Renewal Structure of the Brownian Taut String

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.