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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.24 Stochastic Processes

Jean-François Le Gall

1 Historical Introduction

Stochastic processes are one of the major themes of modern probability theory. Roughly speaking, they are mathematical models that describe the evolution of random phenomena as time goes by. In this article, we shall introduce and illustrate the fundamental ideas of the theory of stochastic processes by concentrating on the single most important example: Brownian motion. We start with a brief historical introduction, in order to provide some motivation for the mathematical theory that follows.

In 1828, the British botanist Robert Brown observed the very irregular and wiggly motion of small particles of pollen suspended in water. Brown pointed out the unpredictable character ...

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