nLab Brownian motion

Idea

Brownian motion is an example of a stochastic process. Brownian motion $B_t$ is a stochastic process on $[0,\infty)$ with the following properties:

• $B_0=0$ almost surely.

• The increments $B_t-B_s$ are independent and normally distributed $N(0,t-s)$ for $t \gt s$.

• The function $t\mapsto B_t$ is continuous.

Related concepts

• Wiener measure

• Brownian loop soup

References

Introduction:

• Tamas Szabados, An elementary introduction to the Wiener process and stochastic integrals, Studia Scientiarum Mathematicarum Hungarica 31 (1996) 249-297 [arXiv:1008.1510]

History:

• Arthur Genthon, The concept of velocity in the history of Brownian motion – From physics to mathematics and back, Eur. Phys. J. H 45 (2020) 49-105 [arXiv:2006.05399, doi:10.1140/epjh/e2020-10009-8]

See also:

• Wikipedia: Brownian motion

• Wikipedia: Wiener process

• Wikipedia: Diffusion