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Viral infections pose a significant risk to public health all around the world. Knowledge of their transmission and epidemiology therefore forms the basis of prevention and control measures. Models are essential components to develop and quantify the characteristics of new pathogens, to evaluate the risks of emerging diseases, and to design strategies to control epidemics. This study investigates the validity of the fractional-order SVEIHR influenza model in predicting seasonal influenza outbreaks in Saudi Arabia using Atangana–Baleanu–Caputo (ABC) operators. The first part of our analysis was examining the positivity, boundedness, and global stability of the disease-free equilibrium of the model. The effective reproduction number of vaccination was calculated, indicating a decrease in the spread of the virus due to vaccination. Additionally, we determined that the reproduction number in the absence of vaccination suggests a higher rate of virus transmission. Therefore, we concluded that a reduction in R0 occurs when vaccination rates increase. Our findings underscore the critical impact of vaccination on controlling the spread of influenza. Then raising coverage of protective vaccines becomes a logical goal for decreasing the basic reproduction number and thus mitigating the epidemic of influenza. We suggest that we extend this model by incorporating additional factors such as antiviral treatments and social behavior changes, which may further influence the dynamics of influenza transmission. [ABSTRACT FROM AUTHOR]

We define a class of not necessarily linear C 0 -semigroups (P t) t ≥ 0 on C b (E) (more generally, on C κ (E) : = 1 κ C b (E) , for some bounded function κ , which is the pointwise limit of a decreasing sequence of continuous functions) equipped with the mixed topology τ 1 M for a large class of topological (not necessarily Polish) state spaces E. If these semigroups are linear, classical theory of operator semigroups on locally convex spaces as well as the theory of bicontinuous semigroups apply to them. In particular, they are infinitesimally generated by their generator (L , D (L)) and thus reconstructable through an Euler formula from their strong derivative at zero in (C b (E) , τ 1 M). In the linear case, we prove that such (P t) t ≥ 0 can be characterized as integral operators given by measure kernels satisfying certain tightness properties. As a consequence, transition semigroups of Markov processes are C 0 -semigroups on (C b (E) , τ 1 M) , if they leave C b (E) invariant and they are jointly weakly continuous in space and time. Hence, they can be reconstructed from their strong derivative at zero and thus have a fully infinitesimal description. This solves an open problem for Markov processes. We show that our results apply to a large number of Markov processes, e.g., those given as the laws of solutions to SDEs and SPDEs, including the stochastic 2D Navier-Stokes equations and the stochastic fast and slow diffusion porous media equations. Furthermore, we introduce the notion of a Markov core operator (L 0 , D (L 0)) for the above generators (L , D (L)) and prove that uniqueness of the Fokker-Planck-Kolmogorov equations corresponding to (L 0 , D (L 0)) for all Dirac initial conditions implies that (L 0 , D (L 0)) is a Markov core operator for (L , D (L)). As a consequence, we identify the Kolmogorov operators of a large number of SDEs on finite and infinite dimensional state spaces as Markov core operators for the infinitesimal generators of the C 0 -semigroups on (C κ (E) , τ κ M) given by their transition semigroups. Furthermore, if each P t is merely convex, we prove that (P t) t ≥ 0 gives rise to viscosity solutions to the Cauchy problem given by its associated (nonlinear) infinitesimal generator. We also show that value functions of optimal control problems, both, in finite and infinite dimensions are particular instances of convex C 0 -semigroups on (C κ (E) , τ κ M). [ABSTRACT FROM AUTHOR]

We give Hoeffding- and Bernstein-type concentration inequalities for the largest eigenvalue of sums of random matrices arising from a Markov chain. We consider time-dependent matrix-valued functions on a general state space, generalizing previous results that had only considered Hoeffding-type inequalities, and only for time-independent functions on a finite state space. In particular, we study a kind of non-commutative moment generating function, provide tight bounds on this object and use a method of Garg et al. (A matrix expander Chernoff bound. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, page 1102–1114, New York, NY, USA, 2018. Association for Computing Machinery) to turn this into tail bounds. Our proof proceeds spectrally, bounding the norm of a certain perturbed operator. In the process we make an interesting connection to dynamical systems and Banach space theory to prove a crucial result on the limiting behaviour of our moment generating function that may be of independent interest. [ABSTRACT FROM AUTHOR]

