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Lipschitz minorants of Brownian motion and Lévy processes

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Abstract

For \(\alpha > 0\), the \(\alpha \)-Lipschitz minorant of a function\(f\!:\!\mathbb{R }\rightarrow \mathbb{R }\) is the greatest function \(m : \mathbb{R }\rightarrow \mathbb{R }\) such that \(m \le f\) and \(|m(s)-m(t)| \le \alpha |s-t|\) for all \(s,t \in \mathbb{R }\), should such a function exist. If \(X=(X_t)_{t \in \mathbb{R }}\) is a real-valued Lévy process that is not pure linear drift with slope \(\pm \alpha \), then the sample paths of \(X\) have an \(\alpha \)-Lipschitz minorant almost surely if and only if \(| \mathbb{E }[X_1] | < \alpha \). Denoting the minorant by \(M\), we investigate properties of the random closed set \(\fancyscript{Z} := \{ t \in \mathbb{R }: M_t = X_t \wedge X_{t-} \}\), which, since it is regenerative and stationary, has the distribution of the closed range of some subordinator “made stationary” in a suitable sense. We give conditions for the contact set \(\fancyscript{Z}\) to be countable or to have zero Lebesgue measure, and we obtain formulas that characterize the Lévy measure of the associated subordinator. We study the limit of \(\fancyscript{Z}\) as \(\alpha \rightarrow \infty \) and find for the so-called abrupt Lévy processes introduced by Vigon that this limit is the set of local infima of \(X\). When \(X\) is a Brownian motion with drift \(\beta \) such that \(|\beta | < \alpha \), we calculate explicitly the densities of various random variables related to the minorant.

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Acknowledgments

We thank an anonymous referee for a number of helpful suggestions. S. N. Evans was supported in part by National Science Foundation Grant DMS-0907630.

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Correspondence to Joshua Abramson.

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Abramson, J., Evans, S.N. Lipschitz minorants of Brownian motion and Lévy processes. Probab. Theory Relat. Fields 158, 809–857 (2014). https://doi.org/10.1007/s00440-013-0497-9

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Keywords

  • Fluctuation theory
  • Regenerative set
  • Subordinator
  • Abrupt process
  • Global minimum
  • \(c\)-Convexity
  • Pasch-Hausdorff envelope

Mathematics Subject Classification (2000)

  • 60G51
  • 60G55
  • 60G17
  • 60J65