Diffusion process

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For the marketing term, see Diffusion of innovations.

In probability theory, a branch of mathematics, a diffusion process is a solution to a stochastic differential equation. It is a continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diffusion processes.

A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with molecules, which is called Brownian motion. The position of the particle is then random; its probability density function as a function of space and time is governed by an advection-diffusion equation.

Mathematical definition[edit]

A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker-Planck equation.[1]

See also[edit]


  1. ^ "9. Diffusion processes" (pdf). Retrieved October 10, 2011.