The Slepian zero set, and Brownian bridge embedded in Brownian motion by a spacetime shift

TL;DR

This paper investigates the structure of the Slepian zero set derived from Brownian motion, revealing a path decomposition and demonstrating the embedding of Brownian bridges within Brownian motion via a random time shift.

Contribution

It provides a detailed analysis of the Slepian zero set and proves the existence of a random time embedding a Brownian bridge within Brownian motion.

Findings

Path decomposition of the Slepian process for 0 ≤ t ≤ 1

Almost sure existence of a random time T in the zero set

Embedding of Brownian bridge within Brownian motion at time T

Abstract

This paper is concerned with various aspects of the Slepian process $(B_{t+1} - B_t, t \ge 0)$ derived from a one-dimensional Brownian motion $(B_t, t \ge 0 )$. In particular, we offer an analysis of the local structure of the Slepian zero set $\{t : B_{t+1} = B_t \}$, including a path decomposition of the Slepian process for $0 \le t \le 1$. We also establish the existence of a random time $T$ such that $T$ falls in the the Slepian zero set almost surely and the process $(B_{T+u} - B_T, 0 \le u \le 1)$ is standard Brownian bridge.