Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

Correlation of lithofacies and depositional environment using Markov chain analysis in Sambipitu formation at Ngalang river, Gunungkidul, Yogyakarta, Indonesia

The research area is located at Ngalang river, Gedangsari sub-district, Gunungkidul Regency, Special Region of Yogyakarta. The research area is part of southern mountain area which is composed of lithology of Sambipitu calcareous sandstone. The depositional process phase in the Sambipitu formation has a unique pattern and is relevant to the previous lithology. As a result, the stratigraphic position and lithological repetition pattern were reviewed using the statistical method (Markov chain). The aim of this research is to use geostatistics to examine the sedimentation trend in order to predict the existence of rock facies in the Sambipitu Formation. In each unit of lithology cycle, geostatistics is expected to assist, to predict and to interpret the significance of subsequent lithology appearances. The research method used was measured stratigraphy, determination of rock age and depositional environment based on fossil identification. In addition, this research used probability matrix in Markov chain analysis. The results of the Markov chain analysis showed that lithology of rock in the upper Sambipitu formation had a non-random transition pattern. The results of statistical calculation showed that the calculation value was greater than the Chi-square table value (333.9>34.38) that the H0 component was rejected. Lithofacies and depositional environment are correlated to several geological aspects such as distribution of rock facies, source of rock, paleobtahymetri, trace fossils and sedimentation process.

Multimodal dynamics of nonhomogeneous absorbing Markov chains evolving at stochastic transition rates

The Tokuyama–Mori projection operator method for a reduced time-convolutionless description of a local temporal behavior of an open quantum system interacting with the weakly dissipative and fluctuating pervasive environment is applied to a Markov chain subject to random transition probabilities. The solution to the problem of the multimodal dynamics of a two-stage absorbing Markov chain with the fluctuating forward rate constant augmented by a symmetric dichotomous stochastic process is found exactly and compared with that of the problem for the same Markov chain with the fluctuating backward rate constant. It is shown that these two different tetramodal solutions cannot generally be reduced to but be complementary to each other. In the limit of very frequent fluctuations in forward/backward rate constants of a two-stage absorbing Markov chain, as well as in the case of a one-stage recurrent Markov chain, both solutions become bimodal and superimposed to one another. However, there is a distinction between using of those solutions for the dynamics of a two-stage absorbing Markov chain in the limit of very rare fluctuations at the critical point, in which the former solution shows the resonance effect exhibiting itself as the stochastic immobilization in an initial state, while the latter demonstrates the deterministic decay to the other state.

Long-time behavior of random and nonautonomous Fisher-KPP equations. Part II. Transition fronts

In the current series of two papers, we study the long-time behavior of the following random Fisher-KPP equation: [Formula: see text] where [Formula: see text], [Formula: see text] is a given probability space, [Formula: see text] is an ergodic metric dynamical system on [Formula: see text], and [Formula: see text] for every [Formula: see text]. We also study the long-time behavior of the following nonautonomous Fisher-KPP equation: [Formula: see text] where [Formula: see text] is a positive locally Hölder continuous function. In the first part of the series, we studied the stability of positive equilibria and the spreading speeds of (1.1) and (1.2). In this second part of the series, we investigate the existence and stability of transition fronts of (1.1) and (1.2). We first study the transition fronts of (1.1). Under some proper assumption on [Formula: see text], we show the existence of random transition fronts of (1.1) with least mean speed greater than or equal to some constant [Formula: see text] and the nonexistence of random transition fronts of (1.1) with least mean speed less than [Formula: see text]. We prove the stability of random transition fronts of (1.1) with least mean speed greater than [Formula: see text]. Note that it is proved in the first part that [Formula: see text] is the infimum of the spreading speeds of (1.1). We next study the existence and stability of transition fronts of (1.2). It is not assumed that [Formula: see text] and [Formula: see text] are bounded above and below by some positive constants. Many existing results in literature on transition fronts of Fisher-KPP equations have been extended to the general cases considered in the current paper. The current paper also obtains several new results.

Random processes with random transitions between stable states

Introduction: Studying random processes with several stable states and random transitions between them is important because it opens a wide range of practical problems. The detailed information structure is not studied well enough, and there is no unified approach to the description and probabilistic analysis of such processes.Purpose: Studying the main probabilistic characteristics of random processes with two stable states, and probabilistic analysis of control over chaotic transitions under various control actions.Results: We show the ways to represent and preliminarily analyze random processes with two stable states on the phase plane and in the pseudophase space. A general probabilistic model for the processes in question is proposed in the form of a two-component probabilistic «mixture» of distributions. A probabilistic analysis was carried out for the principles of control over random transitions between different states. We have defined the basic probabilistic characteristics for the processes in a management action with a variety of spectral-correlation properties and a changeable threshold for random transitions. The Poisson model of a random transition flow is analyzed with an example of «high» threshold levels.Practical relevance: The methods of visual, qualitative and analytical research in studying dynamic systems with several stable states can be combined. The proposed probabilistic models, regardless of the physical nature of the processes under consideration, can be used in problems of probabilistic analysis, control over probabilistic structure of random transitions, and simulation of physical, technical or biological systems with random switching.

A general mathematical framework for understanding the behavior of heterogeneous stem cell regeneration

Stem cell heterogeneity is essential for the homeostasis in tissue development. This paper established a general formulation for understanding the dynamics of stem cell regeneration with cell heterogeneity and random transitions of epigenetic states. The model generalizes the classical G0 cell cycle model, and incorporates the epigenetic states of stem cells that are represented by a continuous multidimensional variable and the kinetic rates of cell behaviors, including proliferation, differentiation, and apoptosis, that are dependent on their epigenetic states. Moreover, the random transition of epigenetic states is represented by an inheritance probability that can be described as a conditional beta distribution. This model can be extended to investigate gene mutation-induced tumor development. The proposed formula is a generalized formula that helps us to understand various dynamic processes of stem cell regeneration, including tissue development, degeneration, and abnormal growth.