Topic
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.
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TL;DR: In this article, the authors examined the effect of small stable regions or islands on the correlation function for the stochastic trajectories of the Hamiltonians of two degrees of freedom.
288 citations
TL;DR: This paper introduces delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation, and shows that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.
Abstract: Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.
287 citations
TL;DR: A general theory using the Levy-Khinchine representation for infinitely divisible cascade processes is presented, which allows for a classification of scaling behaviors of various strongly nonlinear dissipative systems.
Abstract: It is proposed that the statistics of the inertial range of fully developed turbulence can be described by a quantized random multiplicative process. We then show that (i) the cascade process must be a log-infinitely divisible stochastic process (i.e., stationary independent log-increments); (ii) the inertial-range statistics of turbulent fluctuations, such as the coarse-grained energy dissipation, are log-Poisson; and (iii) a recently proposed scaling model [Z.-S. She and E. Leveque 72, 336 (1994)] of fully developed turbulence can be derived. A general theory using the Levy-Khinchine representation for infinitely divisible cascade processes is presented, which allows for a classification of scaling behaviors of various strongly nonlinear dissipative systems.
286 citations
Book•
11 Aug 2014TL;DR: This book offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis and theory is developed in tandem with state-of-the art computational methods through worked examples, exercises, theorems and proofs.
Abstract: Part I. Deterministic Differential Equations: 1. Linear analysis 2. Galerkin approximation and finite elements 3. Time-dependent differential equations Part II. Stochastic Processes and Random Fields: 4. Probability theory 5. Stochastic processes 6. Stationary Gaussian processes 7. Random fields Part III. Stochastic Differential Equations: 8. Stochastic ordinary differential equations (SODEs) 9. Elliptic PDEs with random data 10. Semilinear stochastic PDEs.
284 citations
TL;DR: The usual continuum product based on the Kolmogorov construction together with the Lebesgue measure as well as the usual finitely additive measure-theoretic framework is shown to be not suitable for modeling individual risks.
284 citations