Fokker–Planck equation

File:Linear Potential2.gif

A solution to the one-dimensional Fokker–Planck equation, with both the drift and the diffusion term. In this case the initial condition is a Dirac delta function centered away from zero velocity. Over time the distribution widens due to random impulses.

In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well.<ref>Template:Cite book</ref> The Fokker–Planck equation has multiple applications in information theory, graph theory, data science, finance, economics etc.

It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931.<ref>Template:Cite journal</ref> When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski),<ref>Template:Cite book</ref> and in this context it is equivalent to the convection–diffusion equation. When applied to particle position and momentum distributions, it is known as the Klein–Kramers equation. The case with zero diffusion is the continuity equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion.<ref>Template:Cite book</ref>

The first consistent microscopic derivation of the Fokker–Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov.<ref>N. N. Bogolyubov Jr. and D. P. Sankovich (1994). "N. N. Bogolyubov and statistical mechanics". Russian Math. Surveys 49(5): 19—49. {{#invoke:doi|main}}</ref><ref>N. N. Bogoliubov and N. M. Krylov (1939). Fokker–Planck equations generated in perturbation theory by a method based on the spectral properties of a perturbed Hamiltonian. Zapiski Kafedry Fiziki Akademii Nauk Ukrainian SSR 4: 81–157 (in Ukrainian).</ref>

One dimension

In one spatial dimension x, for an 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 <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>Template:Citation</ref>Template:Equation box 1Template:Hidden begin In the following, use <math>\sigma = \sqrt{2D}</math>.

Define the infinitesimal generator <math>\mathcal{L}</math> (the following can be found in Ref.<ref name=ottinger>Template:Cite book</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>

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 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>Template:Cite journal</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.

Higher dimensions

More generally, if

<math display="block">d\mathbf{X}_t = \boldsymbol{\mu}(\mathbf{X}_t,t)\,dt + \boldsymbol{\sigma}(\mathbf{X}_t,t)\,d\mathbf{W}_t,</math>

where <math>\mathbf{X}_t</math> and <math>\boldsymbol{\mu}(\mathbf{X}_t,t)</math> are Template:Mvar-dimensional vectors, <math>\boldsymbol{\sigma}(\mathbf{X}_t,t)</math> is an <math>N \times M</math> matrix and <math>\mathbf{W}_t</math> is an M-dimensional standard Wiener process, the probability density <math>p(\mathbf{x},t)</math> for <math>\mathbf{X}_t</math> satisfies the Fokker–Planck equationTemplate:Equation box 1with drift vector <math>\boldsymbol{\mu} = (\mu_1,\ldots,\mu_N)</math> and diffusion tensor <math display="inline">\mathbf{D} = \frac{1}{2} \boldsymbol{\sigma\sigma}^\mathsf{T}</math>, i.e.<math display="block">D_{ij}(\mathbf{x},t) = \frac{1}{2}\sum_{k=1}^M \sigma_{ik}(\mathbf{x},t) \sigma_{jk}(\mathbf{x},t).</math>

If instead of an Itô SDE, a Stratonovich SDE is considered,

<math display="block">d\mathbf{X}_t = \boldsymbol{\mu}(\mathbf{X}_t,t)\,dt + \boldsymbol{\sigma}(\mathbf{X}_t,t)\circ d\mathbf{W}_t,</math>

the Fokker–Planck equation will read:<ref name=ottinger/>Template:Rp

<math display="block">\frac{\partial p(\mathbf{x},t)}{\partial t} = -\sum_{i=1}^N \frac{\partial}{\partial x_i} \left[ \mu_i(\mathbf{x},t) \, p(\mathbf{x},t) \right] + \frac{1}{2} \sum_{k=1}^M \sum_{i=1}^{N} \frac{\partial}{\partial x_i} \left\{ \sigma_{ik}(\mathbf{x},t) \sum_{j=1}^{N} \frac{\partial}{\partial x_j} \left[ \sigma_{jk}(\mathbf{x},t) \, p(\mathbf{x},t) \right] \right\}</math>

Generalization

In general, the Fokker–Planck equations are a special case to the general Kolmogorov forward equation

<math display="block">\partial_t \rho = \mathcal{A}^*\rho</math>

where the linear operator <math>\mathcal{A}^*</math> is the Hermitian adjoint to the infinitesimal generator for the Markov process.<ref>Template:Cite book</ref>

Examples

The Fokker–Planck equation encompasses a variety of more specific situations and contexts, which appear as special cases.

Wiener process

A standard scalar Wiener process is generated by the stochastic differential equation

<math display="block">dX_t = dW_t.</math>

Here the drift term is zero and the diffusion coefficient is 1/2. Thus the corresponding Fokker–Planck equation is

<math display="block"> \frac{\partial p(x,t)}{\partial t} = \frac{1}{2} \frac{\partial^2 p(x,t)}{\partial x^2}, </math>

which is the simplest form of a diffusion equation. If the initial condition is <math>p(x,0) = \delta(x)</math>, the solution is

<math display="block"> p(x,t) = \frac{1}{\sqrt{2 \pi t}}e^{-{x^2}/({2t})}.</math>

Boltzmann distribution at the thermodynamic equilibrium

The overdamped Langevin equation

<math>dX_t = -\frac{1}{k_\text{B}T} \left(\nabla_x U\bigg\vert_{x=X_t}\right) dt + dW_t</math>

leads to

<math>\partial_t p = \frac 1 2 \nabla\cdot \left(\frac{p}{k_\text{B}T} \nabla U + \nabla p\right).</math>

The Boltzmann distribution

<math>p(x) \propto e^{- U(x)/k_\text{B} T}</math>

is an equilibrium distribution, and assuming <math>U</math> grows sufficiently rapidly (that is, the potential well is deep enough to confine the particle), the Boltzmann distribution is the unique equilibrium.

Ornstein–Uhlenbeck process

The Ornstein–Uhlenbeck process is a process defined as

<math display="block">dX_t = -a X_t \, dt + \sigma \, dW_t.</math>

with <math>a>0</math>. Physically, this equation can be motivated as follows: a particle of mass <math> m </math> with velocity <math> V_t</math> moving in a medium, e.g., a fluid, will experience a friction force which resists motion whose magnitude can be approximated as being proportional to particle's velocity <math> -a V_t</math> with <math> a = \mathrm{constant} </math>. Other particles in the medium will randomly kick the particle as they collide with it and this effect can be approximated by a white noise term; <math> \sigma (d W_t/dt) </math>. Newton's second law is written as

<math display="block"> m \frac{dV_t}{dt}=-a V_t +\sigma \frac{dW_t}{dt}. </math>

Taking <math> m = 1</math> for simplicity and changing the notation as <math> V_t\rightarrow X_t</math> leads to the Ornstein–Uhlenbeck form. The corresponding Fokker–Planck equation is <math display="block"> \frac{\partial p(x,t)}{\partial t} = a \frac{\partial}{\partial x}\left(x \,p(x,t)\right) + \frac{\sigma^2}{2} \frac{\partial^2 p(x,t)}{\partial x^2}, </math>

The stationary solution <math>(\partial_t p = 0)</math> is <math display="block">p_{\text{ss}}(x) = \sqrt{\frac{a}{\pi \sigma^2}} e^{-{ax^2}/{\sigma^2}}.</math>

Plasma physics

In plasma physics, the distribution function <math>p_s (\mathbf{x},\mathbf{v},t)</math> for a particle species <math>s</math> takes the place of the probability density function. The corresponding Boltzmann equation is given by

<math display="block">\frac{\partial p_s}{\partial t} + \mathbf{v} \cdot \boldsymbol{\nabla} p_s + \frac{Z_s e}{m_s} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right) \cdot \boldsymbol{\nabla}_v p_s = -\frac{\partial}{\partial v_i} \left(p_s \langle\Delta v_i\rangle\right) + \frac{1}{2} \frac{\partial^2}{\partial v_i \, \partial v_j} \left(p_s \langle\Delta v_i \, \Delta v_j\rangle\right),</math>

where the third term includes the particle acceleration due to the Lorentz force and the Fokker–Planck term at the right-hand side represents the effects of particle collisions. The quantities <math>\langle\Delta v_i\rangle</math> and <math>\langle\Delta v_i \, \Delta v_j\rangle</math> are the average change in velocity a particle of type <math>s</math> experiences due to collisions with all other particle species in unit time. Expressions for these quantities are given elsewhere.<ref name="Rosenbluth">Template:Cite journal</ref> If collisions are ignored, the Boltzmann equation reduces to the Vlasov equation.

