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Fokker-Planck Equation

The Fokker-Planck equation (FPE) is a partial differential equation that describes the time evolution of the probability density function of a stochastic process defined by a [[Stochastic Differential Equation (SDE)|stochastic differential equation]]. It provides the deterministic macroscopic description of the stochastic microscopic dynamics, bridging the gap between random particle trajectories and their ensemble distribution.

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1. Core Concept

1.1 From Stochastic Trajectories to Deterministic Density

A single particle governed by an [[Stochastic Differential Equation (SDE)|SDE]] follows a random path:

$$\mathrm{d}{X}_t=\mu (X_t,t)\mathrm{d}t+\sigma (X_t,t)\mathrm{d}{W}_t$$

While each trajectory is random, the probability density $p(x,t)$ of finding the particle at position $x$ at time $t$ evolves deterministically according to the Fokker-Planck equation:

$$\frac{\partial p(x,t)}{\partial t}=-\frac{\partial }{\partial x}[\mu (x,t)p(x,t)]+\frac{1}{2}\frac{{\partial }^2}{\partial x^2}[{\sigma }^2(x,t)p(x,t)]$$

[!NOTE] Key Insight
The Fokker-Planck equation transforms a stochastic trajectory problem into a deterministic density evolution problem — much like how the [[Schrödinger Equation]] governs quantum probability amplitudes. It is the master equation for continuous-state, continuous-time [[Markov Process|Markov processes]].

1.2 Physical Interpretation

Term	Expression	Physical Meaning
Drift term	$-\frac{\partial }{\partial x}[\mu p]$	Probability flows in the direction of deterministic trend
Diffusion term	$\frac{1}{2}\frac{{\partial }^2}{\partial x^2}[{\sigma }^{2}p]$	Probability spreads out due to random noise

• $\mu (x,t)$ = drift coefficient: pushes the distribution’s mean
• $\sigma (x,t)$ = diffusion coefficient: broadens the distribution

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2. One-Dimensional Fokker-Planck Equation

2.1 Standard Form

For an [[Stochastic Differential Equation (SDE)|SDE]] with drift $\mu (x,t)$ and diffusion $\sigma (x,t)$ :

$$\frac{\partial p}{\partial t}=-\frac{\partial }{\partial x}[\mu (x,t)p]+\frac{1}{2}\frac{{\partial }^2}{\partial x^2}[{\sigma }^2(x,t)p]$$

2.2 Probability Current Form

The FPE can be rewritten as a continuity equation (conservation of probability):

$$\frac{\partial p}{\partial t}+\frac{\partial J}{\partial x}=0$$

where the probability current (probability flux) $J(x,t)$ is:

$$J(x,t)=\mu (x,t)p(x,t)-\frac{1}{2}\frac{\partial }{\partial x}[{\sigma }^2(x,t)p(x,t)]$$

[!NOTE] Conservation of Probability
Just like the continuity equation in fluid dynamics ( $\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \mathbf{v})=0$ ), the FPE ensures that total probability is conserved: ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}p(x,t)\mathrm{d}x=1$ for all $t$ .

2.3 Derivation from SDE

Step 1 — The infinitesimal generator $\mathcal{L}$ of the [[Stochastic Differential Equation (SDE)|SDE]] is:

$$\mathcal{L}f(x)=\mu (x,t)\frac{\partial f}{\partial x}+\frac{1}{2}{\sigma }^2(x,t)\frac{{\partial }^{2}f}{\partial x^2}$$

Step 2 — Its adjoint operator ${\mathcal{L}}^{\ast }$ governs the density evolution:

$$\frac{\partial p}{\partial t}={\mathcal{L}}^{\ast }p=-\frac{\partial }{\partial x}[\mu p]+\frac{1}{2}\frac{{\partial }^2}{\partial x^2}[{\sigma }^{2}p]$$

Step 3 — Boundary conditions ensure probability conservation:

