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2018 The argmin process of random walks, Brownian motion and Lévy processes
Jim Pitman, Wenpin Tang
Electron. J. Probab. 23(none): 1-35 (2018). DOI: 10.1214/18-EJP185


In this paper we investigate the argmin process of Brownian motion $B$ defined by $\alpha _t:=\sup \left \{s \in [0,1]: B_{t+s}=\inf _{u \in [0,1]}B_{t+u} \right \}$ for $t \geq 0$. The argmin process $\alpha $ is stationary, with invariant measure which is arcsine distributed. We prove that $(\alpha _t; t \geq 0)$ is a Markov process with the Feller property, and provide its transition kernel $Q_t(x,\cdot )$ for $t>0$ and $x \in [0,1]$. Similar results for the argmin process of random walks and Lévy processes are derived. We also consider Brownian extrema of a given length. We prove that these extrema form a delayed renewal process with an explicit path construction. We also give a path decomposition for Brownian motion at these extrema.


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Jim Pitman. Wenpin Tang. "The argmin process of random walks, Brownian motion and Lévy processes." Electron. J. Probab. 23 1 - 35, 2018.


Received: 22 August 2017; Accepted: 6 June 2018; Published: 2018
First available in Project Euclid: 20 June 2018

zbMATH: 06924672
MathSciNet: MR3827967
Digital Object Identifier: 10.1214/18-EJP185

Primary: 60G50, 60G51, 60J65


Vol.23 • 2018
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