Smoluchowski diffusion equation

The Smoluchowski diffusion equation is effectively equivalent to the convection–diffusion equation. Consider an overdamped Brownian particle under external force <math>F(r)</math>:<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>m\ddot{r} = - \gamma \dot{r} + F(r) + \sigma \xi(t)</math>

where the <math>m\ddot r</math> term is negligible (the meaning of "overdamped"). Thus, it is just

<math>\gamma \, dr = F(r)\, dt + \sigma \, dW_t.</math>

The Fokker–Planck equation for this particle is the Smoluchowski diffusion equation:

<math>\partial_t P(r,t| r_0, t_0) = \nabla \cdot \left[D (\nabla - \beta F(r)) P(r,t| r_0, t_0)\right] </math>

Here, <math>D</math> is the diffusion constant and <math>\beta = 1 / k_\text{B} T</math>. The importance of this equation is it allows for both the inclusion of the effect of temperature on the system of particles and a spatially dependent diffusion constant.

Template:Hidden begin Starting with the Langevin Equation of a Brownian particle in external field <math>F(r)</math>, where <math>\gamma</math> is the friction term, <math>\xi</math> is a fluctuating force on the particle, and <math>\sigma</math> is the amplitude of the fluctuation.

<math display="block">m\ddot{r} = - \gamma \dot{r} + F(r) + \sigma \xi(t)</math>

At equilibrium the frictional force is much greater than the inertial force, <math>\left\vert \gamma \dot{r} \right\vert \gg \left\vert m \ddot{r} \right\vert</math>. Therefore, the Langevin equation becomes,

<math display="block">\gamma \dot{r} = F(r) + \sigma \xi(t)</math>

Which generates the following Fokker–Planck equation,

<math display="block">\partial_t P(r,t|r_0,t_0) = \left(\nabla^2\frac{\sigma^2}{2 \gamma^2} - \nabla \cdot \frac{F(r)}{\gamma}\right) P(r,t|r_0,t_0) </math>

Rearranging the Fokker–Planck equation,

<math display="block">\partial_t P(r,t|r_0,t_0)= \nabla \cdot \left( \nabla D- \frac{F(r)}{\gamma}\right) P(r,t|r_0,t_0)</math>

Where <math>D = \frac{\sigma^2}{2 \gamma^2}</math>. Note, the diffusion coefficient may not necessarily be spatially independent if <math>\sigma</math> or <math>\gamma</math> are spatially dependent.

Next, the total number of particles in any particular volume is given by,

<math display="block">N_V (t| r_0, t_0) = \int_V dr P(r,t|r_0,t_0)</math>

Therefore, the flux of particles can be determined by taking the time derivative of the number of particles in a given volume, plugging in the Fokker–Planck equation, and then applying Gauss's Theorem.

<math display="block">\partial_t N_V (t|r_0, t_0) = \int_V dV \nabla \cdot\left( \nabla D- \frac{F(r)}{\gamma}\right) P(r,t|r_0, t_0) = \int_{\partial V} d\mathbf{a} \cdot j(r,t|r_0, t_0)</math>

<math display="block">j(r,t|r_0, t_0) = \left( \nabla D- \frac{F(r)}{\gamma}\right)P(r,t|r_0, t_0)</math>

In equilibrium, it is assumed that the flux goes to zero. Therefore, Boltzmann statistics can be applied for the probability of a particles location at equilibrium, where <math>F(r) = -\nabla U(r)</math> is a conservative force and the probability of a particle being in a state <math>r</math> is given as <math>P(r,t|r_0, t_0) = \frac{e^{-\beta U(r)}}{Z}</math>.

