Editing Fokker–Planck equation (section)

===Ornstein–Uhlenbeck process===

The [[Ornstein–Uhlenbeck process]] is a process defined as

<math display="block">dX_t = -a X_t \, dt + \sigma \, dW_t.</math>

with <math>a>0</math>. Physically, this equation can be motivated as follows: a particle of mass <math> m </math> with velocity <math> V_t</math> moving in a medium, e.g., a fluid, will experience a friction force which resists motion whose magnitude can be approximated as being proportional to particle's velocity <math> -a V_t</math> with <math> a = \mathrm{constant} </math>.  Other particles in the medium will randomly kick the particle as they collide with it and this effect can be approximated by a white noise term; <math> \sigma (d W_t/dt) </math>. Newton's second law is written as

<math display="block"> m \frac{dV_t}{dt}=-a V_t +\sigma \frac{dW_t}{dt}. </math>

Taking <math> m = 1</math> for simplicity and changing the notation as <math> V_t\rightarrow X_t</math> leads to the Ornstein–Uhlenbeck form. The corresponding Fokker–Planck equation is
<math display="block">
\frac{\partial p(x,t)}{\partial t} = a \frac{\partial}{\partial x}\left(x \,p(x,t)\right) + \frac{\sigma^2}{2} \frac{\partial^2 p(x,t)}{\partial x^2},
</math>

The stationary solution <math>(\partial_t p = 0)</math> is
<math display="block">p_{\text{ss}}(x) = \sqrt{\frac{a}{\pi \sigma^2}} e^{-{ax^2}/{\sigma^2}}.</math>