---
tag: 9HDQ094ADQ
kind: concept
name: Diffusion process
msc: [60J60, 60H10]
specializes: [0QH5ENKZM1]
sources:
  - url: https://en.wikipedia.org/wiki/Diffusion_process
same_as:
  wikidata: Q5275442
  wikipedia: Diffusion process
---

## Summary

A diffusion process is a Markov process with continuous paths, such as Brownian motion with a drift and a variable volatility.
Its local behaviour is described by a drift vector and a diffusion matrix.
Its [[generator|VPA41PWRR1]] is a second-order differential operator, so diffusions connect probability with parabolic and elliptic partial differential equations.
Most diffusions are built as solutions of stochastic differential equations.

## Explanations

### L1 human

Picture a particle of dust in a moving fluid.
The fluid carries it along, which is the drift.
Collisions with molecules shake it at random, which is the diffusion part.
The path is continuous but very rough, like the path of Brownian motion.
The [[Ornstein–Uhlenbeck process|HZM2ASCDY4]], pulled back towards $0$ while being shaken, is a standard example.