<math display="block">j(r,t|r_0, t_0) = \left( \nabla D- \frac{F(r)}{\gamma}\right)\frac{e^{-\beta U(r)}}{Z} = 0</math>

<math display="block">\Rightarrow \nabla D = F(r) \left(\frac{1}{\gamma} - D \beta\right)</math>

This relation is a realization of the fluctuation–dissipation theorem. Now applying <math> \nabla \cdot \nabla </math> to <math>D P(r,t|r_0, t_0)</math> and using the Fluctuation-dissipation theorem,

<math display="block">\begin{align} \nabla \cdot \nabla D P(r,t|r_0,t_0) &= \nabla \cdot D \nabla P(r,t|r_0,t_0)+ \nabla \cdot P(r,t|r_0,t_0) \nabla D \\ &=\nabla \cdot D \nabla P(r,t|r_0,t_0)+\nabla \cdot P(r,t|r_0,t_0) \frac{F(r)}{\gamma} - \nabla \cdot P(r,t|r_0,t_0) D \beta F(r) \end{align}</math>

Rearranging,

<math display="block"> \Rightarrow \nabla \cdot \left( \nabla D- \frac{F(r)}{\gamma}\right)P(r,t|r_0,t_0)= \nabla \cdot D(\nabla-\beta F(r)) P(r,t|r_0,t_0)</math>

Therefore, the Fokker–Planck equation becomes the Smoluchowski equation, <math display="block">\partial_t P(r,t| r_0, t_0) = \nabla \cdot D (\nabla - \beta F(r)) P(r,t| r_0, t_0) </math> for an arbitrary force <math>F(r)</math>.Template:Hidden end

Computational considerations

Brownian motion follows the Langevin equation, which can be solved for many different stochastic forcings with results being averaged (canonical ensemble in molecular dynamics). However, instead of this computationally intensive approach, one can use the Fokker–Planck equation and consider the probability <math>p(\mathbf{v}, t)\,d\mathbf{v}</math> of the particle having a velocity in the interval <math>(\mathbf{v}, \mathbf{v} + d\mathbf{v})</math> when it starts its motion with <math>\mathbf{v}_0</math> at time 0.

File:Linear Potential2.gif

Brownian dynamics simulation for particles in 1-D linear potential compared with the solution of the Fokker–Planck equation

1-D linear potential example

Brownian dynamics in one dimension is simple.<ref name=":0" /><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Theory

Starting with a linear potential of the form <math>U(x) = cx,</math> the corresponding Smoluchowski equation becomes

<math display="block">\partial_t P(x,t| x_0, t_0) = D \partial_x (\partial_x + \beta c) P(x,t| x_0, t_0) .</math>

Here, the diffusion constant <math>D</math> is constant over space and time. The boundary conditions are such that the probability vanishes at <math>x \rightarrow \pm \infin </math> with an initial condition of the ensemble of particles starting in the same place,

<math>P(x,t=t_0|x_0,t_0)= \delta (x-x_0).</math>

Defining <math>\tau = D t </math> and <math>b = \beta c </math> and applying the coordinate transformation,

<math display="block">y = x +\tau b ,\ \ \ y_0= x_0 + \tau_0 b </math>

With

<math>P(x, t, |x_0, t_0) = q(y, \tau|y_0, \tau_0)</math>

the Smoluchowki equation becomes

<math display="block">\partial_\tau q(y, \tau| y_0, \tau_0) =\partial_y^2 q(y, \tau| y_0, \tau_0).</math>

This is the free diffusion equation; it has the solution

<math>q(y, \tau| y_0, \tau_0)= \frac{1}{\sqrt {4 \pi (\tau - \tau_0)}} e^{ -\frac{(y-y_0)^2}{4(\tau-\tau_0)} }</math>

After transforming back to the original coordinates, the probaility distribution is obtained: <math display="block">P(x, t | x_0, t_0)= \frac{1}{\sqrt{4 \pi D (t - t_0)}} \exp {\left[{ -\frac{(x-x_0+ D \beta c(t-t_0))^2}{4D(t-t_0)}} \right]}.</math>