• Natural boundary: $J(x,t)\to 0$ as $|x|\to \mathrm{\infty }$
• Reflecting boundary: $J=0$ at boundary
• Absorbing boundary: $p=0$ at boundary

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3. Key Special Cases

3.1 Pure Diffusion ([[Wiener Process|Wiener Process]] / Brownian Motion)

For $\mu =0$ , $\sigma =1$ (standard [[Wiener Process|Wiener Process]]):

$$\frac{\partial p}{\partial t}=\frac{1}{2}\frac{{\partial }^{2}p}{\partial x^2}$$

This is the heat equation (or diffusion equation). The solution with initial condition $p(x,0)=\delta (x-x_0)$ is:

$$p(x,t)=\frac{1}{\sqrt{2\pi t}}\exp (-\frac{(x-x_0{)}^2}{2t})$$

3.2 Ornstein-Uhlenbeck Process

For $\mu (x)=-\theta x$ (mean-reverting), $\sigma$ constant:

$$\frac{\partial p}{\partial t}=\frac{\partial }{\partial x}[\theta xp]+\frac{{\sigma }^2}{2}\frac{{\partial }^{2}p}{\partial x^2}$$

Stationary solution:

$$p_{\mathrm{\infty }}(x)=\sqrt{\frac{\theta }{\pi {\sigma }^2}}\exp (-\frac{\theta x^2}{{\sigma }^2})\sim \mathcal{N}(0,\frac{{\sigma }^2}{2\theta })$$

3.3 Geometric Brownian Motion

For $\mu (x)=\mu x$ , $\sigma (x)=\sigma x$ (used in finance):

$$\frac{\partial p}{\partial t}=-\frac{\partial }{\partial x}[\mu xp]+\frac{{\sigma }^2}{2}\frac{{\partial }^2}{\partial x^2}[x^{2}p]$$

Solution (log-normal distribution):

$$p(x,t)=\frac{1}{x\sigma \sqrt{2\pi t}}\exp (-\frac{(\ln x-\ln x_0-(\mu -{\sigma }^2/2)t{)}^2}{2{\sigma }^{2}t})$$

3.4 Summary Table

Process	Drift $\mu (x)$	Diffusion $\sigma (x)$	FPE Type
Wiener Process	$0$	$1$	Heat equation
OU Process	$-\theta x$	$\sigma$	Mean-reverting
Geometric BM	$\mu x$	$\sigma x$	Log-normal
Constant drift	$\mu$	$\sigma$	Convection-diffusion
VP-[[Stochastic Differential Equation (SDE)|SDE]]	$-\frac{1}{2}\beta (t)x$	$\sqrt{\beta (t)}$	Variance-preserving
VE-[[Stochastic Differential Equation (SDE)|SDE]]	$0$	$\sqrt{\frac{\mathrm{d}{\sigma }^2}{\mathrm{d}t}}$	Variance-exploding

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4. Multidimensional Fokker-Planck Equation

4.1 General $d$ -Dimensional Form

For a multivariate SDE:

$$\mathrm{d}{\mathbf{X}}_t=\mathit{\mu }({\mathbf{X}}_t,t)\mathrm{d}t+\mathit{\sigma }({\mathbf{X}}_t,t)\mathrm{d}{\mathbf{W}}_t$$

where ${\mathbf{W}}_t\in {\mathbb{R}}^m$ is an $m$ -dimensional [[Wiener Process|Wiener process]], the Fokker-Planck equation becomes:

$$\frac{\partial p}{\partial t}=-\sum _{i=1}^d\frac{\partial }{\partial x_i}\left[{\mu }_{i}p\right]+\frac{1}{2}\sum _{i=1}^d\sum _{j=1}^d\frac{{\partial }^2}{\partial x_i\partial x_j}\left[D_{ij}p\right]$$

where the diffusion matrix is $\mathbf{D}=\mathit{\sigma }{\mathit{\sigma }}^{\mathrm{\top }}\in {\mathbb{R}}^{d\times d}$ .

