Class Of Markov Processes Research Articles

Estimation with right‐censored observations under a semi‐Markov model

The semi-Markov process often provides a better framework than the classical Markov process for the analysis of events with multiple states. The purpose of this paper is twofold. First, we show that in the presence of right censoring, when the right end-point of the support of the censoring time is strictly less than the right end-point of the support of the semi-Markov kernel, the transition probability of the semi-Markov process is nonidentifiable, and the estimators proposed in the literature are inconsistent in general. We derive the set of all attainable values for the transition probability based on the censored data, and we propose a nonparametric inference procedure for the transition probability using this set. Second, the conventional approach to constructing confidence bands is not applicable for the semi-Markov kernel and the sojourn time distribution. We propose new perturbation resampling methods to construct these confidence bands. Different weights and transformations are explored in the construction. We use simulation to examine our proposals and illustrate them with hospitalization data from a recent cancer survivor study.

Connecting Logistic Probability Models With Basic Dynamic Processes

The logistic distribution is a popular probability model yet it is usually not motivated with reference to the processes under study. While its popularity can be attributed to its simplicity, it can also be derived from basic contextual considerations. Although it has been shown that logistic growth is the limiting form of a class of Markov processes on a lattice, the connections of these microscopic models with statistics are less widely appreciated. We review some history of logistic distributions in classical models of infection, and describe how the apparent density dependence emerges as a consequence of a particular lattice embedding. We then review how logistic growth arises from microscopic random behavior. We also derive a “square-logistic” model from basic considerations. Finally, we describe how the underlying discrete model relates to ordinary logistic regression modeling of time dependence.

Dynamics of a Qubit as a Classical Stochastic Process with Time-Correlated Noise: Minimal Measurement Invasiveness

So far it has been shown that the quantum dynamics cannot be described as a classical Markov process unless the number of classical states is uncountably infinite. In this Letter, we present a stochastic model with time-correlated noise that exactly reproduces any unitary evolution of a qubit and requires just four classical states. The invasive updating of only 1 bit during a measurement accounts for the quantum violation of the Leggett-Garg inequalities. Unlike in a pilot-wave theory, the stochastic forces governing the jumps among the four states do not depend on the quantum state but only on the unitary evolution. This model is used to derive a local hidden variable model, augmented by 1 bit of classical communication, for simulating entangled Bell states.

Exponential ergodicity and regularity for equations with Lévy noise

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α -stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument in [8]. Irreducibility and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite dimensional equations, introduced in [27], with Hölder continuous drift and a general, non-degenerate, symmetric α -stable noise, and infinite dimensional parabolic systems, introduced in [29], with Lipschitz drift and cylindrical α -stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [28], with a different method.

Exact Nonequilibrium Steady State of a Strongly Driven OpenXXZChain

An exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].

Diffusive behavior from a quantum master equation

We study a general class of translation invariant quantum Markov evolutions for a particle on \documentclass[12pt]{minimal}\begin{document}${\mathbb Z}^d$\end{document}Zd. The evolution consists of free flow, interrupted by scattering events. We assume spatial locality of the scattering events and exponentially fast relaxation of the momentum distribution. It is shown that the particle position diffuses in the long time limit. This generalizes standard results about central limit theorems for classical (non-quantum) Markov processes.