Simulation

The simulation above was completed using a Brownian dynamics simulation.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Starting with a Langevin equation for the system, <math display="block">m\ddot{x} = - \gamma \dot{x} -c + \sigma \xi(t)</math> where <math>\gamma</math> is the friction term, <math>\xi</math> is a fluctuating force on the particle, and <math>\sigma</math> is the amplitude of the fluctuation. At equilibrium the frictional force is much greater than the inertial force, <math>\left| \gamma \dot{x} \right| \gg \left| m \ddot{x} \right|</math>. Therefore, the Langevin equation becomes, <math display="block">\gamma \dot{x} = -c + \sigma \xi(t)</math>

For the Brownian dynamic simulation the fluctuation force <math>\xi(t)</math> is assumed to be Gaussian with the amplitude being dependent of the temperature of the system <math display="inline">\sigma = \sqrt{2\gamma k_\text{B} T}</math>. Rewriting the Langevin equation,

<math display="block">\frac{dx}{dt}=-D \beta c + \sqrt{2D}\xi(t)</math> where <math display="inline">D = \frac{k_\text{B}T}{\gamma}</math> is the Einstein relation. The integration of this equation was done using the Euler–Maruyama method to numerically approximate the path of this Brownian particle.

Solution

Being a partial differential equation, the Fokker–Planck equation can be solved analytically only in special cases. A formal analogy of the Fokker–Planck equation with the Schrödinger equation allows the use of advanced operator techniques known from quantum mechanics for its solution in a number of cases. Furthermore, in the case of overdamped dynamics when the Fokker–Planck equation contains second partial derivatives with respect to all spatial variables, the equation can be written in the form of a master equation that can easily be solved numerically.<ref>Template:Cite journal</ref> In many applications, one is only interested in the steady-state probability distribution <math> p_0(x)</math>, which can be found from <math display="inline">\frac{\partial p(x,t)}{\partial t} = 0</math>. The computation of mean first passage times and splitting probabilities can be reduced to the solution of an ordinary differential equation which is intimately related to the Fokker–Planck equation.

Particular cases with known solution and inversion

In mathematical finance for volatility smile modeling of options via local volatility, one has the problem of deriving a diffusion coefficient <math>{\sigma}(\mathbf{X}_t,t)</math> consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of the Fokker–Planck equation: Given the density f(x,t) of the option underlying X deduced from the option market, one aims at finding the local volatility <math>{\sigma}(\mathbf{X}_t,t)</math> consistent with f. This is an inverse problem that has been solved in general by Dupire (1994, 1997) with a non-parametric solution.<ref>Bruno Dupire (1994) Pricing with a Smile. Risk Magazine, January, 18–20.</ref><ref>Bruno Dupire (1997) Pricing and Hedging with Smiles. Mathematics of Derivative Securities. Edited by M.A.H. Dempster and S.R. Pliska, Cambridge University Press, Cambridge, 103–111. Template:ISBN.</ref> Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility <math>{\sigma}(\mathbf{X}_t,t)</math> consistent with a solution of the Fokker–Planck equation given by a mixture model.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> More information is available also in Fengler (2008),<ref>Fengler, M. R. (2008). Semiparametric Modeling of Implied Volatility, 2005, Springer Verlag, Template:ISBN</ref> Gatheral (2008),<ref>Jim Gatheral (2008). The Volatility Surface. Wiley and Sons, Template:ISBN.</ref> and Musiela and Rutkowski (2008).<ref>Marek Musiela, Marek Rutkowski. Martingale Methods in Financial Modelling, 2008, 2nd Edition, Springer-Verlag, Template:ISBN.</ref>

The path integral formulation

Every Fokker–Planck equation is equivalent to a path integral. The path integral formulation is an excellent starting point for the application of field theory methods.<ref>Template:Cite book</ref> This is used, for instance, in critical dynamics.

The derivation of the path integral is similar to that used in quantum mechanics. A derivation for the Fokker–Planck equation with one variable <math>x</math> follows. Inserting a delta function and integrating by parts gives:

<math display="block">\begin{align} \frac{\partial }{\partial t} p{\left( x', t\right)} & = - \frac{\partial }{\partial x'} \left[ D_1(x',t) p(x',t) \right] + \frac{\partial^2 }{\partial {x'}^2} \left[ D_2(x',t) p(x',t) \right] \\[1ex] & = \int_{-\infty}^{\infty} dx\left[ \left( D_{1}{\left( x,t\right)} \frac{\partial }{\partial x} + D_2{ \left( x,t\right)} \frac{\partial^2}{\partial x^2}\right) \delta{\left( x' -x\right)} \right] p(x,t). \end{align}</math>

