Patterns in random walks and Brownian motion

TL;DR

This paper investigates the existence of specific continuous paths within Brownian motion, deriving asymptotics for pattern occurrence and establishing some existence and non-existence results through stochastic analysis.

Contribution

It introduces new asymptotic results for pattern embedding in Brownian motion and proves some existence and non-existence theorems using stochastic analysis techniques.

Findings

Asymptotic formulas for expected waiting times of patterns

Existence of certain continuous paths in Brownian motion proved

Non-existence results for other patterns established

Abstract

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. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented.