# Diffusion process

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{{Short description|Solution to a stochastic differential equation}}
{{for|the marketing term|Diffusion of innovations}}
{{one source |date=March 2024}}
In [probability theory](/source/probability_theory) and [statistics](/source/statistics), '''diffusion processes''' are a class of continuous-time [Markov process](/source/Markov_process) with [almost surely](/source/almost_surely) [continuous](/source/continuous_function) sample paths. Diffusion processes are [stochastic](/source/stochastic) in nature and hence are used to model many real-life stochastic systems. [Brownian motion](/source/Brownian_motion), [reflected Brownian motion](/source/reflected_Brownian_motion) and [Ornstein–Uhlenbeck processes](/source/Ornstein%E2%80%93Uhlenbeck_processes) are examples of diffusion processes. It is used heavily in [statistical physics](/source/statistical_physics), [statistical analysis](/source/statistical_analysis), [information theory](/source/information_theory), [data science](/source/data_science), [neural networks](/source/Artificial_neural_network), [finance](/source/finance) and [marketing](/source/marketing).

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 other particles, which is called [Brownian motion](/source/Brownian_motion).  The position of the particle is then random; its [probability density function](/source/probability_density_function) as a [function of space and time](/source/function_of_space_and_time) is  governed by a [convection–diffusion equation](/source/convection%E2%80%93diffusion_equation).

== Mathematical definition ==
A ''diffusion process'' is a [Markov process](/source/Markov_process) with [continuous sample paths](/source/Sample-continuous_process) for which the [Kolmogorov forward equation](/source/Kolmogorov_equations) is the [Fokker–Planck equation](/source/Fokker%E2%80%93Planck_equation).<ref>{{cite web|title=9. Diffusion processes|url=http://math.nyu.edu/faculty/varadhan/stochastic.fall08/sec10.pdf|access-date=October 10, 2011}}</ref>

A diffusion process is defined by the following properties.  Let <math>a^{ij}(x,t)</math> be uniformly continuous coefficients and <math>b^{i}(x,t)</math> be bounded, Borel measurable drift terms. There is a unique family of probability measures <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> (for <math>\tau \ge 0</math>, <math>\xi \in \mathbb{R}^d</math>) on the canonical space <math>\Omega = C([0,\infty), \mathbb{R}^d)</math>, with its Borel <math>\sigma</math>-algebra, such that:

1. (Initial Condition) The process starts at <math>\xi</math> at time <math>\tau</math>: <math>\mathbb{P}^{\xi,\tau}_{a;b}[\psi \in \Omega : \psi(t) = \xi \text{ for } 0 \le t \le \tau] = 1.</math>

2. (Local Martingale Property) For every <math>f \in C^{2,1}(\mathbb{R}^d \times [\tau,\infty))</math>, the process 

<math>M_t^{[f]} = f(\psi(t),t) - f(\psi(\tau),\tau) - \int_\tau^t \bigl(L_{a;b} + \tfrac{\partial}{\partial s}\bigr) f(\psi(s),s)\,ds</math>  is a local martingale under <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> for <math>t \ge \tau</math>, with <math>M_t^{[f]} = 0</math> for <math>t \le \tau</math>.

This family <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> is called the <math>\mathcal{L}_{a;b}</math>-diffusion.

== SDE Construction and Infinitesimal Generator ==

It is clear that if we have an <math>\mathcal{L}_{a;b}</math>-diffusion, i.e. <math>(X_t)_{t \ge 0}</math> on <math>(\Omega, \mathcal{F}, \mathcal{F}_t, \mathbb{P}^{\xi,\tau}_{a;b})</math>, then <math>X_t</math> satisfies the SDE <math>dX_t^i = \frac{1}{2}\,\sum_{k=1}^d \sigma^i_k(X_t)\,dB_t^k + b^i(X_t)\,dt</math>. In contrast, one can construct this diffusion from that SDE if <math>a^{ij}(x,t) = \sum_k \sigma^k_i(x,t)\,\sigma^k_j(x,t)</math> and <math>\sigma^{ij}(x,t)</math>, <math>b^i(x,t)</math> are Lipschitz continuous. 
To see this, let <math>X_t</math> solve the SDE starting at <math>X_\tau = \xi</math>. For <math>f \in C^{2,1}(\mathbb{R}^d \times [\tau,\infty))</math>, apply Itô's formula: <math>df(X_t,t) = \bigl(\frac{\partial f}{\partial t} + \sum_{i=1}^d b^i \frac{\partial f}{\partial x_i} + v \sum_{i,j=1}^d a^{ij}\,\frac{\partial^2 f}{\partial x_i \partial x_j}\bigr)\,dt + \sum_{i,k=1}^d \frac{\partial f}{\partial x_i}\,\sigma^i_k\,dB_t^k.</math> Rearranging gives <math>f(X_t,t) - f(X_\tau,\tau) - \int_\tau^t \bigl(\frac{\partial f}{\partial s} + L_{a;b}f\bigr)\,ds = \int_\tau^t \sum_{i,k=1}^d \frac{\partial f}{\partial x_i}\,\sigma^i_k\,dB_s^k,</math> whose right‐hand side is a local martingale, matching the local‐martingale property in the diffusion definition. The law of <math>X_t</math> defines <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> on <math>\Omega = C([0,\infty), \mathbb{R}^d)</math> with the correct initial condition and local martingale property. Uniqueness follows from the Lipschitz continuity of <math>\sigma\!,\!b</math>. In fact, <math>L_{a;b} + \tfrac{\partial}{\partial s}</math> coincides with the infinitesimal generator <math>\mathcal{A}</math> of this process. If <math>X_t</math> solves the SDE, then for <math>f(\mathbf{x},t) \in C^2(\mathbb{R}^d \times \mathbb{R}^+)</math>, the generator <math>\mathcal{A}</math> is <math>\mathcal{A}f(\mathbf{x},t) = \sum_{i=1}^d b_i(\mathbf{x},t)\,\frac{\partial f}{\partial x_i} + v\sum_{i,j=1}^d a_{ij}(\mathbf{x},t)\,\frac{\partial^2 f}{\partial x_i \partial x_j} + \frac{\partial f}{\partial t}.</math>

== See also ==
* [Stochastic differential equation](/source/Stochastic_differential_equation)
* [Itô calculus](/source/It%C3%B4_calculus)
* [Fokker–Planck equation](/source/Fokker%E2%80%93Planck_equation)
* [Markov process](/source/Markov_process)
* [Diffusion](/source/Diffusion)
* [Itô diffusion](/source/It%C3%B4_diffusion)
* [Jump diffusion](/source/Jump_diffusion)
* [Sample-continuous process](/source/Sample-continuous_process)

== References ==
{{Reflist}}

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Category:Markov processes

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Adapted from the Wikipedia article [Diffusion process](https://en.wikipedia.org/wiki/Diffusion_process) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Diffusion_process?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.