Exact first-passage time distributions for three random diffusivity models

We study the extremal properties of a stochastic process $x_t$ defined by a Langevin equation $\dot{x}= \sqrt{2D_o V (B_t )} \xi_t$, where $\xi$ is a Gaussian white noise with zero mean, $D_0$ is a constant scale factor, and $V (B_t)$ is a stochastic "diffusivity" (noise strength), which i...

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Detalles Bibliográficos
Autores: Grebenkov, D. S., Sposini, V., Metzler, R., Oshanin, G., Seno, F.
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2021
País:España
Institución:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/1250
Acceso en línea:http://hdl.handle.net/20.500.11824/1250
Access Level:acceso abierto
Palabra clave:First-passage time distributions
Random diffusivity models
Anomalous diffusion
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spelling Exact first-passage time distributions for three random diffusivity modelsGrebenkov, D. S.Sposini, V.Metzler, R.Oshanin, G.Seno, F.First-passage time distributionsRandom diffusivity modelsAnomalous diffusionWe study the extremal properties of a stochastic process $x_t$ defined by a Langevin equation $\dot{x}= \sqrt{2D_o V (B_t )} \xi_t$, where $\xi$ is a Gaussian white noise with zero mean, $D_0$ is a constant scale factor, and $V (B_t)$ is a stochastic "diffusivity" (noise strength), which itself is a functional of independent Brownian motion $B_t$. We derive exact, compact expressions in one and three dimensions for the probability density functions (PDFs) of the first passage time (FPT) $t$ from a fixed location $x_0$ to the origin for three different realisations of the stochastic diffusivity: a cut-off case $V (B_t) = \Theta(B_t)$ (Model I), where $\Theta(z)$ is the Heaviside theta function; a Geometric Brownian Motion $V (B_t) = exp(B_t)$ (Model II); and a case with $V (B_t) = B_t^2$ (Model III). We realise that, rather surprisingly, the FPT PDF has exactly the L\'evy-Smirnov form (specific for standard Brownian motion) for Model II, which concurrently exhibits a strongly anomalous diffusion. For Models I and III either the left or right tails (or both) have a different functional dependence on time as compared to the L\'evy-Smirnov density. In all cases, the PDFs are broad such that already the first moment does not exist. Similar results are obtained in three dimensions for the FPT PDF to an absorbing spherical target.202120212021info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/1250reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttps://iopscience.iop.org/article/10.1088/1751-8121/abd42cReconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/12502026-06-19T12:47:47Z
dc.title.none.fl_str_mv Exact first-passage time distributions for three random diffusivity models
title Exact first-passage time distributions for three random diffusivity models
spellingShingle Exact first-passage time distributions for three random diffusivity models
Grebenkov, D. S.
First-passage time distributions
Random diffusivity models
Anomalous diffusion
title_short Exact first-passage time distributions for three random diffusivity models
title_full Exact first-passage time distributions for three random diffusivity models
title_fullStr Exact first-passage time distributions for three random diffusivity models
title_full_unstemmed Exact first-passage time distributions for three random diffusivity models
title_sort Exact first-passage time distributions for three random diffusivity models
dc.creator.none.fl_str_mv Grebenkov, D. S.
Sposini, V.
Metzler, R.
Oshanin, G.
Seno, F.
author Grebenkov, D. S.
author_facet Grebenkov, D. S.
Sposini, V.
Metzler, R.
Oshanin, G.
Seno, F.
author_role author
author2 Sposini, V.
Metzler, R.
Oshanin, G.
Seno, F.
author2_role author
author
author
author
dc.subject.none.fl_str_mv First-passage time distributions
Random diffusivity models
Anomalous diffusion
topic First-passage time distributions
Random diffusivity models
Anomalous diffusion
description We study the extremal properties of a stochastic process $x_t$ defined by a Langevin equation $\dot{x}= \sqrt{2D_o V (B_t )} \xi_t$, where $\xi$ is a Gaussian white noise with zero mean, $D_0$ is a constant scale factor, and $V (B_t)$ is a stochastic "diffusivity" (noise strength), which itself is a functional of independent Brownian motion $B_t$. We derive exact, compact expressions in one and three dimensions for the probability density functions (PDFs) of the first passage time (FPT) $t$ from a fixed location $x_0$ to the origin for three different realisations of the stochastic diffusivity: a cut-off case $V (B_t) = \Theta(B_t)$ (Model I), where $\Theta(z)$ is the Heaviside theta function; a Geometric Brownian Motion $V (B_t) = exp(B_t)$ (Model II); and a case with $V (B_t) = B_t^2$ (Model III). We realise that, rather surprisingly, the FPT PDF has exactly the L\'evy-Smirnov form (specific for standard Brownian motion) for Model II, which concurrently exhibits a strongly anomalous diffusion. For Models I and III either the left or right tails (or both) have a different functional dependence on time as compared to the L\'evy-Smirnov density. In all cases, the PDFs are broad such that already the first moment does not exist. Similar results are obtained in three dimensions for the FPT PDF to an absorbing spherical target.
publishDate 2021
dc.date.none.fl_str_mv 2021
2021
2021
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/1250
url http://hdl.handle.net/20.500.11824/1250
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://iopscience.iop.org/article/10.1088/1751-8121/abd42c
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
reponame_str BIRD. BCAM's Institutional Repository Data
collection BIRD. BCAM's Institutional Repository Data
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