4.2 Compact Operator Notation

$$\frac{\partial p}{\partial t}=-\mathrm{\nabla }\cdot (\mathit{\mu }p)+\frac{1}{2}\mathrm{\nabla }\cdot (\mathrm{\nabla }\cdot (\mathbf{D}p))$$

or in terms of the probability current $\mathbf{J}$ :

$$\frac{\partial p}{\partial t}+\mathrm{\nabla }\cdot \mathbf{J}=0,\mathbf{J}=\mathit{\mu }p-\frac{1}{2}\mathrm{\nabla }\cdot (\mathbf{D}p)$$

4.3 Isotropic Noise Special Case

When $\mathit{\sigma }=\sigma \mathbf{I}$ (independent noise in each dimension):

$$\frac{\partial p}{\partial t}=-\mathrm{\nabla }\cdot (\mathit{\mu }p)+\frac{{\sigma }^2}{2}{\mathrm{\nabla }}^{2}p$$

This is the form most commonly encountered in [[Diffusion Model|diffusion models]].

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5. Stationary Distribution

5.1 Equilibrium Condition

For a time-homogeneous [[Stochastic Differential Equation (SDE)|SDE]] ( $\mu$ and $\sigma$ independent of $t$ ), the stationary distribution $p_{\mathrm{\infty }}(x)$ satisfies $\frac{\partial p_{\mathrm{\infty }}}{\partial t}=0$ :

$$0=-\frac{\mathrm{d}}{\mathrm{d}x}[\mu (x)p_{\mathrm{\infty }}(x)]+\frac{1}{2}\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}[{\sigma }^2(x)p_{\mathrm{\infty }}(x)]$$

5.2 Closed-Form Solution (1D)

Integrating once with zero probability current ( $J=0$ ) gives:

$$\mu (x)p_{\mathrm{\infty }}(x)=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}x}[{\sigma }^2(x)p_{\mathrm{\infty }}(x)]$$

Solution:

$$p_{\mathrm{\infty }}(x)=\frac{C}{{\sigma }^2(x)}\exp \left(2{\int }_{x_0}^x\frac{\mu (y)}{{\sigma }^2(y)}\mathrm{d}y\right)$$

where $C$ is a normalization constant such that $\int p_{\mathrm{\infty }}(x)\mathrm{d}x=1$ .

5.3 Potential Form

If $\mu (x)=-\frac{\mathrm{d}U(x)}{\mathrm{d}x}$ (gradient of a potential $U$ ) and $\sigma$ is constant:

$$p_{\mathrm{\infty }}(x)=C\exp (-\frac{2U(x)}{{\sigma }^2})$$

This is the Gibbs-Boltzmann distribution with “temperature” $\frac{{\sigma }^2}{2}$ .

[!NOTE] Physical Analogy
The stationary distribution corresponds to thermal equilibrium in statistical mechanics. The drift $\mu (x)=-\mathrm{\nabla }U(x)$ drives the system toward lower potential energy, while diffusion ( $\sigma$ ) adds thermal fluctuations.

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6. Forward and Backward Kolmogorov Equations

Aspect	Forward (Fokker-Planck)	Backward (Kolmogorov)
Variable	Future state $x$ (at time $t$ )	Initial state $x_0$ (at time $0$ )
Operates on	Probability density $p(x,t\mid x_0)$	Expectation $u(x_0,t)=\mathbb{E}[f(X_t)\mid X_0=x_0]$
Equation	$\frac{\partial p}{\partial t}={\mathcal{L}}^{\ast }p$	$\frac{\partial u}{\partial t}=\mathcal{L}u$
Boundary condition	Given initial distribution	Given final payoff $u(x,T)=f(x)$
Application	Density evolution, diffusion models	Option pricing (Feynman-Kac), hitting probabilities

6.1 Backward Kolmogorov Equation

The backward equation governs the evolution of expectations:

$$\frac{\partial u(x,t)}{\partial t}=\mu (x)\frac{\partial u}{\partial x}+\frac{1}{2}{\sigma }^2(x)\frac{{\partial }^{2}u}{\partial x^2}$$

with terminal condition $u(x,T)=f(x)$ . Then $u(x_0,0)=\mathbb{E}[f(X_T)\mid X_0=x_0]$ .