The aim of this paper is to examine the stability of a class of nonlinear stochastic operators on a finite‐dimensional simplex. We identify a number of properties of the set of the class of stochastic operators, including the stability of each operator. We then use these operators to define non‐homogeneous Markov measures that rely on initial data and carry out an investigation into the absolute continuity and singularity of these measures. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new type of pseudo-differential equations naturally connected with non-archimedean pseudo-differential operators and whose symbols are new classes of negative definite functions in the p-adic context and in arbitrary dimension. These equations are proposed as a mathematical models to study the spreading of infectious diseases (say COVID-19) through a random walk on a complex energy landscape and taking into account social clusters in a situation of extreme social isolation. [ABSTRACT FROM AUTHOR]

Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function f:K→R$f\colon K \rightarrow \mathbb {R}$ defined on a convex domain K⊆Rd$K \subseteq \mathbb {R}^{d}$ and any random variable X$X$ taking values in K$K$, E[f(X)]⩾f(E[X])$\mathbb {E}[f(X)] \geqslant f(\mathbb {E}[X])$. In this paper, sharp upper and lower bounds on E[f(X)]$\mathbb {E}[f(X)]$, termed 'graph convex hull bounds', are derived for arbitrary functions f$f$ on arbitrary domains K$K$, thereby extensively generalizing Jensen's inequality. The derivation of these bounds necessitates the investigation of the convex hull of the graph of f$f$, which can be challenging for complex functions. On the other hand, once these inequalities are established, they hold, just like Jensen's inequality, for anyK$K$‐valued random variable X$X$. Therefore, these bounds are of particular interest in cases where f$f$ is relatively simple and X$X$ is complicated or unknown. Both finite‐ and infinite‐dimensional domains and codomains of f$f$ are covered as well as analogous bounds for conditional expectations and Markov operators. [ABSTRACT FROM AUTHOR]

We study Markov operators T, A, and T * of general Markov chains on an arbitrary measurable space. The operator, T, is defined on the Banach space of all bounded measurable functions. The operator A is defined on the Banach space of all bounded countably additive measures. We construct an operator T * , topologically conjugate to the operator T, acting in the space of all bounded finitely additive measures. We prove the main result of the paper that, in general, a Markov operator T * is quasi-compact if and only if T is quasi-compact. It is proved that the conjugate operator T * is quasi-compact if and only if the Doeblin condition (D) is satisfied. It is shown that the quasi-compactness conditions for all three Markov operators T, A, and T * are equivalent to each other. In addition, we obtain that, for an operator T * to be quasi-compact, it is necessary and sufficient that it does not have invariant purely finitely additive measures. A strong uniform reversible ergodic theorem is proved for the quasi-compact Markov operator T * in the space of finitely additive measures. We give all the proofs for the most general case. A detailed analysis of Lin's counterexample is provided. [ABSTRACT FROM AUTHOR]

In this work, we characterize the fixed points of the doubly stochastic operators defined on discrete spaces $ \ell ^p(I) , $ ℓ p (I) , where I is a nonempty set and $ p\in [1,+\infty). $ p ∈ [ 1 , + ∞). We use our results to characterize all Markov chains with infinite state space and doubly stochastic transition probability matrix that have at least one stationary distribution vector. [ABSTRACT FROM AUTHOR]

