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Cannizzaro, Giuseppe and Hairer, Martin (2023) The Brownian Castle. Communications on Pure and Applied Mathematics, 76 (10). doi:10.1002/cpa.22085 ISSN 1097-0312.
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Official URL: https://doi.org/10.1002/cpa.22085
Abstract
We introduce a 1+1‐dimensional temperature‐dependent model such that the classical ballistic deposition model is recovered as its zero‐temperature limit. Its ∞‐temperature version, which we refer to as the 0‐Ballistic Deposition (0‐BD) model, is a randomly evolving interface which, surprisingly enough, does not belong to either the Edwards–Wilkinson (EW) or the Kardar–Parisi–Zhang (KPZ) universality class. We show that 0‐BD has a scaling limit, a new stochastic process that we call Brownian Castle (BC) which, although it is “free”, is distinct from EW and, like any other renormalisation fixed point, is scale‐invariant, in this case under the 1:1:2 scaling (as opposed to 1:2:3 for KPZ and 1:2:4 for EW). In the present article, we not only derive its finite‐dimensional distributions, but also provide a “global” construction of the Brownian Castle which has the advantage of highlighting the fact that it admits backward characteristics given by the (backward) Brownian Web (see [37, 16]). Among others, this characterisation enables us to establish fine pathwise properties of BC and to relate these to special points of the Web. We prove that the Brownian Castle is a (strong) Markov and Feller process on a suitable space of càdlàg functions and determine its long‐time behaviour. Finally, we give a glimpse to its universality by proving the convergence of 0‐BD to BC in a rather strong sense. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.
Item Type: | Journal Article | |||||||||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | |||||||||||||||
SWORD Depositor: | Library Publications Router | |||||||||||||||
Library of Congress Subject Headings (LCSH): | Brownian motion processes, Stochastic analysis | |||||||||||||||
Journal or Publication Title: | Communications on Pure and Applied Mathematics | |||||||||||||||
Publisher: | John Wiley & Sons Australia, Ltd | |||||||||||||||
ISSN: | 1097-0312 | |||||||||||||||
Official Date: | October 2023 | |||||||||||||||
Dates: |
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Volume: | 76 | |||||||||||||||
Number: | 10 | |||||||||||||||
DOI: | 10.1002/cpa.22085 | |||||||||||||||
Status: | Peer Reviewed | |||||||||||||||
Publication Status: | Published | |||||||||||||||
Access rights to Published version: | Open Access (Creative Commons) | |||||||||||||||
Copyright Holders: | © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. | |||||||||||||||
Date of first compliant deposit: | 9 December 2022 | |||||||||||||||
Date of first compliant Open Access: | 9 December 2022 | |||||||||||||||
RIOXX Funder/Project Grant: |
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