Importance of Brownian motion

Historically, Brownian motion has been associated with the analysis of movements that evolve over time in such a disorderly fashion that it seems difficult to predict their evolution, even in a very short time interval. It plays a central role in the theory of random processes, not least because in many theoretical or applied problems, Brownian motion or the diffusions deduced from it provide simple limit models on which many calculations can be carried out.

In 1827, English botanist Robert Brown described the erratic movement of fine organic particles suspended in a gas or fluid [1]. In the 19th century, several physicists subsequently recognized that this movement was highly irregular and did not seem to admit of a tangent; we could therefore not speak of its speed, nor a fortiori apply the laws of mechanics to it.

In 1900, Bachelier introduced...