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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
Kenichi Nanbu1
TL;DR: In this article, the authors proposed a method to determine the velocities of simulated molecules after a small time increment was derived from the Boltzmann equation, which was shown to give an exact solution of the Boltzman equation.
Abstract: The stochastic law that prescribes the velocities of simulated molecules after a small time increment was derived from the Boltzmann equation. The scheme to determine the velocity of a molecule after a small time increment is divided into three steps. The first step gives the collision probability of the molecule without specifying its collision partner. The second step gives a conditional probability distribution. If the molecule is accepted in the first step as a colliding molecule, its collision partner is sampled from this probability distribution. The last step gives a probability density from which the direction of the relative velocity after collision is sampled, and hence the step gives the post-collision velocity of the molecule. It is shown that the use of the present simulation scheme gives an exact solution of the Boltzmann equation.

360 citations

Book
19 Nov 2014
TL;DR: In this article, the Fokker-Planck Equation is modelled with Stochastic Differential Equations (SDE) and the Langevin Equation (LDE).
Abstract: Stochastic Processes.- Diffusion Processes.- Introduction to Stochastic Differential Equations.- The Fokker-Planck Equation.- Modelling with Stochastic Differential Equations.- The Langevin Equation.- Exit Problems for Diffusions.- Derivation of the Langevin Equation.- Linear Response Theory.- Appendix A Frequently Used Notations.- Appendix B Elements of Probability Theory.

358 citations

Journal ArticleDOI
TL;DR: It is shown that the Hodgkin-Huxley equations can be approximated by a one-dimensional bistable Langevin equation and spontaneous action potentials can arise from the channel fluctuations and are analogous to escape by a particle over a potential barrier.

358 citations

Journal ArticleDOI
TL;DR: A direct test for deterministic dynamics can be established by measurement of average directional vectors in a coarse-grained d-dimensional embedding of a time series, and examples are given to show the clear differences between deterministic and stochastic dynamics.
Abstract: A direct test for deterministic dynamics can be established by measurement of average directional vectors in a coarse-grained d-dimensional embedding of a time series. Theoretical analysis of the statistical properties of a random time series using the same embedding technique is possible by consideration of classical results concerning random walks in d dimensions. Examples are given to show the clear differences between deterministic dynamics, such as may be generated by chaotic systems, and stochastic dynamics.

357 citations

Journal ArticleDOI
TL;DR: In this paper, passivity analysis is conducted for discrete-time stochastic neural networks with both Markovian jumping parameters and mixed time delays by introducing a Lyapunov functional that accounts for the Mixed time delays.
Abstract: In this paper, passivity analysis is conducted for discrete-time stochastic neural networks with both Markovian jumping parameters and mixed time delays. The mixed time delays consist of both discrete and distributed delays. The Markov chain in the underlying neural networks is finite piecewise homogeneous. By introducing a Lyapunov functional that accounts for the mixed time delays, a delay-dependent passivity condition is derived in terms of the linear matrix inequality approach. The case of Markov chain with partially unknown transition probabilities is also considered. All the results presented depend upon not only discrete delay but also distributed delay. A numerical example is included to demonstrate the effectiveness of the proposed methods.

355 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