantee a full escape, due to trajectories that can return several times inside the basin of attraction after crossing the boundary, before eventually escaping far away. We apply these results to study neuronal networks that can generate bursting events. To analyze interburst durations and their statistics, we study the phase space of a mean-field model, based on synaptic short-term changes, that exhibit burst and interburst dynamics. We find that the interburst corresponds to an escape with multiple reentries inside the basin of attraction. To conclude, escaping far away from a basin of attraction is not equivalent to reaching the boundary, thus providing an explanation for non-Poissonian long interburst durations present in neuronal dynamics. [ABSTRACT FROM AUTHOR]