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The probability-density equation for the Ornstein-Uhlenbeck process with drift $-v$ and diffusion coefficient $\sqrt{2}$. The unit-covariance Gaussian density is stationary because its Fokker-Planck probability current vanishes. This normalization makes mean velocity decay at rate one and energy excess decay at rate two.

Put $q(x)=\sigma (x{)}^{2}p(x)$. The explicit density formula and local integrability assumption giveHence its Fokker-Planck probability current $bp-q^{\prime }/2$ is zero. The stationary adjoint equation is $L^{\ast }p=-(bp{)}^{\prime }+q^{\prime \prime }/2=0$. We prove the required integral identity with cutoffs, avoiding an unstated boundary condition on derivatives of $u$.

Choose smooth $0\le {\chi }_R\le 1$, equal to one on $[-R,R]$, supported in $[-2R,2R]$, with $|{\chi }_R^{\prime }|\le C_1/R$ and $|{\chi }_R^{\prime \prime }|\le C_2/R^2$. Since $pLu=(q{u}_x{)}^{\prime }/2$, integrate the backward equation in time and integrate by parts twice in space:These integrations have compact support, as permitted in the question. Bounded coefficients and normalized $p$ giveThe right side is bounded in absolute value bywhich tends to zero. Dominated convergence on the left therefore provesThis is the cutoff proof of invariance for a zero-flux diffusion density. In particular, if an independent initial state has density $p$, part (a) and Fubini give a constant expectation of this test function at every time. This consequence alone is sufficient for the final coupling argument.

For the normalized standard Gaussian density,The two terms cancel, so the Ornstein-Uhlenbeck Fokker-Planck equation has stationary density for a Fokker-Planck equationEquivalently its Fokker-Planck probability current $-\nabla \gamma -v\gamma$ vanishes identically. The normalization follows from the Gaussian integral in each coordinate.

Here $\Gamma >0$ is a kinetic coefficient, and the additive Gaussian white noise must be centered. A rigorous starting point is a finite spatial discretization or finite set of modes of the coarse-grained field. For each field coordinate, the Fokker-Planck probability current has the continuum notationThe equilibrium Boltzmann distribution ${\mathcal{P}}_{\mathrm{eq}}\propto e^{-\beta F}$ has $\delta {\mathcal{P}}_{\mathrm{eq}}/\delta \varphi =-\beta (\delta F/\delta \varphi ){\mathcal{P}}_{\mathrm{eq}}$. Detailed balance for this time-even field requires zero configuration-space current. ThereforeThis is the Model A fluctuation-dissipation relation. Conversely, that noise strength makes the equilibrium current vanish and gives reversible relaxational dynamics when the equilibrium measure is normalizable.

Alternatively, the Onsager--Machlup path probability for Model A dynamics compares the squared required noise histories $\dot{\varphi }+\Gamma \delta F/\delta \varphi$ and $-\dot{\varphi }+\Gamma \delta F/\delta \varphi$. Their difference gives $\log ({\mathbb{P}}_F/{\mathbb{P}}_B)=-2\Gamma \Delta F/{\sigma }^2$, using the functional chain rule. The midpoint convention and time-reversal invariance of a path Jacobian justify cancellation of the common Jacobian. Comparing with the conditional path ratio gives the same result. The spatial white-noise continuum is a coarse-grained idealization: retain a microscopic cutoff when pointwise fluctuations or ultraviolet divergences matter.

With $A_i=a_{ij}{x}_j$ and constant $B_{ij}=b_{ij}$, the Fokker-Planck probability current becomesIn the stationary state we impose the stronger zero-current condition $J_i=0$, corresponding to detailed balance.

For a sufficiently regular Itô diffusion with $D=\sigma {\sigma }^T/2$, a density $\pi$ is stationary if and only if it solves the stationary Fokker-Planck equationwith integrable Fokker-Planck probability current and boundary conditions that make its outward flux vanish. Here $(\nabla \cdot (D\pi ){)}_i={\sum }_j{\partial }_j(D_{ij}\pi )$.