We consider transfer operators for topological Markov shift (TMS) with countable states and with holes which are $ 2 $-cylinders. We found that, if the closed system of the shift had an irreducible transition matrix and the potential was a weaker Lipschitz continuous and summable, then we obtained a version of Ruelle-Perron-Frobenius theorem and quasi-compactness of the associated Ruelle transfer operator. The escape rate of the open system was also calculated. We also found that the Ruelle operator of summable potential on topologically transitive TMS has a spectral gap property. As another example, we applied the main results to the transfer operators associated to graph iterated function systems. [ABSTRACT FROM AUTHOR]

The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota's results on the author's research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differential equations. [ABSTRACT FROM AUTHOR]

Starting from the analysis of the lack of positivity of the Cattaneo heat equation, this work addresses the thermodynamic relevance of the positivity constraint in irreversible thermodynamics, that is at least as significant as the entropic constraints. The fulfillment of this condition in hyperbolic models leads to the parametrization of the concentration fields with respect to internal variables associated with the microscopic dynamics. Using Brownian motion theory as a landmark example for deriving macroscopic transport equations from the equations of motion at the particle/molecular level, we discuss two typical problems involving hydrodynamic interactions at the microscale: surface chemical reactions at a solid interface of a diffusing reactant, and mass-balance equations in a complex viscoelastic fluid, in which the physics of the interaction leads either to overcoming the parabolic diffusion model or to considering the parametrization of the concentration with respect to the degrees of freedom associated with the relaxation dynamics of the solvent fluid.

Air pollution is a significant global environmental challenge that can cause health problems affecting everyone on the planet without any geographical boundary. It has a substantial impact directly or indirectly not only on human health but on social and economic activities as well. Researchers worldwide are working to evaluate and predict the air quality index using advanced computational models. These models have drastically altered how we think about and approach API prediction (Air Pollution Index). The primary intent of this paper is to highlight the suitable machine learning models on remotely sensed Sentinel-5P sensor data in assessing and predicting air pollution before, during, and after the lockdown enforced due to the COVID-19 pandemic. This work includes the assessment and prediction of API using four air-polluting parameters- Nitrogen Dioxide, Ozone, Carbon Monoxide, and Sulphur Dioxide in four metropolitan cities of India - Kolkata, Mumbai, Delhi, and Chennai. The paper used Markov Chain as an operator for predicting the AQI state and verified it using ground-level and satellite data. The model's accuracy was estimated using the predicted dataset RMSE (Root Mean Square Error). The outcome of the prediction model was also validated with actual data, which substantiates the finding that during this lockdown period of the COVID-19 pandemic, NO 2 concentration was reduced significantly due to less traffic and energy production from industries. [ABSTRACT FROM AUTHOR]

Starting from the analysis of the lack of positivity of the Cattaneo heat equation, this work addresses the thermodynamic relevance of the positivity constraint in irreversible thermodynamics, that is at least as significant as the entropic constraints. The fulfillment of this condition in hyperbolic models leads to the parametrization of the concentration fields with respect to internal variables associated with the microscopic dynamics. Using Brownian motion theory as a landmark example for deriving macroscopic transport equations from the equations of motion at the particle/molecular level, we discuss two typical problems involving hydrodynamic interactions at the microscale: surface chemical reactions at a solid interface of a diffusing reactant, and mass-balance equations in a complex viscoelastic fluid, in which the physics of the interaction leads either to overcoming the parabolic diffusion model or to considering the parametrization of the concentration with respect to the degrees of freedom associated with the relaxation dynamics of the solvent fluid. [ABSTRACT FROM AUTHOR]

In this paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense. [ABSTRACT FROM AUTHOR]

In this work, we consider a class of centred kernel quadratic stochastic operators. We prove that in this class a centred kernel quadratic stochastic operator convergences almost surely and in L² with an exponential L²-rate to its limit distribution. We propose an approximation scheme for this class of quadratic stochastic operators and describe three algorithms for simulating them. We consider in detail an example where the kernel is a Guassian one. [ABSTRACT FROM AUTHOR]