Entropy and Irreversibility in Classical and Quantum Mechanics

Review of the irreversibility problem in modern physics with new researches is given. Some characteristics of the Markov chains are specified and the important property of monotonicity of a probability is formulated. Using one thin inequality, the behavior of relative entropy in the classical case is considered. Further we pass to studying of the irreversibility phenomena in quantum problems. By new method is received the Lindblad’s equation and its physical essence is explained. Deep analogy between the classical Markov processes and development described by the Lindblad’s equation is conducted. Using method of comparison of the Lind-blad’s equation with the linear Langevin equation we receive a system of differential equations, which are more general, than the Caldeira-Leggett equation. Here we consider quantum systems without inverse influ-ence on a surrounding background with high temperature. Quantum diffusion of a single particle is consid-ered and possible ways of the permission of the Schrödinger’s cat paradox and the role of an external world for the phenomena with quantum irreversibility are discussed. In spite of previous opinion we conclude that in the equilibrium environment is not necessary to postulate the processes with collapses of wave functions. Besides, we draw attention to the fact that the Heisenberg’s uncertainty relation does not always mean the restriction is usually the product of the average values of commuting variables. At last, some prospects in the problem of quantum irreversibility are discussed.

Multifractional Markov Processes in Heterogeneous Domains

In this article, we introduce a class of Markov processes whose transition probability densities are defined by multifractional pseudodifferential evolution equations on compact domains with variable local dimension. The infinitesimal generators of these Markov processes are given by the trace of strongly elliptic pseudodifferential operators of variable order on such domains. The results derived provide a pseudomultifractal version of some existing special classes of multifractional Markov processes. In particular, pseudostable processes are defined on domains with variable local dimension in this framework. In the case where the local dimension of the domain and the local Hölder exponents of the transition probability densities are constant, the existing results on fractal versions of Lévy processes are recovered.

Quantum walks: Decoherence and coin-flipping games

We investigate the global chirality distribution of the quantum walk on the line when decoherence is introduced either through simultaneous measurements of the chirality and particle position, or as a result of broken links. The first mechanism drives the system towards a classical diffusive behavior. This is used to build new quantum games, similar to the spin-flip game. The second mechanism involves two different possibilities: (a) All the quantum walk links have the same probability of being broken. (b) Only the quantum walk links on a half-line are affected by random breakage. In case (a) the decoherence drives the system to a classical Markov process, whose master equation is equivalent to the dynamical equation of the quantum density matrix. This is not the case in (b) where the asymptotic global chirality distribution unexpectedly maintains some dependence with the initial condition. Explicit analytical equations are obtained for all cases.

The Compositional Construction of Markov Processes

We describe a symmetric monoidal category whose arrows are automata in which the actions have probabilities. The endomorphisms of the identity for the tensor are classical finite Markov processes. The operations of the category permit the compositional description Markov processes. We illustrate by describing a Markov process with 12 n states, which represents a model of the classical Dining Philosopher problem with n dining philosophers, showing how to calculate the probability of reaching deadlock in k steps. A straightforward application of the Perron-Frobenius Theorem yields that this probability tends to 1 as k tends to infinity.

On Markov diffusion processes with delayed reflection from boundaries of a segment

A continuous semi-Markov process with values in a closed interval is considered. This process coincides with a Markov diffusion process inside the interval. Thus, violation of the Markov property is only possible at the boundary of the interval. We prove a sufficient condition under which a semi-Markov process is Markov. We show that, in addition to Markov processes with instantaneous reflection from the boundary of the interval. there exists a class of Markov processes with delayed reflection from the boundary. Such a process has a positive average measure of time at which its trajectory belongs to the boundaries. This gives a different proof of a similar result by Gikhman and Skorokhod of 1968. Bibliography: 5 titles.

Time-Symmetric Quantum Theory of Smoothing

Smoothing is an estimation technique that takes into account both past and future observations and can be more accurate than filtering alone. In this Letter, a quantum theory of smoothing is constructed using a time-symmetric formalism, thereby generalizing prior work on classical and quantum filtering, retrodiction, and smoothing. The proposed theory solves the important problem of optimally estimating classical Markov processes coupled to a quantum system under continuous measurements, and is thus expected to find major applications in future quantum sensing systems, such as gravitational wave detectors and atomic magnetometers.

Power-law-distributed level crossings define fractal behavior

A relationship is established between the autocorrelation function of continuous Gaussian and non-Gaussian stochastic processes and the discrete process that describes their zero or level crossings. Random fractals occur when the distribution for the number of crossings is described by a class of Markov processes whose singlefold statistics are the discrete analog of the Lévy-stable continuous probability densities.

