Fokker-Planck Equation: Derivation and Intuition

Overview

The Fokker-Planck equation (also called the forward Kolmogorov equation) describes how the probability distribution of a stochastic process evolves over time. It is the bridge between individual particle dynamics (described by SDEs) and collective probability evolution.

This document derives the Fokker-Planck equation from first principles and explains why probability distributions must obey it.

---

Referenced From

• Main Document: docs/diffusion/reverse_process/reverse_process_derivation.md — Uses the Fokker-Planck equation to derive the reverse SDE
• Related: notebooks/diffusion/02_sde_formulation/supplements/07_fokker_planck_equation.md

---

Table of Contents

1. The Setting
2. Intuitive Picture
3. Derivation via Infinitesimal Evolution
4. Kramers-Moyal Expansion (Rigorous)
5. Physical Interpretation
6. Connection to Conservation Laws
7. Example: Simple Diffusion
8. Why This Matters for Reverse SDEs

---

The Setting

Stochastic Differential Equation

Consider a \(d\)-dimensional stochastic process \(x(t) \in \mathbb{R}^d\) governed by:

\[ dx = f(x,t)\,dt + g(t)\,dw \]

where:

• \(f(x,t) \in \mathbb{R}^d\) is the drift (deterministic force)
• \(g(t) \in \mathbb{R}\) is the diffusion coefficient (noise amplitude)
• \(w(t) \in \mathbb{R}^d\) is standard Brownian motion

Probability Distribution

At each time \(t\), the random variable \(x(t)\) has a probability distribution:

\[ p_t(x) = p(x, t) \]

where \(\int p_t(x)\,dx = 1\).

The Question

How does \(p_t(x)\) evolve over time?

The Fokker-Planck equation answers this:

\[ \boxed{\frac{\partial p_t}{\partial t} = -\nabla \cdot (f p_t) + \frac{1}{2}g(t)^2 \nabla^2 p_t} \]

---

Intuitive Picture

Before diving into the math, let's build intuition.

What Changes Probability at a Point?

Imagine a point \(x\) in space. The probability density \(p_t(x)\) at this point can change due to:

1. Drift (advection): Particles drift from nearby points to \(x\), or from \(x\) to elsewhere
2. Diffusion (spreading): Particles randomly wander in and out of \(x\)

These two mechanisms give rise to the two terms in the Fokker-Planck equation.

Analogy: Heat Equation

You can think of probability like heat: - Drift term: Like a wind blowing heat from one place to another - Diffusion term: Like heat spreading from hot to cold regions

The Fokker-Planck equation is essentially a heat equation with an additional drift/advection term.

---

Derivation via Infinitesimal Evolution

We'll derive the equation by considering how probability evolves over an infinitesimal time step \(\Delta t\).

Step 1: Chapman-Kolmogorov Equation

The probability at time \(t + \Delta t\) is related to the probability at time \(t\) by:

\[ p_{t+\Delta t}(x) = \int p_t(x') \, p(x, t+\Delta t \mid x', t) \, dx' \]

where \(p(x, t+\Delta t \mid x', t)\) is the transition probability from \(x'\) at time \(t\) to \(x\) at time \(t + \Delta t\).

Interpretation: To find the probability of being at \(x\) at time \(t + \Delta t\), we sum over all possible starting points \(x'\) at time \(t\), weighted by their probability.

Step 2: Taylor Expand the Left Side

\[ p_{t+\Delta t}(x) = p_t(x) + \frac{\partial p_t}{\partial t} \Delta t + O(\Delta t^2) \]

Step 3: Understand the Transition Probability

For the SDE \(dx = f(x,t)\,dt + g(t)\,dw\), the change over \(\Delta t\) is:

\[ \Delta x = x(t + \Delta t) - x(t) = f(x,t) \Delta t + g(t) \sqrt{\Delta t} \, \xi \]

where \(\xi \sim \mathcal{N}(0, I)\) is a standard normal random variable.

This means: - Mean displacement: \(\mathbb{E}[\Delta x \mid x] = f(x,t) \Delta t\) - Covariance: \(\text{Cov}[\Delta x \mid x] = g(t)^2 \Delta t \, I\)

Step 4: Express Transition Probability

For small \(\Delta t\), the transition from \(x'\) to \(x\) is approximately:

\[ p(x \mid x', \Delta t) \approx \delta(x - x' - f(x',t)\Delta t) * \mathcal{N}(0, g(t)^2 \Delta t \, I) \]

where \(*\) denotes convolution.