We prove that the generator of the L 2 implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a Hilbert bimodule, extending earlier results by Cipriani and Sauvageot for tracially symmetric semigroups and the second-named author for GNS-symmetric semigroups. This result hinges on the introduction of a new completely positive map on the algebra of bounded operators on the GNS Hilbert space. This transformation maps symmetric Markov operators to symmetric Markov operators and is essential to obtain the required inner product on the Hilbert bimodule. [ABSTRACT FROM AUTHOR]

We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori--Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori's projection operator, which is a higher-order approximate Koopman learning method. We show that more expressive nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori's projection operator and the most optimal yet computationally infeasible Zwanzig's projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural network--based regressions, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature. [ABSTRACT FROM AUTHOR]

We obtain various new limit theorems for random walks on SL 2 (C) under low moment conditions. For non-elementary measures with a finite second moment we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of É. Le Page. For measures with a finite third moment, we obtain the Local Limit Theorem for the matrix coefficients, improving a recent result of Grama-Quint-Xiao and the authors, and Berry–Esseen bounds with optimal rate O (1 / n) for the norm cocycle and the matrix coefficients. The main tool is a detailed study of the spectral properties of the Markov operator and its purely imaginary perturbations acting on different function spaces. We introduce, in particular, a new function space derived from the Sobolev space W 1 , 2 that provides uniform estimates. [ABSTRACT FROM AUTHOR]

In this paper, we consider general Markov chains with discrete time in an arbitrary measurable (phase) space. Markov chains are given by a classical transition function that generates a pair of conjugate linear Markov operators in a Banach space of measurable bounded functions and in a Banach space of bounded finitely additive measures. We study sequences of Cesaro means of powers of Markov operators on the set of finitely additive probability measures. It is proved that the set of all limit measures (points) of such sequences in the weak topology generated by the preconjugate space is non-empty, weakly compact, and all of them are invariant for this operator. We also show that the well-known Doeblin condition (D) for the ergodicity of a Markov chain is equivalent to condition (∗) : all invariant finitely additive measures of the Markov chain are countably additive, i.e., there are no invariant purely finitely additive measures. We give all the proofs for the most general case. [ABSTRACT FROM AUTHOR]

In this paper, we derive stability margins for optimal and inverse optimal stochastic feedback regulators. Specifically, gain, sector, and disk margin guarantees are obtained for discrete-time nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal controllers that minimise a nonlinear-nonquadratic performance criterion. Furthermore, using the newly developed notion of stochastic dissipativity we derive a return difference inequality to provide connections between stochastic dissipativity and optimality of nonlinear controllers for discrete-time stochastic dynamical systems. In particular, using extended Kalman-Yakubovich-Popov conditions characterising stochastic dissipativity we show that our optimal feedback control law satisfies a return difference inequality predicated on a difference operator of a controlled Markov dispersion process and is stochastically dissipative with respect to a specific quadratic supply rate. [ABSTRACT FROM AUTHOR]

We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in Mazurenko and Zarichnyi (2018) [24]. The main feature developed here is a metrization of the space of idempotent measures using the embedding of the space of idempotent measures to the space of fuzzy sets. The metric obtained induces a topology stronger than the canonical pointwise convergence topology. A key result is the existence of a bijection between idempotent measures and fuzzy sets and a conjugation between the Markov operator of an IFS on idempotent measures and the fuzzy fractal operator of the associated Fuzzy IFS. This allows to prove that the Markov operator for idempotent measures is a contraction w.r.t. the induced metric and, from this, to obtain a convergence theorem and algorithms that draw pictures of invariant measures as grayscale images. [ABSTRACT FROM AUTHOR]

