## Electronic Journal of Probability

### The argmin process of random walks, Brownian motion and Lévy processes

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 60, 35 pp.

Dates
Accepted: 6 June 2018
First available in Project Euclid: 20 June 2018

https://projecteuclid.org/euclid.ejp/1529460158

Digital Object Identifier
doi:10.1214/18-EJP185

Mathematical Reviews number (MathSciNet)
MR3827967

Zentralblatt MATH identifier
06924672

#### Citation

Pitman, Jim; Tang, Wenpin. The argmin process of random walks, Brownian motion and Lévy processes. Electron. J. Probab. 23 (2018), paper no. 60, 35 pp. doi:10.1214/18-EJP185. https://projecteuclid.org/euclid.ejp/1529460158

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