Editing Fokker–Planck equation (section)

==One dimension==
In one spatial dimension ''x'', for an [[Itô calculus|Itô process]] driven by the standard [[Wiener process]] <math>W_t</math> and described by the [[stochastic differential equation]] (SDE)
<math display="block">dX_t = \mu(X_t, t) \,dt + \sigma(X_t, t) \,dW_t</math>
with [[Drift velocity|drift]] <math>\mu(X_t, t)</math> and [[diffusion]] coefficient <math>D(X_t, t) = \sigma^2(X_t, t)/2</math>, the Fokker–Planck equation for the probability density <math>p(x, t)</math> of the random variable <math>X_t</math> is <ref>{{Citation |title=The Fokker–Planck Equation: Methods of Solution and Applications |last=Risken |first=H. |volume=Second Edition, Third Printing |pages=72 |date=1996 |publication-date=1996}}</ref>{{Equation box 1|cellpadding|border|indent=:|equation=<math> \frac{\partial}{\partial t} p(x, t) = -\frac{\partial}{\partial x}\left[\mu(x, t) p(x, t)\right] + \frac{\partial^2}{\partial x^2}\left[D(x, t) p(x, t)\right]. </math>|border colour=#0073CF|background colour=#F5FFFA}}{{hidden begin
|title = Link between the Itô SDE and the Fokker–Planck equation
}}
In the following, use <math>\sigma = \sqrt{2D}</math>.

Define the [[Infinitesimal generator (stochastic processes)|infinitesimal generator]] <math>\mathcal{L}</math> (the following can be found in Ref.<ref name=ottinger>{{cite book|last=Öttinger|first=Hans Christian|title=Stochastic Processes in Polymeric Fluids|date=1996|publisher=Springer-Verlag|location=Berlin-Heidelberg|isbn=978-3-540-58353-0|page=75}}</ref>):
<math display="block">
 \mathcal{L}p(X_t) = \lim_{\Delta t \to 0} \frac1{\Delta t}\left(\mathbb{E}\big[p(X_{t + \Delta t}) \mid X_t = x \big] - p(x)\right).
</math>

The ''transition probability'' <math>\mathbb{P}_{t, t'}(x \mid x')</math>, the probability of going from <math>(t', x')</math> to <math>(t, x)</math>, is introduced here; the expectation can be written as
<math display="block">
 \mathbb{E}(p(X_{t + \Delta t}) \mid X_t = x) = \int p(y) \, \mathbb{P}_{t + \Delta t,t}(y \mid x) \,dy.
</math>
Now we replace in the definition of <math>\mathcal{L}</math>, multiply by <math>\mathbb{P}_{t, t'}(x \mid x')</math> and integrate over <math>dx</math>. The limit is taken on
<math display="block">
 \int p(y) \int \mathbb{P}_{t + \Delta t, t}(y \mid x)\,\mathbb{P}_{t, t'}(x \mid x') \,dx \,dy - \int p(x) \, \mathbb{P}_{t, t'}(x \mid x') \,dx.
</math>
Note now that
<math display="block">
 \int \mathbb{P}_{t + \Delta t, t}(y \mid x) \, \mathbb{P}_{t, t'}(x \mid x') \,dx = \mathbb{P}_{t + \Delta t, t'}(y \mid x'),
</math>
which is the Chapman–Kolmogorov theorem. Changing the dummy variable <math>y</math> to <math>x</math>, one gets
<math display="block">
\begin{align}
 \int p(x) \lim_{\Delta t \to 0} \frac1{\Delta t} \left( \mathbb{P}_{t + \Delta t, t'}(x \mid x') - \mathbb{P}_{t, t'}(x \mid x') \right) \,dx,
\end{align}
</math>
which is a time derivative. Finally we arrive to
<math display="block">
 \int [\mathcal{L}p(x)] \mathbb{P}_{t, t'}(x \mid x') \,dx = \int p(x) \, \partial_t \mathbb{P}_{t, t'}(x \mid x') \,dx.
</math>
From here, the Kolmogorov backward equation can be deduced. If we instead use the adjoint operator of <math>\mathcal{L}</math>, <math>\mathcal{L}^\dagger</math>, defined such that
<math display="block">
 \int [\mathcal{L}p(x)] \mathbb{P}_{t, t'}(x \mid x') \,dx = \int p(x) [\mathcal{L}^\dagger \mathbb{P}_{t, t'}(x \mid x')] \,dx,
</math>
then we arrive to the Kolmogorov forward equation, or Fokker–Planck equation, which, simplifying the notation <math>p(x, t) = \mathbb{P}_{t, t'}(x \mid x')</math>, in its differential form reads
<math display="block">
 \mathcal{L}^\dagger p(x, t) = \partial_t p(x, t).
</math>

