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2018 Stationary distributions of the Atlas model
Li-Cheng Tsai
Electron. Commun. Probab. 23: 1-10 (2018). DOI: 10.1214/18-ECP112

Abstract

In this article we study the Atlas model, which consists of Brownian particles on $ \mathbb{R} $, independent except that the Atlas (i.e., lowest ranked) particle $ X_{(1)}(t) $ receives drift $ \gamma dt $, $ \gamma \in \mathbb{R} $. For any fixed shape parameter $ a>2\gamma _- $, we show that, up to a shift $ \frac{a} {2}t $, the entire particle system has an invariant distribution $ \nu _a $, written in terms an explicit Radon-Nikodym derivative with respect to the Poisson point process of density $ ae^{a\xi } d\xi $. We further show that $ \nu _a $ indeed has the product-of-exponential gap distribution $ \pi _a $ derived in [ST17]. As a simple application, we establish a bound on the fluctuation of the Atlas particle $ X_{(1)}(t) $ uniformly in $ t $, with the gaps initiated from $ \pi _a $ and $ X_{(1)}(0)=0 $.

Citation

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Li-Cheng Tsai. "Stationary distributions of the Atlas model." Electron. Commun. Probab. 23 1 - 10, 2018. https://doi.org/10.1214/18-ECP112

Information

Received: 14 February 2017; Accepted: 29 January 2018; Published: 2018
First available in Project Euclid: 23 February 2018

zbMATH: 1391.60199
MathSciNet: MR3771768
Digital Object Identifier: 10.1214/18-ECP112

Subjects:
Primary: 60J60
Secondary: 60H10

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