Renewal Structure of the Brownian Taut String

TL;DR

This paper uncovers a renewal structure for the Brownian taut string, enabling explicit formulas for energy limits and establishing a CLT for energy fluctuations, advancing understanding of Brownian motion constrained by a fixed-width tube.

Contribution

It introduces a natural renewal structure linked to Brownian h-extrema, leading to explicit energy limit formulas and a CLT for the taut string's energy fluctuations.

Findings

Derived an explicit expression for the energy limit constant.

Established a CLT for the fluctuations of the energy spent.

Connected the renewal structure to the time decomposition of Brownian motion.

Abstract

In a recent paper, M. Lifshits and E. Setterqvist introduced the taut string of a Brownian motion $w$, defined as the function of minimal quadratic energy on $[0,T]$ staying in a tube of fixed width $h>0$ around $w$. The authors showed a Law of Large Number (L.L.N.) for the quadratic energy spent by the string for a large time $T$. In this note, we exhibit a natural renewal structure for the Brownian taut string, which is directly related to the time decomposition of the Brownian motion in terms of its $h$-extrema (as first introduced by Neveu and Pitman). Using this renewal structure, we derive an expression for the constant in the L.L.N. given by M. Lifshits and E. Setterqvist. In addition, we provide a Central Limit Theorem (C.L.T.) for the fluctuations of the energy spent by the Brownian taut string.