We present and develop different approaches to study the asymptotic behavior of the distribution functions in the odd continued fractions case. Firstly, by considering the transition operator of the Markov chain associated with these expansions on a certain Banach space of complex-valued functions of bounded variation, we make a brief survey of the solution in the Gauss-Kuzmin-type problem. Secondly, we use the method of Szüsz to obtain a similar asymptotic result and to give a good estimate of the convergence rate involved. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of iterates of bounded linear operators (not necessarily positive), acting on Banach spaces, is studied. Through the Dobrushin ergodicity coefficient, we generalize some ergodic theorems obtained earlier for classical Markov semigroups acting on L 1 (or positive operators on abstract state spaces). [ABSTRACT FROM AUTHOR]

The present review paper provides recent results on convexity and its applications to the constrained extension of linear operators, motivated by the existence of subgradients of continuous convex operators, the Markov moment problem and related Markov operators, approximation using the Krein–Milman theorem, related optimization, and polynomial approximation on unbounded subsets. In many cases, the Mazur–Orlicz theorem also leads to Markov operators as solutions. The common point of all these results is the Hahn–Banach theorem and its consequences, supplied by specific results in polynomial approximation. All these theorems or their proofs essentially involve the notion of convexity. [ABSTRACT FROM AUTHOR]

This article is devoted to studying the non-commutative Poisson boundary associated with \Big (B\big (\mathcal {F}(\mathcal {H})\big), P_{\omega }\Big) where \mathcal {H} is a separable Hilbert space (finite or infinite-dimensional), \dim \mathcal {H} > 1, with an orthonormal basis \mathcal {E}, B\big (\mathcal {F}(\mathcal {H})\big) is the algebra of bounded linear operators on the full Fock space \mathcal {F}(\mathcal {H}) defined over \mathcal {H}, \omega = \{\omega _e: e \in \mathcal {E} \} is a sequence of strictly positive real numbers such that \sum _e \omega _e = 1 and P_{\omega } is the Markov operator on B\big (\mathcal {F}(\mathcal {H})\big) defined by \begin{align*} P_{\omega }(x) = \sum _{e \in \mathcal {E}} \omega _e l_e^* x l_e, \ x \in B\big (\mathcal {F}(\mathcal {H})\big), \end{align*} where, for e \in \mathcal {E}, l_e denotes the left creation operator associated with e. We observe that the non-commutative Poisson boundary associated with \Big (B\big (\mathcal {F}(\mathcal {H})\big), P_{\omega }\Big) is \sigma-weak closure of the Cuntz algebra \mathcal {O}_{\dim \mathcal {H}} generated by the right creation operators. We prove that the Poisson boundary is an injective factor of type III for any choice of \omega. Moreover, if \mathcal {H} is finite-dimensional, we completely classify the Poisson boundary in terms of its Connes' S invariant and curiously they are type III _{\lambda } factors with \lambda belonging to a certain small class of algebraic numbers. [ABSTRACT FROM AUTHOR]

In this study, we consider general Markov chains (MC) defined by a transition probability (kernel) that is finitely additive. These Markov chains were constructed by S. Ramakrishnan within the concepts and symbolism of game theory. Here, we study these MCs by using the operator approach. In our work, the state space (phase space) of the MC has any cardinality and the sigma-algebra is discrete. The construction of a phase space allows us to decompose the Markov kernel (and the Markov operators that it generates) into the sum of two components: countably additive and purely finitely additive kernels. We show that the countably additive kernel is atomic. Some properties of Markov operators with a purely finitely additive kernel and their invariant measures are also studied. A class of combined finitely additive MC and two of its subclasses are introduced, and the properties of their invariant measures are proven. Some asymptotic regularities of such MCs were revealed. [ABSTRACT FROM AUTHOR]

We consider a random billiard map, the one in which the standard specular reflection rule is replaced by a random reflection given by a Markov operator. We exhibit an invariant measure for random billiards on general tables. In the special case of a circular table we show that almost every (random) orbit is dense in the boundary as well as in the circular ring formed between the circle boundary and the random caustic. We additionally prove Strong Knudsen's Law for a particular case of families of absolutely continuous measures with respect to Liouville. We also calculate the Lyapunov exponent for the random billiard map in the circle. [ABSTRACT FROM AUTHOR]

