The Library
Brownian motion with asymptotically normal reflection in unbounded domains : from transience to stability
Tools
Bresar, Miha, Mijatović, Aleksandar and Wade, Andrew (2023) Brownian motion with asymptotically normal reflection in unbounded domains : from transience to stability. Annals of Probability . ISSN 0091-1798. (In Press)
|
PDF
WRAP-Brownian-motion-asymptotically-reflection-unbounded-domains-24.pdf - Accepted Version - Requires a PDF viewer. Download (727Kb) | Preview |
Official URL: https://imstat.org/journals-and-publications/annal...
Abstract
We quantify the asymptotic behaviour of multidimensional drifltess diffusions in domains unbounded in a single direction, with asymptotically normal reflections from the boundary. We identify the critical growth/contraction rates of the domain that separate stability, null recurrence and transience. In the stable case we prove existence and uniqueness of the invariant distribution and establish the polynomial rate of decay of its tail. We also establish matching polynomial upper and lower bounds on the rate of convergence to stationarity in total variation. All exponents are explicit in the model parameters that determine the asymptotics of the growth rate of the domain, the interior covariance, and the reflection vector field.
Proofs are probabilistic, and use upper and lower tail bounds for additive functionals up to return times to compact sets, for which we develop
novel sub/supermartingale criteria, applicable to general continuous semimartingales. Narrowing domains fall outside of the standard literature, in part because boundary local time can accumulate arbitrarily rapidly. Establishing Feller continuity (essential for characterizing stability) thus requires an extension of the usual approach.
Our recurrence/transience classification extends previous work on strictly
normal reflections, and expands the range of phenomena observed across all dimensions. For all recurrent cases, we provide quantitative information through upper and lower bounds on tails of return times to compact sets.
Item Type: | Journal Article | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Subjects: | Q Science > QA Mathematics | |||||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | |||||||||||||||
Library of Congress Subject Headings (LCSH): | Brownian motion processes, Stochastic analysis, Stochastic processes -- Mathematical models, Probabilities -- Mathematical models | |||||||||||||||
Journal or Publication Title: | Annals of Probability | |||||||||||||||
Publisher: | Institute of Mathematical Statistics | |||||||||||||||
ISSN: | 0091-1798 | |||||||||||||||
Official Date: | 29 March 2023 | |||||||||||||||
Dates: |
|
|||||||||||||||
Status: | Peer Reviewed | |||||||||||||||
Publication Status: | In Press | |||||||||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
Date of first compliant deposit: | 11 April 2024 | |||||||||||||||
Date of first compliant Open Access: | 11 April 2024 | |||||||||||||||
Grant number: | EP/V009478/1; & EP/V009478/1; EP/W006227/1; EP/W00657X/1 | |||||||||||||||
RIOXX Funder/Project Grant: |
|
|||||||||||||||
Related URLs: | ||||||||||||||||
Open Access Version: |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |
Downloads
Downloads per month over past year