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.…
Verfasserangaben: | 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 |
Titel des übergeordneten Werks (Englisch): | Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe |
Schriftenreihe (Bandnummer): | Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe (1142) |
Publikationstyp: | Postprint |
Sprache: | Englisch |
Datum der Erstveröffentlichung: | 09.02.2021 |
Erscheinungsjahr: | 2020 |
Veröffentlichende Institution: | Universität Potsdam |
Datum der Freischaltung: | 19.04.2021 |
Freies Schlagwort / Tag: | Brownian motion; diffusion; extremal values; maximum and range; random diffusivity |
Ausgabe: | 1142 |
Seitenanzahl: | 24 |
Quelle: | New Journal of Physics 23 (2021) Art. 023014 DOI: 10.1088/1367-2630/abd313 |
Organisationseinheiten: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie |
DDC-Klassifikation: | 5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik |
Peer Review: | Referiert |
Publikationsweg: | Open Access / Green Open-Access |
Lizenz (Deutsch): | CC-BY - Namensnennung 4.0 International |
Externe Anmerkung: | Bibliographieeintrag der Originalveröffentlichung/Quelle |