On a class of Markov processes

In this paper a definition of partial ordering of classes of one-dimensional Markov processes is given. Theorem 1 gives the necessary and sufficient condition for the existence of the partial order relationship between two given classes of one-dimensional Markov processes. Theorem 2 gives the sufficient condition for the existence of the partial order relationship between two given classes of Markov processes with a discrete parameter.

In this article a class of Markov processes on the ring of finite adeles of the rational numbers are introduced. A class of non-Archimedean metrics on \(\mathbb{A}_{f}\) are chosen in order to describe this ring as a general polyadic ring and to introduce a family of pseudodifferential operators and parabolic-type equations on \({L^2}(\mathbb{A}_{f})\). The fundamental solutions of these parabolic equations determine transition functions of time and space homogeneous Markov processes on \(\mathbb{A}_{f}\) which are invariant under multiplication by units. Considering the infinite place ℝ, we extend these results to the complete ring of adeles.

In a probabi!istic treatment of semi-linear parabolic equations key steps were to construct a certain class of Markov processes on a big state space and to prove the branching property of semi-groups of the processes (cf. ikeda-NagasawaWatanabe [1]). In this paper branching property will be characterized in terms of multiDlicativity (or decomposability) of semigroup of a Markov process on a multipliqat~ve state space (cf. § I for definitions). This characterization will then be applied to branching Markov processes with age and sign in § 2~ and to a branching Markov process treated recently by Sirao [7] in § 3.

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

A class of Markov processes in continuous time, with local transition rules, acting on colourings of a lattice, is defined. An algorithm is described for dynamic simulation of such processes. The computation time for the next state is O(logb), where b is the number of possible next states. This technique is used to give some evidence that the limiting shape for a random growth process in the plane with exponential distribution is approximately a circle.

Stochastic processes of temporary equilibria

Let {λn,μn} be a set of birth-death parameters, A the associated generator (1.4.2) and {X(t): 0 ≤ t < ∞} a birth-death process with generator A. In this and the following chapters we shall be concerned with natural birth-death processes only, i.e., A is assumed to satisfy the conditions C(A) and D(A). The state -1 will be disregarded and the term transition matrix will be used for the matrix P(⋅) = (pij(⋅)), where i, j = 0, 1,....... Since the properties to be discussed in this chapter are independent of the initial distribution of the process, we shall often identify the birth-death process {X(t)} with its transition matrix P(.). KARLIN and McGREGOR (1957a, 1959b) have proved the following important feature of the transition matrix P(⋅) of a natural birth-death process: (2.1.1) P(t) is strictly totally positive (STP) for t > 0, which means that every subdeterminant of P(t) is strictly positive for t > 0. KARLIN and McGREGOR (1959a) showed that property (2.1.1) is characteristic for birth-death processes in the class of Markov processes as defined in section 1.1. An immediate conclusion from (2.1.1) is (2.1.2) pij(t) > 0 i, j = 0, 1,...; t > 0.KeywordsTransition MatrixSpectral FunctionMoment ProblemConjugate PointClassical PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider an open quantum system, with dissipation applied only to a part of its degrees of freedom, evolving via a quantum Markov dynamics. We demonstrate that, in the Zeno regime of large dissipation, the relaxation of the quantum system towards a pure quantum state is linked to the evolution of a classical Markov process towards a single absorbing state. The rates of the associated classical Markov process are determined by the original quantum dynamics. Extension of this correspondence to absorbing states with internal structure allows us to establish a general criterion for having a Zeno-limit nonequilibrium stationary state of arbitrary finite rank. An application of this criterion is illustrated in the case of an open XXZ spin-1/2 chain dissipatively coupled at its edges to baths with fixed and different polarizations. For this system, we find exact nonequilibrium steady-state solutions of ranks 1 and 2.

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.

In this paper we consider a general class of Markov processes with discrete time parameter {X n : n = 0, 1, 2, …}, where the transitions from one state to another are determined by: The functions h j (·) are strictly increasing, and in general the Markov processes have dependent increments. In section 1 the model is described more precisely. Considering the sequence of distribution functions F n(x) = Pr[X n ≤ x] (n = 0, 1, 2, ….) a numerical method is given by which upper and lower bounds can be determined for F n(x) (n ≥ n o). Numerical results are given for some special cases, and the determination of an alarm level in a flame failure control system is described as an application. The results show that the upper and lower bounds can lie very close together.

The additive-increase multiplicative-decrease (AIMD) schemes designed to control congestion in communication networks are investigated from a probabilistic point of view. Functional limit theorems for a general class of Markov processes that describe these algorithms are obtained. The asymptotic behaviour of the corresponding invariant measures is described in terms of the limiting Markov processes. For some special important cases, including TCP congestion avoidance, an important autoregressive property is proved. As a consequence, the explicit expression of the related invariant probabilities is derived. The transient behaviour of these algorithms is also analysed.

The problem considered is non-linear filtering of Gaussian observations of a Markov jump-diffusion with an embedded Markov chain, that is described by stochastic differential equations driven by Brownian motions and a random Poisson measure. The modelling potential of this class of Markov processes is illustrated by some simple realistic examples.

We study some regularity properties in locally stationary Markov models which are fundamental for controlling the bias of nonparametric kernel estimators. In particular, we provide an alternative to the standard notion of derivative process developed in the literature and that can be used for studying a wide class of Markov processes. To this end, for some families of $V$-geometrically ergodic Markov kernels indexed by a real parameter $u$, we give conditions under which the invariant probability distribution is differentiable with respect to $u$, in the sense of signed measures. Our results also complete the existing literature for the perturbation analysis of Markov chains, in particular when exponential moments are not finite. Our conditions are checked on several original examples of locally stationary processes such as integer-valued autoregressive processes, categorical time series or threshold autoregressive processes.

Csaki et al.(5) have given strong approximations of continuous additive functional of Brownian motion. We establish here an extension of these results for a large class of Markov processes.

A quantum logic gate of particular interest to both electrical engineers and game theorists is the quantum multiplexer. This shared interest is due to the facts that an arbitrary quantum logic gate may be expressed, up to arbitrary accuracy, via a circuit consisting entirely of variations of the quantum multiplexer, and that certain one player games, the history dependent Parrondo games, can be quantized as games via a particular variation of the quantum multiplexer. However, to date all such quantizations have lacked a certain fundamental game theoretic property. The main result in this dissertation is the development of quantizations of history dependent quantum Parrondo games that satisfy this fundamental game theoretic property. Our approach also yeilds fresh insight as to what should be considered as the proper quantum analogue of a classical Markov process and gives the first game theoretic measures of multiplexer behavior.