Diffusion process

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 continuous sample paths.

A sample path of a diffusion process mimics the trajectory of a molecule, which is embedded in a flowing fluid and at the same time subjected to random displacements due to collisions with other molecules, i.e. Brownian motion. The position of this molecule is then random; its probability density function is governed by an advection-diffusion equation.

Mathematical definition

A diffusion process is any Markov process with continuous paths defined by a transition probability function p(s,x;t,dy) satisfying the Chapman-Kolmogorov equation.^[1]

See also

• Sample-continuous process
• Itō diffusion
• Jump diffusion

References

1. ^ "9. Diffusion processes" (pdf). http://math.nyu.edu/faculty/varadhan/stochastic.fall08/sec10.pdf. Retrieved October 10, 2011.