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Continuous paths in Brownian motion and related problems


This thesis is composed of six chapters, which mainly deals with embedding continuous paths in Brownian motion. It is adapted from two publications \cite{PT15a, PT15b}, joint with Jim Pitman.

We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion.

By various stochastic analysis arguments (path decomposition, It\^{o} excursion theory, potential theory...), we are able to prove some of these existence and non-existence results:


\begin{tabular}{| c | c | c | c | c |}


$Z$ & $e$ & $V(b^{\lambda})$ & $m$ & $R$ \ \hline

Embedding into $B$ & No & No & Yes & Yes \ \hline



where $e$ is a normalized Brownian excursion, $V(b)$ is the Vervaat transform of Brownian bridge ending at $\lambda$, $m$ is a Brownian meander, and $R$ is the three dimensional Bessel process of unit length.

The question of embedding a Brownian bridge into Brownian motion is more chanllenging. After explaining why some simple or traditional approaches do not work, we make use of recent work of Last and Thorisson on shift couplings of stationary random measures to prove the result. These can be applied after a thorough analysis of the Slepian zero set $\{t \geq 0; B_t = B_{t+1}\}$.

We also discuss the potential theoretical aspect of embedding continuous paths in a random process. A list of open problems is presented.

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