Fokker-Planck equation

Set

context	$n\in\mathbb N$
context	$\mu:C(\mathbb R^n,\mathbb R^n)$
context	$\mathsf{D}:C^2(\mathbb R^n,\mathbb R^{n^2})$
range	$::\mu(\mathbf{x})$
range	$::\mathsf{D}(\mathbf{x})$

definiendum

$f \in \mathrm{it}$

postulate	$f:C^2(\mathbb R^n\times\mathbb R,\mathbb R)$
range	$::f(\mathbf{x},t)$

postulate

$\frac{\partial }{\partial t} f = -\mathrm{div} (\mu \cdot f) + \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{\partial^2}{\partial x_i \, \partial x_j} (\mathsf{D}_{ij}\cdot f)$

Discussion

Among other things, the Fokker-Planck equation also describes the evolution of the probability density of a Wiener process.

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Reference

Wikipedia: Fokker-Planck equation

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Fokker-Planck equation

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Fokker-Planck equation

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