6.2 Feynman-Kac Formula

The backward equation extends to include a potential (discount) term:

$$\frac{\partial u}{\partial t}+\mu \frac{\partial u}{\partial x}+\frac{{\sigma }^2}{2}\frac{{\partial }^{2}u}{\partial x^2}-r(x)u=0$$

with solution:

$$u(x,t)=\mathbb{E}[f(X_T)\exp (-{\int }_t^{T}r(X_s)\mathrm{d}s)|X_t=x]$$

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7. Connection to Diffusion Models

7.1 Forward Process as Fokker-Planck Evolution

In [[Diffusion Model|diffusion models]], the forward process follows an [[Stochastic Differential Equation (SDE)|SDE]]:

$$\mathrm{d}{x}_t=f(t)x_t\mathrm{d}t+g(t)\mathrm{d}{W}_t$$

The Fokker-Planck equation describing $p_t(x)$ is:

$$\frac{\partial p_t(x)}{\partial t}=-{\mathrm{\nabla }}_x\cdot [f(t)x{p}_t(x)]+\frac{1}{2}g(t{)}^2{\mathrm{\nabla }}_x^2{p}_t(x)$$

The initial condition is the data distribution $p_0(x)=p_{\text{data}}(x)$ , and after sufficient time $T$ , $p_T(x)\approx \mathcal{N}(0,{\sigma }_T^{2}I)$ .

7.2 VP-SDE and VE-SDE Specific Forms

Variance-Preserving (VP) [[Stochastic Differential Equation (SDE)|SDE]] ( $f(t)=-\frac{1}{2}\beta (t)$ , $g(t)=\sqrt{\beta (t)}$ ):

$$\frac{\partial p_t}{\partial t}=\frac{1}{2}\beta (t){\mathrm{\nabla }}_x\cdot (x{p}_t)+\frac{1}{2}\beta (t){\mathrm{\nabla }}_x^2{p}_t$$

Variance-Exploding (VE) [[Stochastic Differential Equation (SDE)|SDE]] ( $f(t)=0$ , $g(t)=\sqrt{\frac{\mathrm{d}{\sigma }^2(t)}{\mathrm{d}t}}$ ):

$$\frac{\partial p_t}{\partial t}=\frac{1}{2}\frac{\mathrm{d}{\sigma }^2(t)}{\mathrm{d}t}{\mathrm{\nabla }}_x^2{p}_t$$

7.3 Connection to [[Probability Flow ODE]]

The Fokker-Planck continuity form:

$$\frac{\partial p_t}{\partial t}=-{\mathrm{\nabla }}_x\cdot [{\mathbf{v}}_t(x)p_t(x)]$$

directly defines the [[Probability Flow ODE]] velocity field:

$${\mathbf{v}}_t(x)=f(t)x-\frac{1}{2}g(t{)}^2{\mathrm{\nabla }}_x\log p_t(x)$$

This means the [[Probability Flow ODE]] and the forward [[Stochastic Differential Equation (SDE)|SDE]] share the same Fokker-Planck equation — and therefore the same marginal densities $p_t(x)$ at all times.

[!NOTE] Key Bridge
The Fokker-Planck equation is the mathematical bridge connecting three equivalent descriptions of diffusion models:

1. SDE (stochastic trajectories)
2. Probability Flow ODE (deterministic trajectories)
3. Score matching (density gradients)

7.4 Score Function Role

In the Fokker-Planck framework, the [[Score Function|score function]] ${\mathrm{\nabla }}_x\log p_t(x)$ appears naturally as the term that shifts the probability current from pure diffusion toward the data distribution in the reverse process.

