Academic literature on the topic 'Chapman-Kolmogorov equation'

The Chapman–Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Fractional integrals can be used to describe the fractal media. Using fractional integrals, the fractional generalization of the Chapman–Kolmogorov equation is obtained. From the fractional Chapman–Kolmogorov equation, the Fokker–Planck equation is derived.

Nonlinear Markov processes are known to arise from diffusion approximations of nonlinear master equations and generalized Boltzmann equations and have been studied in their own merit in the context of nonlinear Fokker–Planck equations. Nonlinear Markov processes can account for collective phenomena, collective oscillations, and collective chaotic behavior. Despite the importance of nonlinear Markov processes for the physics of stochastic processes, relatively little is known about the Chapman–Kolmogorov equation of nonlinear Markov processes. The manuscript derives the exact solution of the Chapman–Kolmogorov equation for a recently proposed deterministic Markov model. Numerical solutions of the Chapman–Kolmogorov equation are used to demonstrate the nonlinear semi-group property of conditional probabilities of nonlinear Markov processes.

Uniformization transforms a pseudo-Poisson process with unequal intensities (leaving rates) into one with uniform intensity. Self-transitions is the price to pay. Two intensities arise when one considers an absorbing barrier of a Markov process as a body in its own right: a pair of Markov processes intertwined by an extended Chapman–Kolmogorov equation naturally arises. We show that Sauer's two-state space empathy theory handles such intertwined processes. The price of self-transitions is also avoided.

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Abstract In this dissertation we will study how extended Poisson process can be applied to construct discrete probabilistic models. An Extended Poisson Process is a continuous time stochastic process with the state space being the natural numbers, it is obtained as a generalization of homogeneous Poisson process where transition rates depend on the current state of the process. From its transition rates and Chapman-Kolmogorov di¤erential equations, we can determine the probability distribution at any …xed time of the process. Conversely, given any probability distribution on the natural numbers, it is possible to determine uniquely a sequence of transition rates of an extended Poisson process such that, for some instant, the unidimensional probability distribution coincides with the provided probability distribution. Therefore, we can conclude that extended Poisson process is as a very ‡exible framework on the analysis of discrete data, since it generalizes all probabilistic discrete models. We will present transition rates of extended Poisson process which generate Poisson, Binomial and Negative Binomial distributions and determine maximum likelihood estima- tors, con…dence intervals, and hypothesis tests for parameters of the proposed models. We will also perform a bayesian analysis of such models with informative and noninformative prioris, presenting posteriori summaries and comparing these results to those obtained by means of classic inference.
Nesta dissertação veremos como o proceso de Poisson estendido pode ser aplicado à construção de modelos probabilísticos discretos. Um processo de Poisson estendido é um processo estocástico a tempo contínuo com espaço de estados igual ao conjunto dos números naturais, obtido a partir de uma generalização do processo de Poisson homogê- neo onde as taxas de transição dependem do estado atual do processo. A partir das taxas de transição e das equações diferenciais de Chapman-Kolmogorov pode-se determinar a distribuição de probabilidades para qualquer tempo …xado do processo. Reciprocamente, dada qualquer distribuição de probabilidades sobre o conjunto dos números naturais é pos- sível determinar, de maneira única, uma seqüência de taxas de transição de um processo de Poisson estendido tal que, para algum instante, a distribução unidimensional do processo coincide com a dada distribuição de probabilidades. Portanto, o processo de Poisson es- tendido se apresenta como uma ferramenta bastante ‡exível na análise de dados discretos, pois generaliza todos os modelos probabilísticos discretos. Apresentaremos as taxas de transição dos processos de Poisson estendido que ori- ginam as distribuições de Poisson, Binomial e Binomial Negativa e determinaremos os estimadores de máxima verossimilhança, intervalos de con…ança e testes de hipóteses dos parâmetros dos modelos propostos. Faremos também uma análise bayesiana destes mod- elos com prioris informativas e não informativas, apresentando os resumos a posteriori e comparando estes resultados com aqueles obtidos via inferência clássica.