More precisely, using Gaussian approximation:

\[ p(x \mid x', \Delta t) \approx \frac{1}{(2\pi g^2 \Delta t)^{d/2}} \exp\left(-\frac{|x - x' - f(x',t)\Delta t|^2}{2g^2 \Delta t}\right) \]

Step 5: Substitute into Chapman-Kolmogorov

\[ p_{t+\Delta t}(x) = \int p_t(x') \, p(x \mid x', \Delta t) \, dx' \]

Now we use a clever trick: expand around \(x' = x\) (nearby points contribute most).

Let \(\delta x = x - x'\), so \(x' = x - \delta x\):

\[ p_{t+\Delta t}(x) = \int p_t(x - \delta x) \, p(x \mid x - \delta x, \Delta t) \, d(\delta x) \]

Step 6: Taylor Expand \(p_t(x - \delta x)\)

\[ p_t(x - \delta x) = p_t(x) - \delta x \cdot \nabla p_t(x) + \frac{1}{2} \sum_{i,j} \delta x_i \delta x_j \frac{\partial^2 p_t}{\partial x_i \partial x_j} + \ldots \]

Using index notation:

\[ p_t(x - \delta x) \approx p_t(x) - \sum_i \delta x_i \frac{\partial p_t}{\partial x_i} + \frac{1}{2} \sum_{i,j} \delta x_i \delta x_j \frac{\partial^2 p_t}{\partial x_i \partial x_j} \]

Step 7: Compute Moments of \(\delta x\)

From the transition probability, conditioned on starting at \(x' = x - \delta x \approx x\):

First moment:

$$

\mathbb{E}[\delta x] = f(x,t) \Delta t + O(\Delta t^2) $$

Second moment (each component):

$$

\mathbb{E}[\delta x_i \delta x_j] = \begin{cases} g(t)^2 \Delta t + f_i f_j \Delta t^2 & \text{if } i = j \ f_i f_j \Delta t^2 & \text{if } i \neq j \end{cases} $$

For small \(\Delta t\), the \(\Delta t^2\) terms are negligible, so:

\[ \mathbb{E}[\delta x_i \delta x_j] \approx g(t)^2 \Delta t \, \delta_{ij} \]

where \(\delta_{ij}\) is the Kronecker delta.

Step 8: Substitute Moments

\[ p_{t+\Delta t}(x) = \int \left[p_t(x) - \sum_i \delta x_i \frac{\partial p_t}{\partial x_i} + \frac{1}{2} \sum_{i,j} \delta x_i \delta x_j \frac{\partial^2 p_t}{\partial x_i \partial x_j}\right] p(\delta x \mid x, \Delta t) \, d(\delta x) \]

Taking expectations over \(\delta x\):

\[ p_{t+\Delta t}(x) = p_t(x) - \sum_i \mathbb{E}[\delta x_i] \frac{\partial p_t}{\partial x_i} + \frac{1}{2} \sum_{i,j} \mathbb{E}[\delta x_i \delta x_j] \frac{\partial^2 p_t}{\partial x_i \partial x_j} \]

Substitute the moments:

\[ p_{t+\Delta t}(x) = p_t(x) - \sum_i f_i(x,t) \Delta t \frac{\partial p_t}{\partial x_i} + \frac{1}{2} \sum_i g(t)^2 \Delta t \frac{\partial^2 p_t}{\partial x_i^2} \]

Step 9: Rearrange and Take Limit

\[ p_{t+\Delta t}(x) - p_t(x) = -\sum_i f_i(x,t) \frac{\partial p_t}{\partial x_i} \Delta t + \frac{1}{2} g(t)^2 \sum_i \frac{\partial^2 p_t}{\partial x_i^2} \Delta t \]

Divide by \(\Delta t\) and take \(\Delta t \to 0\):

\[ \frac{\partial p_t}{\partial t} = -\sum_i \frac{\partial}{\partial x_i} \left[f_i(x,t) p_t(x)\right] + \frac{1}{2} g(t)^2 \sum_i \frac{\partial^2 p_t}{\partial x_i^2} \]

Step 10: Vector Notation

Using \(\nabla \cdot\) for divergence and \(\nabla^2\) for Laplacian:

\[ \boxed{\frac{\partial p_t}{\partial t} = -\nabla \cdot (f p_t) + \frac{1}{2}g(t)^2 \nabla^2 p_t} \]

This is the Fokker-Planck equation!