On the possibility of extinction in a class of Markov processes in economics

We analyze the possibility of eventual extinction of a replenishable economic asset (natural resource or capital) whose stocks follow a stationary Markov process with zero as an absorbing state. In particular, the stochastic process of stocks is determined by a given sequence of i.i.d. random variables with bounded support and a positive-valued transition function that maps the current level of the stock and the current realization of the random variable to the next period’s stock. Such processes arise naturally in stochastic dynamic models of economic growth and exploitation of natural resources. Under a minimal set of assumptions, the paper identifies conditions for almost sure extinction from all initial stocks as well as conditions under which the stocks enter every neighborhood of zero infinitely often almost surely. Our results emphasize the crucial role played by the nature of the transition function under the worst realization of the random shock and clarifies the role of the “average” rate of growth in the context of extinction.

Lower Bounds of Mixing Rate for a Class of Markov Processes

Lower bounds of $\alpha$-mixing rate are established for particular models from a class of stationary Markov processes under recurrence conditions. The bounds are compared to previously known upper bounds of $\beta$-mixing rate in cases of polynomial and subexponential convergence. In the latter case, it is shown that the bounds are sharp.

Dual Nonlinear Filters and Entropy Production

This paper makes connections between nonlinear filtering and the entropic properties of Markov processes. It starts by developing information flow models for continuous-time, discrete-state filtering problems, identifying rates of information supply and dissipation. Time reversal yields a dual filtering problem in which these flows are interchanged. The dual problem comprises a diffusion signal with nonlinear dynamics, and observations of the point process variety, but yields a finite-dimensional nonlinear filter. The paper goes on to define an entropic time derivative for a general class of Markov processes and relates the entropic derivatives of the signal and filter to the rates of information supply and dissipation. This leads to the definition of a rate of interactive entropy production, which measures the time asymmetry of the interaction between the signal and filter. This asymmetry is of the same nature as that occurring in the theory of nonequilibrium statistical mechanics based on stochastic dynamics. In this context, the interaction between the signal and filter is nondissipative—a property intimately connected with the existence of a dual problem.

Limiting Densities for a Class of Markov Processes

Limiting Densities for a Class of Markov Processes

Some results about the sample path properties of Markov processes with independent self-similar components

This paper considers a special class of operator self-similar processes Markov processes {X(t),t≥0} with independent self-similar components, that is,X(t)=(X1(t),…,Xd(t)), where {Xi(t),t≥0},i=1,2,…,d ared independent real valued self-similar Markov processes. By means of Borel-Cantelli lemma, we give two results about asymptotic property ast→∞ of sample paths for two special classes of Markov processes with independent self-similar components.

Wavelet Galerkin pricing of American options on Lévy driven assets

The price of an American-style contract on assets driven by a class of Markov processes containing, in particular, Lévy processes of pure jump type with infinite jump activity is expressed as the solution of a parabolic variational integro-differential inequality (PIDI). A Galerkin discretization in logarithmic price using a wavelet basis is presented. Log-linear complexity in each time-step is achieved by wavelet compression of the moment matrix of the price process’ jump measure and by wavelet preconditioning of the large matrix LCPs at each time-step. Efficiency is demonstrated by numerical experiments for pricing American put contracts on various jump-diffusion and pure jump models. Failure of the smooth pasting principle is observed for American put contracts for certain finite variation pure jump price processes.

Les équations de la chaleur et de Poisson pour le laplacien généralisé de Jacobi–Dunkl

We show that the heat equation for the Jacobi–Dunkl operator, has a solution in terms of a semigroup of Markovian operators with strictly positive kernel. This result is used to solve the Poisson equation and to introduce a new class of Markov processes on the real line. To cite this article: F. Chouchene et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).