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Stochastic process

About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.


Papers
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Journal ArticleDOI
TL;DR: The mathematical theory behind the simple random walk is introduced and how this relates to Brownian motion and diffusive processes in general and a reinforced random walk can be used to model movement where the individual changes its environment.
Abstract: Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

1,313 citations

Journal ArticleDOI
TL;DR: This paper presents a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process.
Abstract: In many stochastic models, particularly in queueing theory, Poisson arrivals both observe (see) a stochastic process and interact with it. In particular cases and/or under restrictive assumptions it has been shown that the fraction of arrivals that see the process in some state is equal to the fraction of time the process is in that state. In this paper, we present a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process.

1,226 citations

Journal ArticleDOI
P. Mazur1
TL;DR: In this article, it was shown that the non-Gaussian-Markoff process for Brownian motion derived on a statistical mechanical basis by Prigogine and Balescu, and Prigogueine and Philippot, is related through a transformation of variables to the Gaussian Markoff process of the conventional phenomenological theory of Brownian motions.

1,193 citations

Book
27 Sep 2018
TL;DR: A broad range of illustrations is embedded throughout, including classical and modern results for covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, machine learning, compressed sensing, and sparse regression.
Abstract: High-dimensional probability offers insight into the behavior of random vectors, random matrices, random subspaces, and objects used to quantify uncertainty in high dimensions Drawing on ideas from probability, analysis, and geometry, it lends itself to applications in mathematics, statistics, theoretical computer science, signal processing, optimization, and more It is the first to integrate theory, key tools, and modern applications of high-dimensional probability Concentration inequalities form the core, and it covers both classical results such as Hoeffding's and Chernoff's inequalities and modern developments such as the matrix Bernstein's inequality It then introduces the powerful methods based on stochastic processes, including such tools as Slepian's, Sudakov's, and Dudley's inequalities, as well as generic chaining and bounds based on VC dimension A broad range of illustrations is embedded throughout, including classical and modern results for covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, machine learning, compressed sensing, and sparse regression

1,190 citations

Book
01 Jan 1990
TL;DR: In this paper, a comprehensive account of statistical linearization with related techniques allowing the solution of a very wide variety of practical non-linear random vibration problems is given, and the principal value of these methods is that they are readily generalized to deal with complex mechanical and structural systems and complex types of excitation such as earthquakes.
Abstract: Interest in the study of random vibration problems using the concepts of stochastic process theory has grown rapidly due to the need to design structures and machinery which can operate reliably when subjected to random loads, for example winds and earthquakes. This is the first comprehensive account of statistical linearization - powerful and versatile methods with related techniques allowing the solution of a very wide variety of practical non-linear random vibration problems. The principal value of these methods is that unlike other analytical methods, they are readily generalized to deal with complex mechanical and structural systems and complex types of excitation such as earthquakes.

1,174 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023159
2022355
2021985
20201,151
20191,119
20181,115