The <math>x</math>-derivatives act only on the <math>\delta</math>-function, not on <math>p(x,t)</math>. Performing an integral over a time interval <math>\varepsilon</math> gives

<math display="block">p(x', t + \varepsilon) =\int_{-\infty}^\infty \, \mathrm{d}x\left(\left( 1+\varepsilon \left[ D_1(x,t) \frac \partial {\partial x} + D_2(x,t) \frac{\partial^2}{\partial x^2}\right]\right) \delta(x' - x) \right) p(x,t)+O(\varepsilon^2).</math>

The Dirac <math>\delta</math>-function can be represented by the Fourier integral as

<math display="block">\delta{\left( x' - x\right)} = \int_{-\infty}^{\infty} \frac{\mathrm{d} k}{2\pi} e^{-ik {\left( x - x'\right)}}</math>

which yields <math display="block">\begin{align} p(x', t+\varepsilon) & = \int_{-\infty}^\infty \mathrm{d}x \int_{-\infty}^{\infty} \frac{\mathrm{d}k}{2\pi} \left(1-\varepsilon \left[ ik D_1(x,t) +k^2 D_2(x,t) \right] \right) e^{-ik (x - x')}p(x,t) +O(\varepsilon^2) \\[5pt] & =\int_{-\infty}^\infty \mathrm{d}x \int_{-\infty}^{\infty} \frac{\mathrm{d}k}{2\pi}\exp \left( -\varepsilon \left[ ik\frac{(x'- x)}{\varepsilon} +ik D_1(x,t) +k^2 D_2(x,t) \right] \right) p(x,t) +O(\varepsilon^2). \end{align}</math>

This equation expresses <math>p(x', t+\varepsilon)</math> as functional of <math>p(x,t)</math>. Iterating <math>(t'-t)/\varepsilon</math> times and performing the limit <math>\varepsilon \rightarrow 0</math> gives a path integral with action

<math display="block">S=-\int \mathrm{d}t\left[ ik D_1 (x,t) + k^2 D_2 (x,t) +ik\frac{\partial x}{\partial t} \right].</math>

The variable <math>k</math> conjugate to <math>x</math> is called the "response variable".<ref name="Janssen">Template:Cite journal</ref>

Although formally equivalent, different problems may be solved more easily in the Fokker–Planck equation or the path integral formulation. The equilibrium distribution for instance may be obtained more directly from the Fokker–Planck equation.

Analytical Theory of Mean First Passage Time

In the theory of stochastic processes, the mean first passage time (MFPT) is the expected time for a stochastic trajectory to reach a specified boundary or target region for the first time. For a diffusion process governed by the stochastic differential equation (SDE)

<math>dX_t = b(X_t)\,dt + \sqrt{2\varepsilon}\,\sigma(X_t)\,dW_t,</math>

the evolution of the probability density <math>p(x,t)</math> is described by the Fokker–Planck equation:

<math> \frac{\partial p}{\partial t} = -\nabla \cdot (b(x)p) + \varepsilon \nabla \cdot \left( D(x) \nabla p \right), </math>

where <math>D(x) = \sigma(x)\sigma(x)^T</math> is the diffusion tensor, and <math>\varepsilon \ll 1</math> is the noise intensity.

To compute the MFPT <math>u(x) = \mathbb{E}_x[\tau]</math>, where <math>\tau</math> is the first exit time from a domain <math>\Omega</math>, one solves the backward Kolmogorov equation, also known as the Dynkin equation:

<math> \mathcal{L} u(x) = -1, \quad x \in \Omega; \qquad u(x) = 0, \quad x \in \partial\Omega, </math>

with generator

<math> \mathcal{L} = b(x) \cdot \nabla + \varepsilon \sum_{i,j} D_{ij}(x) \frac{\partial^2}{\partial x_i \partial x_j}. </math>

Boundary Layers and WKB Asymptotics

In the small noise regime (<math>\varepsilon \to 0</math>), solutions typically exhibit boundary layers near <math>\partial\Omega</math>, where escape occurs. The MFPT can be approximated using the WKB (Wentzel–Kramers–Brillouin) ansatz:

