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Linear and Non-linear Fokker–Planck Equations

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Let \( \smash{ {\hat X} } \) denote a stochastic process defined on the space O and the time interval \( { [t_0,\infty] } \), where t 0 denotes the initial time of the process. We assume that the process \( \smash{ {\hat X} } \) can be described in terms of a random variable \( \smash{ X \in \Omega } \). More precisely, let X(t) denote the time-dependent evolution of the random variable X for \( \smash{ t \ge t_0 } \). Then, we assume that the process \( \smash{ {\hat X} } \) can be described in terms of the infinitely large set of realizations \( \smash{ X^{(i)}(t) } \) of X(t) with \( \smash{ i=1,2,\dots } \). The realizations \( \smash{ i=1,2,\dots } \) constitute a statistical ensemble. At every time t the probability density P of \( \smash{ {\hat X} } \) can be computed from the realizations \( \smash{ X^{(i)}(t) } \), that is, from the ensemble by means of

$$ P(x,t) = \left\langle{\delta(x-X(t))}\right\rangle \:, $$
(1)...

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Abbreviations

Linear:

Here, linear with respect to a probability density.

Nonlinear:

Here, nonlinear with respect to a probability density.

Markov process:

Process for which it is sufficient to have information about the presence in order to make best predictions about the future. Additional information about the past will not improve the predictions.

Martingale process \( \smash{ {\hat Z} } \) :

Process for which the future mean value \( \smash{ \langle {Z(t+\Delta t)}\rangle } \) of a set of realizations \( \smash{ Z^{(i)} } \) that is passing at presence t through a certain common state z is the state z: \( \smash{ \langle {Z(t+\Delta t)}\rangle_{Z(t)=z} = z } \). Additional information about states \( \smash{ z^{\prime} } \) visited at times \( \smash{ t^{\prime} } \) prior to t is irrelevant.

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  1. University of Connecticut, Storrs, USA

    Till D. Frank

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  1. Till D. Frank
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Frank, T.D. (2009). Linear and Non-linear Fokker–Planck Equations. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_311

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