Created: August 28, 2022
Modified: August 28, 2022

Fokker-Planck

This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.

Given a diffusion process specified by the stochastic differential equation

dx_t = \alpha_t(x_t) dt + \beta_t(x_t) dW_t,

the Fokker-Planck equation aka Kolmogorov forward equation describes the time evolution of the transition density $p_{t|s}(x_t | x_s)$ :

\frac{\partial}{\partial t}p_t(x_t) = -\frac{\partial}{\partial x_t} \left\{\alpha_t(x_t) p_t(x_t)\right\} + \frac{1}{2}\frac{\partial^2}{\partial x_t^2}\left\{\beta^2_t(x_t)p_t(x_t)\right\}

To see this, consider the discrete-time case where

x_{t+1} \sim \mathcal{N}\left(x_t + \alpha_t(x_t), \beta_t^2(x_t)\right)

Fokker-Planck

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Fokker-Planck

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