Exact distributions of the maximum and range of random diffusivity processes
- We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of theWe study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.…
Author details: | Denis S. GrebenkovORCiD, Vittoria SposiniORCiD, Ralf MetzlerORCiDGND, Gleb OshaninORCiDGND, Flavio SenoORCiD |
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URN: | urn:nbn:de:kobv:517-opus4-503976 |
DOI: | https://doi.org/10.25932/publishup-50397 |
ISSN: | 1866-8372 |
Title of parent work (English): | Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe |
Publication series (Volume number): | Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe (1142) |
Publication type: | Postprint |
Language: | English |
Date of first publication: | 2021/02/09 |
Publication year: | 2020 |
Publishing institution: | Universität Potsdam |
Release date: | 2021/04/19 |
Tag: | Brownian motion; diffusion; extremal values; maximum and range; random diffusivity |
Issue: | 1142 |
Number of pages: | 24 |
Source: | New Journal of Physics 23 (2021) Art. 023014 DOI: 10.1088/1367-2630/abd313 |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie |
DDC classification: | 5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik |
Peer review: | Referiert |
Publishing method: | Open Access / Green Open-Access |
License (German): | CC-BY - Namensnennung 4.0 International |
External remark: | Bibliographieeintrag der Originalveröffentlichung/Quelle |