Remains the issue of defining explicitly <math>\mathcal{L}</math>. This can be done taking the expectation from the integral form of the [[Itô's lemma]]:
<math display="block">
 \mathbb{E}\big(p(X_t)\big) = p(X_0) + \mathbb{E}\left(\int_0^t \left(\partial_t + \mu\partial_x + \frac{\sigma^2}{2}\partial_x^2 \right) p(X_{t'}) \,dt'\right).
</math>

The part that depends on <math>dW_t</math> vanished because of the martingale property.

Then, for a particle subject to an Itô equation, using
<math display="block">
 \mathcal{L} = \mu\partial_x + \frac{\sigma^2}{2}\partial_x^2,
</math>
it can be easily calculated, using integration by parts, that
<math display="block">
 \mathcal{L}^\dagger = -\partial_x(\mu \cdot) + \frac12 \partial_x^2(\sigma^2 \cdot),
</math>
which bring us to the Fokker–Planck equation:
<math display="block">
 \partial_t p(x, t) = -\partial_x \big(\mu(x, t) \cdot p(x, t)\big) + \partial_x^2\left(\frac{\sigma(x, t)^2}{2} \, p(x,t)\right).
</math>

{{hidden end}}

While the Fokker–Planck equation is used with problems where the initial distribution is known, if the problem is to know the distribution at previous times, the [[Feynman–Kac formula]] can be used, which is a consequence of the Kolmogorov backward equation.

The stochastic process defined above in the Itô sense can be rewritten within the [[Stratonovich integral|Stratonovich convention]] as a Stratonovich SDE:
<math display="block">dX_t = \left[\mu(X_t, t) - \frac{1}{2} \frac{\partial}{\partial X_t}D(X_t, t)\right] \,dt + \sqrt{2 D(X_t, t)} \circ dW_t.</math>
In this form, a noise-induced drift term due to diffusion gradient effects is explicitly visible, arising when the noise is state-dependent. This formulation is commonly used in physics, as it makes for a more intuitive connection to physical processes. It is equivalent to the Itô SDE; any Itô SDE can be converted to Stratonovich form, and ''vice versa''.

The zero-drift equation with constant diffusion can be considered as a model of classical [[Brownian motion]]:
<math display="block">\frac{\partial}{\partial t} p(x, t) = D_0\frac{\partial^2}{\partial x^2}\left[p(x, t)\right].</math>

This model has discrete spectrum of solutions if the condition of fixed boundaries is added for <math>\{0 \leq x \leq L\}</math>:
<math display="block">\begin{align}
p(0, t) &= p(L, t) = 0, \\
p(x, 0) &= p_0(x).
\end{align}</math>

It has been shown<ref name=kam2014>{{cite journal | last = Kamenshchikov | first = S. | title = Clustering and Uncertainty in Perfect Chaos Systems| journal = Journal of Chaos | volume = 2014 | pages = 1–6 | year = 2014 | doi=10.1155/2014/292096| arxiv = 1301.4481 | s2cid = 17719673 | doi-access = free }}</ref> that in this case an analytical spectrum of solutions allows deriving a local uncertainty relation for the coordinate-velocity phase volume:
<math display="block"> \Delta x \, \Delta v \geq D_0. </math>
Here <math>D_0</math> is a minimal value of a corresponding diffusion spectrum <math>D_j</math>, while <math>\Delta x</math> and <math>\Delta v</math> represent the uncertainty of coordinate–velocity definition.