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8. Numerical Methods

8.1 Finite Difference Method

Discretize the spatial domain and approximate derivatives:

# 1D Fokker-Planck solver (finite difference)
def fokker_planck_fd(mu, sigma, x_grid, t_grid, p0):
    """
    Solve ∂p/∂t = -∂/∂x[μp] + (1/2)∂²/∂x²[σ²p]
    Using Crank-Nicolson (implicit, stable)
    """
    dx = x_grid[1] - x_grid[0]
    dt = t_grid[1] - t_grid[0]
    N = len(x_grid)
    
    p = p0.copy()
    p_history = [p0.copy()]
    
    for n in range(1, len(t_grid)):
        # Build tridiagonal matrix for Crank-Nicolson
        A = build_crank_nicolson_matrix(mu, sigma, dx, dt, N)
        B = build_rhs_matrix(mu, sigma, dx, dt, N)
        
        p = solve_linear_system(A, B @ p)
        p_history.append(p.copy())
    
    return p_history

8.2 Monte Carlo Approach

Instead of solving the PDE, simulate many SDE trajectories and estimate the density:

def fokker_planck_mc(mu, sigma, x0, T, n_paths=10000):
    """Estimate density at time T via Monte Carlo simulation."""
    X_T = np.zeros(n_paths)
    
    for i in range(n_paths):
        X = x0
        t = 0
        dt = 0.001
        while t < T:
            dW = np.sqrt(dt) * np.random.randn()
            X += mu(X, t) * dt + sigma(X, t) * dW
            t += dt
        X_T[i] = X
    
    # Estimate density from samples
    density, bins = np.histogram(X_T, bins=100, density=True)
    return bins, density

8.3 Method Comparison

Method	Accuracy	Speed	Dimension	Best For
Finite Difference	High	Slow	$d\le 3$	Low-dim, high precision
Finite Element	High	Medium	$d\le 3$	Complex geometries
Monte Carlo	Low-Medium	Fast	Any $d$	High dimensions
Spectral Methods	Very High	Fast	$d\le 2$	Smooth, periodic problems
Deep Learning	Medium	Medium	Any $d$	High-dim, irregular domains

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9. Mathematical Properties

9.1 Conservation and Positivity

1. Probability conservation: $\frac{\mathrm{d}}{\mathrm{d}t}\int p(x,t)\mathrm{d}x=0$
2. Positivity preservation: If $p(x,0)\ge 0$ , then $p(x,t)\ge 0$ for all $t>0$
3. Smoothing property: FPE instantly smooths any initial distribution ( $t>0\Rightarrow p(x,t)\in C^{\mathrm{\infty }}$ under ellipticity)

9.2 Connection to [[Itô’s Lemma]]

The Fokker-Planck equation can be derived from [[Itô’s Lemma]] by considering the expected time evolution of a test function $\phi (x)$ :

$$\frac{\mathrm{d}}{\mathrm{d}t}\mathbb{E}[\phi (X_t)]=\mathbb{E}[\mathcal{L}\phi (X_t)]$$

where $\mathcal{L}$ is the infinitesimal generator. Integration by parts yields the FPE.

9.3 Ergodic Behavior

For an ergodic [[Markov Process|Markov process]]:

• Unique stationary distribution $p_{\mathrm{\infty }}$ exists
• $p(x,t)\to p_{\mathrm{\infty }}(x)$ as $t\to \mathrm{\infty }$
• Convergence rate determined by spectral gap of ${\mathcal{L}}^{\ast }$

9.4 Entropy Production

The Fokker-Planck equation satisfies an $H$ -theorem: the relative entropy (KL divergence) to the stationary distribution decreases monotonically:

$$\frac{\mathrm{d}}{\mathrm{d}t}{D}_{\text{KL}}(p_t\parallel p_{\mathrm{\infty }})\le 0$$

This implies the system irreversibly approaches equilibrium — a manifestation of the Second Law of Thermodynamics for stochastic systems.

