FokkerPlanck
The Fokker-Planck equation is a partial differential equation that describes the time evolution of the probability density function of a continuous stochastic process. It is closely associated with diffusion and drift phenomena and is named after Adriaan Fokker and Max Planck, who developed and analyzed diffusion-type descriptions in the early 20th century. The equation can be derived from an underlying stochastic differential equation, such as a Langevin equation, and it provides a deterministic description of how probabilities spread over time.
In one spatial dimension, if a stochastic process Xt satisfies dXt = a(Xt,t) dt + b(Xt,t) dWt, where
∂t p(x,t) = - ∂x [ a(x,t) p(x,t) ] + (1/2) ∂x^2 [ b(x,t)^2 p(x,t) ].
Equivalently, with A(x,t) as drift and D(x,t) as diffusion,
∂t p = - ∂x [ A p ] + ∂x^2 [ D p ].
The (1/2) factor reflects Ito calculus conventions.
In multiple dimensions, with drift vector A(x,t) and diffusion matrix D(x,t), the equation becomes
∂t p = - ∑i ∂i [ Ai p ] + ∑i ∑j ∂i ∂j [ Dij p ].
The Fokker-Planck equation governs many physical, chemical, biological, and financial systems where stochastic dynamics lead to
Typical uses include finding stationary distributions, describing relaxation to equilibrium, and analyzing time-dependent probabilities in diffusion-reaction
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