We work out the theory of fractional isomorphism of graphons as a generalization to the classical theory of fractional isomorphism of finite graphs. The generalization is given in terms of homomorphism densities of finite trees and it is characterized in terms of distributions on iterated degree measures, Markov operators, weak isomorphism of a conditional expectation with respect to invariant sub-σ-algebras and isomorphism of certain quotients of given graphons. [ABSTRACT FROM AUTHOR]

The central limit theorem and law of the iterated logarithm for Markov chains corresponding to random walks on the space Homeo( S 1 ) of circle homeomorphisms for centered Lipschitz functions and every starting point are proved. [ABSTRACT FROM AUTHOR]

This paper is intended to establish the connection between stochastic dynamical systems with non-Gaussian noises and stochastic dynamical systems with Gaussian noises, by considering the convergence behavior of invariant probability measures for stochastic real Ginzburg-Landau equation driven by cylindrical α -stable Lévy process, in the limit α → 2. Indeed, we prove that the invariant measure of stochastic real Ginzburg-Landau equation on torus and driven by cylindrical α -stable Lévy processes converges to the invariant measure of stochastic real Ginzburg-Landau equation forced by cylindrical Brownian motions under the Wasserstein distance, as α tends to 2. We state our strategy as below. First, by using an abstract result given by Hairer and Mattingly, we prove that a type of Wasserstein distance is contracting for the dual operator of Markov semigroup associated with the limit equation which forced by Brownian motions. Then, through server priori uniform moment estimates and a convergence result on stochastic convolutions of cylindrical subordinated Brownian motions, we establish a strong convergence result on the solution of stochastic real Ginzburg-Landau equation driven by cylindrical subordinated Brownian motions conditioned on the initial data is distributed as an invariant measure, in the limit α → 2. Finally, by the Monge-Kantorovich duality, we prove that the invariant probability measure of stochastic real Ginzburg-Landau equation driven by cylindrical α -stable Lévy processes converges to the invariant measure of stochastic real Ginzburg-Landau equation forced by Brownian motions under the Wasserstein distance. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global optimality and convergence rate of an off-policy actor critic algorithm in the tabular setting without using density ratio to correct the discrepancy between the state distribution of the behavior policy and that of the target policy. Our work goes beyond existing works on the optimality of policy gradient methods in that existing works use the exact policy gradient for updating the policy parameters while we use an approximate and stochastic update step. Our update step is not a gradient update because we do not use a density ratio to correct the state distribution, which aligns well with what practitioners do. Our update is approximate because we use a learned critic instead of the true value function. Our update is stochastic because at each step the update is done for only the current state action pair. Moreover, we remove several restrictive assumptions from existing works in our analysis. Central to our work is the finite sample analysis of a generic stochastic approximation algorithm with time-inhomogeneous update operators on time-inhomogeneous Markov chains, based on its uniform contraction properties. [ABSTRACT FROM AUTHOR]

We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and obtain several asymptotic results as well as finite sample error bounds. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition. [ABSTRACT FROM AUTHOR]

We consider generalized definitions of mixing and exactness for random dynamical systems in terms of Markov operator cocycles. We first give six fundamental definitions of mixing for Markov operator cocycles in view of observations of the randomness in environments, and reduce them into two different groups. Secondly, we give the definition of exactness for Markov operator cocycles and show that Lin’s criterion for exactness can be naturally extended to the case of Markov operator cocycles. Finally, in the class of asymptotically periodic Markov operator cocycles, we show the Lasotaâ€"Mackey type equivalence between mixing, exactness and asymptotic stability. [ABSTRACT FROM AUTHOR]

Main goal of this paper is to formulate possibly simple and easy to verify criteria on existence of the unique attracting probability measure for stochastic process induced by generalized iterated function systems with probabilities (GIFSPs). To do this, we study the long-time behavior of trajectories of Markov-type operators acting on product of spaces of Borel measures on arbitrary Polish space. Precisely, we get the desired geometric rate of convergence of sequences of measures under the action of such operator to the unique distribution in the Hutchinson–Wasserstein distance. We apply the obtained results to study limiting behavior of random trajectories of GIFSPs as well as stochastic difference equations with multiple delays. [ABSTRACT FROM AUTHOR]

The iterates of Markov operators are studied in Ergodic Theory as well as in Approximation Theory. We investigate the limit of S m (m → ∞) for Markov operators S of a special form occurring often in Approximation Theory. In relation with the iterates, the fixed points and the invariant probability measures of S are determined. Several examples illustrate the general results. Two classical operators for which the iterates do not converge are also presented. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general subordinators or inverses to general subordinators. Our analysis is based on Bernstein symbols and Dirichlet forms, where the symbols characterize the time changes, and the Dirichlet forms characterize the Markov processes. [ABSTRACT FROM AUTHOR]

The iterates of an operator and their limit are studied in ergodic theory, approximation theory and other related fields. In this paper we continue the study of iterates of certain Markov operators acting on C[0,1]. We provide new sufficient conditions under which a Markov operator L is uniquely ergodic, i.e., it admits a unique invariant probability measure ν . The determination of ν is reduced to solving an algebraic system of linear equations. Then ν is used in order to express the limit of the iterates of L. Another useful tool in this study is the eigenstructure of L. The general results are applied to several families of classical Markov operators. [ABSTRACT FROM AUTHOR]

In the present paper, we deal with asymptotical stability of Markov operators acting on abstract state spaces (i.e. an ordered Banach space, where the norm has an additivity property on the cone of positive elements). Basically, we are interested in the rate of convergence when a Markov operator T satisfies the uniform P-ergodicity, i.e. |Tn − P| → 0 , here P is a projection. We have showed that T is uniformly P-ergodic if and only if |Tn − P| ≤ Cβn, 0. In this paper, we prove that such a β is characterized by the spectral radius of T − P. Moreover, we give Deoblin's kind of conditions for the uniform P-ergodicity of Markov operators. [ABSTRACT FROM AUTHOR]

We present a Markov-chain analysis of blockwise-stochastic algorithms for solving partially block-separable optimization problems. Our main contributions to the extensive literature on these methods are statements about the Markov operators and distributions behind the iterates of stochastic algorithms, and in particular the regularity of Markov operators and rates of convergence of the distributions of the corresponding Markov chains. This provides a detailed characterization of the moments of the sequences beyond just the expected behavior. This also serves as a case study of how randomization restores favorable properties to algorithms that iterations of only partial information destroys. We demonstrate this on stochastic blockwise implementations of the forward–backward and Douglas–Rachford algorithms for nonconvex (and, as a special case, convex), nonsmooth optimization. [ABSTRACT FROM AUTHOR]

Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them--as filtered through an observable of the system--is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap--defined as the distance between the subdominant RP resonance and the unit circle--plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño-Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.

In the past decade, many Bayesian shrinkage models have been developed for linear regression problems where the number of covariates, p, is large. Computation of the intractable posterior is often done with three-block Gibbs samplers (3BG), based on representing the shrinkage priors as scale mixtures of Normal distributions. An alternative computing tool is a state of the art Hamiltonian Monte Carlo (HMC) method, which can be easily implemented in the Stan software. However, we found both existing methods to be inefficient and often impractical for large p problems. Following the general idea of Rajaratnam et al., we propose two-block Gibbs samplers (2BG) for three commonly used shrinkage models, namely, the Bayesian group lasso, the Bayesian sparse group lasso, and the Bayesian fused lasso models. We demonstrate with simulated and real data examples that the Markov chains underlying 2BG's converge much faster than that of 3BG's, and no worse than that of HMC. At the same time, the computing costs of 2BG's per iteration are as low as that of 3BG's, and can be several orders of magnitude lower than that of HMC. As a result, the newly proposed 2BG is the only practical computing solution to do Bayesian shrinkage analysis for datasets with large p. Further, we provide theoretical justifications for the superior performance of 2BG's. We establish geometric ergodicity of Markov chains associated with the 2BG for each of the three Bayesian shrinkage models. We also prove, for most cases of the Bayesian group lasso and the Bayesian sparse group lasso model, the Markov operators for the 2BG chains are trace-class. Whereas for all cases of all three Bayesian shrinkage models, the Markov operator for the 3BG chains is not even Hilbert–Schmidt. for this article are available online. [ABSTRACT FROM AUTHOR]

