Fokker–Planck equation - Wikipedia
stochastic-processespartial-differential-equationsstatistical-mechanicsprobability
Abstraction: PDE describing time evolution of probability density under drift and diffusion
Key points:
- Describes the time evolution of the probability density function of a particle's position or velocity under drag and random forces (Brownian motion)
- Named after Adriaan Fokker (1914) and Max Planck (1917); also independently discovered by Kolmogorov (1931) as the "Kolmogorov forward equation"
- Derived from Ito stochastic differential equations; equivalent to the Smoluchowski diffusion equation for overdamped systems
- In one dimension: ∂p/∂t = -∂[D₁p]/∂x + ∂²[D₂p]/∂x² where D₁ is drift and D₂ is diffusion coefficient
- Applications span statistical mechanics, finance (local volatility/option pricing), plasma physics, and data science
- Stationary solution found by setting ∂p/∂t = 0; overdamped Langevin equation yields Boltzmann equilibrium distribution
Connections: Fokker Planck Equation · Stochastic Processes · Brownian Motion · Probability Density
Source: http://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation