Renewal structure of the Brownian taut string

Abstract

In a recent paper Lifshits and Setterqvist (2015), 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 (1989)). Using this renewal structure, we derive an expression for the constant in the L.L.N. given in Lifshits and Setterqvist (2015) . In addition, we provide an invariance principle for the (renormalized) energy spent by the Brownian taut string.