Dissertação de Mestrado em Matemática apresentada à Faculdade de Ciências e Tecnologia
Uma das principais características da Teoria de Polinómios Ortogonais é a sua é a sua ubiquidade; desde a sua relação com áreas fundamentais da matemática, como a Teoria de Operadores ou os Problemas de Momentos, até à sua aplicação a àreas do conhecimento como a física quântica e, tal como neste caso, à análise de certos tipos de processos estocásticos. O presente trabalho focar-se-á, especificamente, na linguagem de polinómios ortogonais como ferramenta na análise transitória de Processos de Nascimento e Morte, um caso particular de uma Cadeia de Markov a tempo contínuo. O primeiro capítulo terá como objectivo fornecer ao leitor uma bagagem sólida no que diz respeito aos conceitos introdutórios e resultados clássicos da teoria de polinómios ortogonais, assim como construir as fundações para o trabalho subsequente. O conceito de ortogonalidade de polinómios será dado relativamente a um funcional linear e muitos dos resultados clássicos, como os teoremas de Favard e Christoffel-Darboux, serão discutidos neste contexto. Ainda neste capítulo o conceito de ortogonalidade de polinómios é trabalhado relativamente a uma função de distribuição, o que será útil nos capítulos seguintes. No capítulo 2 começaremos por analisar o conceito de Cadeia de Markov a tempo contínuo demonstrando como é possível estudar as probabilidades de transição entre estados, deste tipo de processos estocásticos, através dos sistemas de equações diferenciais lineares de Chapman-Kolmogorov e, mais concretamente, como podemos decrevê-los no contexto de um processo de nascimento e morte. Assim, será possível introduzir o leitor às secções seguintes deste mesmo capítulo, onde finalmente a teoria de polinómios ortogonais será utilizada como ferramenta na análise destes mesmo sistemas. Neste contexto, começaremos por identificar os polinómios que iremos associar a um processo de nascimento e morte genérico e apresentar uma candidata a solução dos sistemas de Chapman-Kolmogorov, com base nestes mesmos polinómios. Discutir-se-á ainda a existência de ortogonalidade para os polinómios com que escolhemos trabalhar e por fim será apresentado, com todo o detalhe, o processo pelo qual as probabilidades de transição de um processo de nascimento e morte podem ser dadas neste contexto. O capítulo 3 irá, de certa forma, concretizar o trabalho previamente realizado. Teremos assim como objectivo determinar as probabilidades de transição de casos concretos de processos de nascimento e morte, mais especificamente de modelos de fila de espera como 1 ou 2 servidores. Este trabalho consistirá essencialmente, graças ao que concluímos no capítulo 2, no estudo da ortogonalidade dos polinómios que associaremos a estes mesmos modelos. Para tal, será importante ter em consideração alguns conceitos teóricos adicionais, como o Teorema de Markov, e analizaremos ainda um trabalho efectudado no contexto dos polinómios de Chebychev, como ponto de partida para o raciocínio subsequente. O percurso será a partir daí análogo para os dois modelos que nos propusémos a estudar, com naturais distinções a partir do momento em que é possível concretizar uma candidata a função peso de ortogonalidade para cada um dos casos. Serão, ao longo deste capítulo, discutidos alguns resultados teóricos (devidamente referenciados) cujas demonstrações serão omitidas, mantendo assimo foco do trabalho aqui realizado nas especificidades dos casos em estudo.
One of the mains characteristics of the Theory of Orthogonal Polynomials is it's ubiquity; form it's relation with fundamental field of mathematics, like Operator Theory or the Moments Problems, to it's application to other fields of knowledge like quantum physics or, like in this case, to the analysis of certain types of stochastic processes. The presented work will focus specifically in the language of orthogonal polynomials as a tool in the transient analysis of Birth-Death Processes, a specific case of time-continuous Markov Chains. The first chapter will introduce the reader to the introductory concepts and classic results of the theory of orthogonal polynomials; it will also build the foundations for the subsequent work. The concept of polynomial orthogonality will be given with respect to a linear functional, and most classic results, like Favard's and Christoffel-Darboux's theorems, will be discussed in this context. Still in this chapter the concept of orthogonality for polynomials will be discussed with respect to a distribution function, what will be useful in the next chapters.In chapter 2 we will begin by analyzing the concept of time-continuous Markov Chain showing how the probabilities of transitions between states, of this type of stochastic processes, can be studied through the systems of linear differential equations of Chapman-Kolmogorov and, more specifically, how we can describe these systems in the context of a birth-death process. With this, it will be possible to introduce the reader to the next section of this chapter, where finally the theory of orthogonal polynomials will be used as a tool in the analysis of the referred systems. In this context, we will start by identifying the polynomials that we will associate to a generic birth-death process and present a candidate for the solution of the Chapman-Kolmogorov systems, based on these polynomials. The existence of orthogonality for the referred polynomials will be discussed and finally it will be presented, with all detail, the process by how the transition probabilities for a birth-death process can be given in this context.The third chapter will, in a way, materialize the previous work. Our objective will be to determine the transition probabilities of concrete cases of birth-death processes, more specifically queueing models with 1 or 2 servers. Given what we concluded in chapter 2, this work will consist essentially in the study of the orthogonality of the polynomials we will associate to these models. For this, it will be important to consider some additional theoretical concepts, like the Markov Theorem, and to analyze a work previously done in the context of the Chebychev polynoimials as a starting point for the subsequent arguments. From then on, the work will be analogous for both queueing models, with natural distinctions from the moment we can specify a candidate for a weight function of orthogonality for each case. Through this chapter some theoretical results will be discussed (rightly referenced) where the proofs will be omitted, for the sake of focusing in specificities of each case of study.
Outro - O trabalho aqui desenvolvido foi parcialmente realizado no âmbito do projeto MobiWise: From mobile sensing to mobility advising (P2020 SAICTPAC / 0011/2015), co-financiado pelo COMPETE 2020, Portugal 2020 - Programa Operacional de Competitividade e Internacionalização (POCI), do ERDF (Fundo Europeu de Desenvolvimento Regional) da União Europeia e da Fundação portuguesa para a Ciência e Tecnologia (FCT).