For the displayed parametrization, assumewhere $Z<\infty$, that $D(x)$ is symmetric positive semidefinite, and that $Q(x)$ is antisymmetric. Substituting $\nabla \pi =-\pi \nabla H$ and the stated $\Gamma$ into the probability current cancels all $D$ terms. The remaining divergence isbecause the second derivatives are symmetric in $i,j$ whereas $Q_{ij}=-Q_{ji}$. Thus these conditions, together with the boundary and regularity assumptions, imply stationarity. More generally, the divergence equation above is the exact necessary and sufficient condition; within this construction, $\pi \propto e^{-H}$ and antisymmetric $Q$ are the standard way to satisfy it.

This is a birth-death process with birth and death ratesIts transition diagram has the two arrowsThe gain into state $n$ comes from a death in state $n+1$ or a birth in state $n-1$, while the loss is the sum of both rates out of state $n$. The birth-death master equation is consequently

Multiplying by $n$, summing over the nonnegative integers, and shifting the summation indices shows that each birth contributes $+1$ and each death contributes $-1$. Thus the first-moment equation of a birth-death process givesAt a stationary state, use the variance identity $⟨n^2⟩=⟨n{⟩}^2+(\Delta n{)}^2$. Solving the resulting quadratic equation givesThe minus root is inadmissible whenever it is negative, namely when $(\Delta n{)}^2<\gamma /\alpha$. Equality would give zero mean, which is also incompatible with a positive immigration rate $\gamma$, so for $\gamma >0$ the minus branch requires $(\Delta n{)}^2>\gamma /\alpha$ even to be a possible mean.

For a continuum approximation, write $b(n)=\gamma +\beta n$ and $d(n)=\alpha n(n-1)$. Applying the Kramers-Moyal expansion to the two gain terms and retaining derivatives through second order gives the Fokker-Planck equationwhere the negative drift and infinitesimal jump variance areThis truncation requires the typical population and the scale on which $P$ varies to be much larger than the unit jump size, with the rates varying smoothly across that scale.

The positive zero of $g$ isso in particular $n_{\ast }\sim \beta /\alpha$ as $\alpha /\beta \to 0$. Put $x=n-n_{\ast }$. The linear noise approximation useswhereand, because $b(n_{\ast })=d(n_{\ast })$,In a stationary state with zero Fokker-Planck probability current,Its normalized solution is the normal distributionIt follows that

These estimates agree with the exact stationary first-moment relation on its plus branch to leading order: inserting $(\Delta n{)}^2\sim \beta /\alpha$ gives $⟨n⟩\sim \beta /\alpha$. Moreover,Thus the stationary mass lies far from the boundary $n=0$, is narrow relative to its mean, yet changes across many lattice sites. These are precisely the large-population and slow-variation conditions needed for the diffusion approximation.

The two-dimensional Fokker-Planck equation iswith Fokker-Planck probability currentThe point initial condition iswhere $\delta$ is the Dirac delta function. The reflecting boundary condition for a diffusion iswith $\mathbf{n}$ the outward unit normal. This zero-flux condition conserves the integral of the probability density function over the square.

SetThe stated curl-free condition says that $C_j=({\kappa }^{-1}{)}_{ji}{a}_i$ is a gradient, $C=\nabla \Psi$. DefineThen $a=-D\nabla V$, and the Fokker-Planck probability current becomesThis is the potential condition for a Fokker--Planck equation.

A reflecting boundary confines a diffusion without losing probability. Its forward condition is zero outward Fokker-Planck probability current, $J\cdot n=0$, and its backward condition is a zero conormal derivative.

A probability density left unchanged by the Fokker-Planck equation ${\partial }_{t}f=L^{\ast }f$. Equivalently its Fokker-Planck probability current has zero divergence; the current itself need not vanish. For the Ornstein-Uhlenbeck Fokker-Planck equation, the normalized standard Gaussian density has zero current and is stationary.