---
tag: 22735FGJV6
kind: concept
name: Wiener process
msc: [60J65, 60G15]
specializes: [22WEM6RTW0, 8NT5CFG2ZR]
sources:
  - url: https://en.wikipedia.org/wiki/Wiener_process
same_as:
  wikidata: Q1056809
  wikipedia: Wiener process
---

## Summary

The Wiener process, or standard Brownian motion, is the continuous-time [[stochastic process|0QH5ENKZM1]] with continuous paths and independent, normally distributed increments.
It is the scaling limit of random walks and the basic model of random motion.
It is a [[Gaussian process|22WEM6RTW0]], a [[Lévy process|8NT5CFG2ZR]], a [[martingale|0D0GBNDKDH]] and a Markov process at the same time.

## Explanations

### L1 human

Picture a pollen grain jiggling in water, or a stock price wobbling from second to second.
The Wiener process is the mathematical model of such motion: at every instant it moves by a small random amount, independent of the past.
Over a time span of length $t$ it moves by a normal amount with mean $0$ and variance $t$.
Its path is continuous but, almost surely, so rough that it has no derivative at any point.
If you take a fair coin-tossing walk with $n$ steps per unit of time and step size $1/\sqrt{n}$, you get a Wiener process as $n \to \infty$.