To simulate the dynamic behaviors of large molecular systems, approaches that solve ordinary differential equations such as molecular dynamics (MD) simulation may become inefficient. The kinetic Monte Carlo (KMC) method as the alternative has been widely used in simulating rare events such as chemical reactions or phase transitions. Yet lack of complete knowledge of transitions and the associated rates is one major challenge for accurate KMC predictions. In this paper, a reliable KMC (R-KMC) mechanism is proposed to improve the robustness of KMC results, where propensities are interval estimates instead of precise numbers and sampling is based on random sets instead of random numbers. A multi-event algorithm is developed and implemented. The weak convergence of the multi-event algorithm towards traditional KMC is demonstrated with a proposed generalized Chapman-Kolmogorov Equation.

The entry-time approach to dynamic reliability is based upon computational solution of the Chapman-Kolmogorov (generalized state-transition) equations underlying a certain class of marked point processes. Previous work has verified a particular finite-difference approach to computational solution of these equations. The objective of this work is to illustrate the potential application of the entry-time approach to risk-informed asset management (RIAM) decisions regarding maintenance or replacement of major systems within a plant. Results are presented in the form of plots, with replacement/maintenance period as a parameter, of expected annual revenue, along with annual variance and annual skewness as indicators of associated risks. Present results are for a hypothetical system, to illustrate the capability of the approach, but some considerations related to potential application of this approach to nuclear power plants are discussed.

Aging has become a more important consideration since nuclear power plants have been in operation for decades. However, the current probabilistic risk assessment (PRA) and generation risk assessment (GRA) models are based on the assumption of constant failure rate. The consistent treatment of the dynamics of the evolution of aging and maintenance is critical to system reliability. The entry-time approach to dynamic reliability is based upon computational solution of the Chapman-Kolmogorov (generalized state-transition) equations underlying a certain class of marked point processes. Previous work has verified a particular finite-difference approach to computational solution of these equations. The objective of this work is to illustrate the potential application of the entry-time approach to PRA and GRA regarding maintenance or replacement of major systems within a plant. Results are presented in the form of plots, with replacement/maintenance period as a parameter, of associated risks and probabilities. Further recommendations for application of the entry-time approach to dynamic PRA/GRA are given at the end of the article.