---

Kramers-Moyal Expansion (Rigorous)

The derivation above can be made rigorous using the Kramers-Moyal expansion.

Jump Moments

Define the jump moments:

\[ M^{(n)}_i(x, t) = \lim_{\Delta t \to 0} \frac{1}{\Delta t} \mathbb{E}\left[(\Delta x_i)^n \mid x(t) = x\right] \]

Kramers-Moyal Theorem

The evolution of the probability distribution is:

\[ \frac{\partial p_t}{\partial t} = \sum_{n=1}^\infty \frac{(-1)^n}{n!} \sum_{i_1, \ldots, i_n} \frac{\partial^n}{\partial x_{i_1} \cdots \partial x_{i_n}} \left[M^{(n)}_{i_1 \cdots i_n} p_t\right] \]

Fokker-Planck Approximation

For continuous diffusion processes (like those generated by SDEs with smooth coefficients), the Kramers-Moyal expansion terminates at \(n=2\):

\[ M^{(n)} = 0 \quad \text{for } n \geq 3 \]

This gives:

\[ \frac{\partial p_t}{\partial t} = -\sum_i \frac{\partial}{\partial x_i} [M^{(1)}_i p_t] + \frac{1}{2} \sum_{i,j} \frac{\partial^2}{\partial x_i \partial x_j} [M^{(2)}_{ij} p_t] \]

For Our SDE

From \(dx = f(x,t)\,dt + g(t)\,dw\):

\[ M^{(1)}_i = f_i(x,t) \]

\[ M^{(2)}_{ij} = g(t)^2 \delta_{ij} \]

Substituting:

\[ \frac{\partial p_t}{\partial t} = -\sum_i \frac{\partial}{\partial x_i} [f_i p_t] + \frac{1}{2} g(t)^2 \sum_i \frac{\partial^2 p_t}{\partial x_i^2} \]

This is the Fokker-Planck equation.

---

Physical Interpretation

Let's break down each term:

The Full Equation

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

Term 1: Rate of Change

\[ \frac{\partial p_t}{\partial t} \]

Meaning: How fast probability density is changing at point \(x\) at time \(t\).

Term 2: Drift (Advection)

\[ -\nabla \cdot (f p_t) = -\sum_i \frac{\partial}{\partial x_i} [f_i(x,t) p_t(x)] \]

Meaning: Probability flux due to deterministic drift.

Expanded form:

\[ -\nabla \cdot (f p_t) = -f \cdot \nabla p_t - p_t \nabla \cdot f \]

• \(-f \cdot \nabla p_t\): Advection of probability along the drift field
• \(-p_t \nabla \cdot f\): Change in probability due to compression/expansion of the drift field

Physical picture: Probability "flows" along the drift field \(f(x,t)\), like wind blowing particles.

Term 3: Diffusion (Spreading)

\[ \frac{1}{2}g(t)^2 \nabla^2 p_t = \frac{1}{2}g(t)^2 \sum_i \frac{\partial^2 p_t}{\partial x_i^2} \]

Meaning: Probability spreads from high-density to low-density regions.

Sign convention:

• \(\nabla^2 p_t > 0\): Concave up → probability flows in (increases)
• \(\nabla^2 p_t < 0\): Concave down → probability flows out (decreases)

Physical picture: Random motion causes probability to diffuse, like heat spreading in a metal rod.

---

Connection to Conservation Laws

Continuity Equation

The Fokker-Planck equation can be written as a continuity equation:

\[ \frac{\partial p_t}{\partial t} + \nabla \cdot J = 0 \]

where \(J\) is the probability current (flux):

\[ J = f p_t - \frac{1}{2}g(t)^2 \nabla p_t \]

Components:

• \(f p_t\): Drift current (probability flowing along drift)
• \(-\frac{1}{2}g(t)^2 \nabla p_t\): Diffusion current (Fick's law, probability flowing down gradients)

Conservation of Probability

Integrating over all space:

\[ \frac{d}{dt} \int p_t(x)\,dx = \int \frac{\partial p_t}{\partial t}\,dx = -\int \nabla \cdot J\,dx = 0 \]

(using divergence theorem, assuming \(J \to 0\) at infinity).