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10. Core Formula Cards

[!QUOTE] 1D Fokker-Planck Equation

$$\frac{\partial p}{\partial t}=-\frac{\partial }{\partial x}[\mu (x,t)p]+\frac{1}{2}\frac{{\partial }^2}{\partial x^2}[{\sigma }^2(x,t)p]$$

[!QUOTE] Probability Current Form

$$\frac{\partial p}{\partial t}+\frac{\partial J}{\partial x}=0,J=\mu p-\frac{1}{2}\frac{\partial }{\partial x}[{\sigma }^{2}p]$$

[!QUOTE] Multidimensional FPE

$$\frac{\partial p}{\partial t}=-\sum _i\frac{\partial }{\partial x_i}[{\mu }_{i}p]+\frac{1}{2}\sum _{i,j}\frac{{\partial }^2}{\partial x_i\partial x_j}[D_{ij}p]$$

[!QUOTE] Stationary Distribution (1D, zero flux)

$$p_{\mathrm{\infty }}(x)=\frac{C}{{\sigma }^2(x)}\exp \left(2{\int }_{x_0}^x\frac{\mu (y)}{{\sigma }^2(y)}\mathrm{d}y\right)$$

[!QUOTE] Heat Equation ([[Wiener Process|Wiener Process]] / $\mu =0,\sigma =1$ )

$$\frac{\partial p}{\partial t}=\frac{1}{2}\frac{{\partial }^{2}p}{\partial x^2}$$

[!QUOTE] Infinitesimal Generator

$$\mathcal{L}f(x)=\mu (x)\frac{\partial f}{\partial x}+\frac{1}{2}{\sigma }^2(x)\frac{{\partial }^{2}f}{\partial x^2}$$ $$\frac{\partial p}{\partial t}={\mathcal{L}}^{\ast }p$$

[!QUOTE] Backward Kolmogorov Equation

$$\frac{\partial u}{\partial t}=\mu (x)\frac{\partial u}{\partial x}+\frac{1}{2}{\sigma }^2(x)\frac{{\partial }^{2}u}{\partial x^2}$$

[!QUOTE] FPE in [[Diffusion Model|Diffusion Models]]

$$\frac{\partial p_t}{\partial t}=-\mathrm{\nabla }\cdot [f(t)x{p}_t]+\frac{1}{2}g(t{)}^2{\mathrm{\nabla }}^2{p}_t$$

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11. Summary

Aspect	Description
What it describes	Time evolution of probability density under an SDE
Type	Second-order parabolic PDE
Input	Drift $\mu (x,t)$ , diffusion $\sigma (x,t)$ , initial density $p(x,0)$
Output	Density $p(x,t)$ at all future times
Key property	Conservation of total probability
Role in diffusion models	Bridges SDE description to Probability Flow ODE
Physical analog	Continuity equation, heat equation, Fick’s law of diffusion
Named after	Adriaan Fokker (1914) and Max Planck (1917)

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Related Concepts

• [[Stochastic Differential Equation (SDE)]]
• [[Wiener Process|Wiener Process]]
• [[Probability Flow ODE]]
• [[Diffusion Model]]
• [[Itô’s Lemma]]
• [[Itô Integral]]
• [[Markov Process]]
• [[Martingale]]
• [[Score Function]]
• [[Kolmogorov Equations]]
• [[Neural ODE]]
• [[Langevin Dynamics]]
• [[Feynman-Kac Formula]]

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References

• Paper: On the Theory of Brownian Motion (Fokker, 1914)
• Paper: Über einen Satz der statistischen Dynamik (Planck, 1917)
• Book: The Fokker-Planck Equation: Methods of Solution and Applications — Hannes Risken
• Book: Stochastic Processes in Physics and Chemistry — Van Kampen
• Book: Stochastic Differential Equations: An Introduction with Applications — Bernt Øksendal
• Paper: Score-Based Generative Modeling through SDEs (Song et al., 2021)
• Paper: Maximum Likelihood Training of Score-Based Diffusion Models (Song et al., 2021)
• Blog: Fokker-Planck Equation and Diffusion Models — AI papers summary
• Course: MIT 18.S096 Topics in Mathematics with Applications in Finance
• Course: CS236 Deep Generative Models (Stanford)