<math> u(x) \sim A(x) \exp\left( \frac{S(x)}{\varepsilon} \right), </math>

where <math>S(x)</math> is the quasi-potential or minimum action required for escape, and <math>A(x)</math> is a transport coefficient. The function <math>S(x)</math> solves a Hamilton–Jacobi equation and represents the most likely escape path under small random perturbations. These techniques were developed in the analytical framework of Zeev Schuss.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

In systems with limit cycle attractors, such as those seen in oscillatory biological systems, the exit time distribution deviates from the classical Poisson law, the survival probability <math>P(t)</math> of the process decaying from the limit cycle is modulated in time:

<math> P(t) = \mathbb{P}(\tau > t) \not\sim e^{-\lambda t}, </math>

Instead, the survival probability contains oscillatory terms reflecting the periodic nature of the attractor. The conditional exit time density <math>f(t)</math> is no longer exponential and is better described through an expansion in Hermite polynomials:

<math> Pr(\tau >t) =\sum_{n=1}^\infty c_n \exp (- \lambda_n t) \cos (\omega n t) </math>,

where the coefficients <math>c_n</math> reflect how strongly the exit probability deviates from the exponential form due to phase preference along the limit cycle. This expansion reveals that escape occurs with higher probability at particular phases of the cycle, breaking the memoryless (Markovian) nature of classical escape theory.<ref>Template:Cite journal</ref> The rate of escape depends not only on the noise intensity but also on geometric and dynamical anisotropies along the attractor. This phenomenon is particularly relevant for modeling neuronal excitability, biological clocks, and cardiac rhythms, where timing and variability of transitions are tightly regulated but also susceptible to random perturbations.

Asymptotic MFPT in Gradient Systems

In systems with small noise and a drift given by the gradient of a potential, <math>b(x) = -\nabla\phi(x),</math> the stochastic process

<math> dX_t = -\nabla \phi(X_t)\,dt + \sqrt{2\varepsilon}\,dW_t </math>

models the overdamped Langevin dynamics of a particle in a potential landscape <math>\phi(x)</math>. The associated mean first passage time <math>u(x),</math> which satisfies the backward Kolmogorov equation:

<math> \varepsilon \Delta u(x) - \nabla \phi(x) \cdot \nabla u(x) = -1, </math>

subject to <math>u=0</math> on the exit boundary <math>\partial\Omega_a \subset \partial\Omega,</math> has the following asymptotic solution in the limit <math>\varepsilon\to 0,</math> when <math>x</math> is near a local minimum <math>x_0</math> of <math>\phi</math> and escape occurs over a saddle point <math>x_s</math> of the potential:

<math> \mathbb{E}[\tau] \sim \frac{2\pi}{\sqrt{|\det H(x_s)|}} \cdot \frac{e^{[\phi(x_s) - \phi(x_0)]/\varepsilon}}{\sqrt{\det H(x_0)}}, </math>

where:

• <math>H(x_0)</math> is the Hessian matrix of <math>\phi</math> at the stable point <math>x_0</math>,
• <math>H(x_s)</math> is the Hessian at the saddle point <math>x_s,</math> with one negative eigenvalue,
• <math>\phi(x_s) - \phi(x_0)</math> is the energy barrier or quasi-potential difference the system must cross.

This formula generalizes Kramers' escape time to n-dimensional gradient systems and shows the exponential sensitivity of MFPT to potential barriers, with prefactors determined by second-order variations (local curvatures) of the potential at critical points. This result connects with large deviation theory and WKB asymptotics, where the action functional (or quasi-potential) governs the probability of rare events. It underpins modern approaches to metastability in physics, chemistry, and biology—such as chemical reaction rates, ion channel gating, or noise-induced switching in gene networks.

See also

• Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations
• Boltzmann equation
• Chapman–Kolmogorov equation
• Convection–diffusion equation
• Klein–Kramers equation
• Kolmogorov backward equation
• Kolmogorov equation
• Langevin equation
• Master equation
• Mean-field game theory
• Ornstein–Uhlenbeck process
• Vlasov equation

Further reading

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