Result: Total probability is conserved, as it must be!

---

Example: Simple Diffusion

Setup

Consider pure diffusion with no drift:

\[ dx = \sigma \, dw \]

where \(\sigma\) is constant.

Fokker-Planck Equation

\[ \frac{\partial p_t}{\partial t} = \frac{\sigma^2}{2} \nabla^2 p_t \]

This is the heat equation!

Solution

Starting from a point mass \(p_0(x) = \delta(x - x_0)\), the solution is:

\[ p_t(x) = \frac{1}{(2\pi \sigma^2 t)^{d/2}} \exp\left(-\frac{|x - x_0|^2}{2\sigma^2 t}\right) \]

This is a Gaussian with: - Mean: \(x_0\) (stays at starting point) - Variance: \(\sigma^2 t\) (spreads linearly with time)

Verification: Substitute this solution into the heat equation — it works!

---

Example: Ornstein-Uhlenbeck Process

Setup

Consider the SDE:

\[ dx = -\theta x \, dt + \sigma \, dw \]

where:

• \(-\theta x\): Drift toward origin (like a spring)
• \(\sigma\): Constant diffusion

Fokker-Planck Equation

\[ \frac{\partial p_t}{\partial t} = \nabla \cdot (\theta x p_t) + \frac{\sigma^2}{2} \nabla^2 p_t \]

Stationary Distribution

At equilibrium (\(\partial p / \partial t = 0\)):

\[ 0 = \nabla \cdot (\theta x p) + \frac{\sigma^2}{2} \nabla^2 p \]

Solving this ODE:

\[ p_\infty(x) = \mathcal{N}\left(0, \frac{\sigma^2}{2\theta}\right) \]

Interpretation: The drift pulls particles toward the origin, while diffusion spreads them out. The equilibrium is a balance between these forces.

---

Why This Matters for Reverse SDEs

Forward Process

The forward SDE:

\[ dx = f(x,t)\,dt + g(t)\,dw \]

generates a probability evolution governed by:

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

Reverse Process

To reverse the process, we need to find an SDE whose Fokker-Planck equation gives backward evolution:

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

(note the sign flips).

Key Insight

The Fokker-Planck equation can be rewritten using the score function \(\nabla \log p_t\):

\[ \nabla^2 p_t = \nabla \cdot (\nabla p_t) = \nabla \cdot (p_t \nabla \log p_t) \]

This allows us to express the diffusion term in terms of the score, leading to the effective drift:

\[ \tilde{f} = f - \frac{1}{2}g^2 \nabla \log p_t \]

The reverse SDE uses this effective drift to reverse the probability evolution.

See: reverse_process_derivation.md for the full derivation.

---

Summary

What We Learned

1. Fokker-Planck Equation: Describes how probability distributions evolve for stochastic processes
2. Two Mechanisms: Drift (advection) and diffusion (spreading)
3. Derivation: From infinitesimal evolution using Chapman-Kolmogorov and Taylor expansion
4. Physical Meaning: Continuity equation for probability flux
5. Connection to Reverse SDEs: The score function appears when rewriting the diffusion term

The Equation

\[ \boxed{\frac{\partial p_t}{\partial t} = -\nabla \cdot (f p_t) + \frac{1}{2}g(t)^2 \nabla^2 p_t} \]

Drift term: \(-\nabla \cdot (f p_t)\) — Probability flows along \(f\)

Diffusion term: \(\frac{1}{2}g^2 \nabla^2 p_t\) — Probability spreads

---

References

Classic Texts

• Risken (1989): "The Fokker-Planck Equation: Methods of Solution and Applications"
• Gardiner (2009): "Stochastic Methods: A Handbook for the Natural and Social Sciences"
• Øksendal (2003): "Stochastic Differential Equations: An Introduction with Applications"

Papers

• Fokker (1914): "Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld" — Original work
• Planck (1917): "Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie"
• Kolmogorov (1931): "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" — Forward equation

Related Documents

• Reverse Process: reverse_process_derivation.md
• Forward Process: forward_process_derivation.md
• Supplement in Notebook: notebooks/diffusion/02_sde_formulation/supplements/07_fokker_planck_equation.md