We develop an analytical framework and numerical approach to obtain the coefficient of self-diffusivity for the transport of a rarefied gas in channels in the limit of large Knudsen number. This framework provides a method for determining the influence of channel surface microstructure on the value of diffusivity that is particularly effective when the microstructure exhibits relatively low roughness. This method is based on the observation that the Markov transition (scattering) operator determined by the microstructure, under the condition of weak surface scattering, has a universal form given, up to a multiplicative constant, by the classical Legendre differential operator. We also show how characteristic numbers of the system--namely, geometric parameters of the microstructure, the spectral gap of a Markov operator, and the tangential momentum accommodation coefficient of a commonly used model of surface scattering--are all related. Examples of microstructures are investigated to illustrate the relation of these quantities numerically and analytically. [ABSTRACT FROM AUTHOR]

We are concerned with a family of Markov operators and some modifications of them. Differences involving such operators are studied. We compare their Voronovskaya formulas and determine the limits of their iterates. [ABSTRACT FROM AUTHOR]

This paper focuses on the state estimation problem for a type of coupled neural networks with multiple time delays and markovian jumping communication topologies. To avoid unnecessary resources consuming, a novel state estimator is designed based on event-triggered mechanism, in which the control input of each node is only updated when the measurement output error exceeds a predefined threshold. The event-triggering time sequence is a subset of the switching time sequence, which can naturally excludes the Zeno-behavior. By utilizing an appropriate Lyapunov-Krasovskii functional, as well as the weak infinitesimal operator of Markov process and some algebraic inequalities, an easy-to-check sufficient criterion is derived to ensure the exponential ultimate boundedness of the estimation error. Finally, a simulation example is presented to illustrate the applications and effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In the present paper, we study infinite dimensional orthogonality preserving the second-order nonlinear Markov operators. It is proved that subjectivity of the second-order nonlinear Markov operators is equivalent to the orthogonality preserves in the class of π-Volterra operators. Moreover, a full description of such kind of operators has been found in terms of heredity coefficients. Besides, we are able to represent these operators their canonical forms. Furthermore, some properties of orthogonality preserving the second-order nonlinear operators and their fixed points are studied. [ABSTRACT FROM AUTHOR]

We specify the structure of completely positive operators and quantum Markov semigroup generators that are symmetric with respect to a family of inner products, also providing new information on the order structure and extreme points in some previously studied cases. [ABSTRACT FROM AUTHOR]

Normally, one thinks of probabilistic transition systems as taking an initial probability distribution over the state space into a new probability distribution representing the system after a transition. We, however, take a dual view of Markov processes as transformers of bounded measurable functions. This is very much in the same spirit as a "predicate-transformer" view, which is dual to the state-transformer view of transition systems. We redevelop the theory of labelled Markov processes from this viewpoint; in particular, we explore approximation theory. We obtain three main results. (i) It is possible to define bisimulation on general measure spaces and show that it is an equivalence relation. The logical characterization of bisimulation can be done straightforwardly and generally. (ii) A new and flexible approach to approximation based on averaging can be given. This vastly generalizes and streamlines the idea of using conditional expectations to compute approximations. (iii) We show that there is a minimal process bisimulation-equivalent to a given process, and this minimal process is obtained as the limit of the finite approximants. [ABSTRACT FROM AUTHOR]