# Fokker–Planck Equations

The **Fokker–Planck equation** is a partial differential equation describing how uncertainty evolves under a stochastic differential equation (SDE). Although individual sample paths of an SDE are irregular, their *distribution* evolves deterministically. The Fokker–Planck equation formalizes this “deterministic evolution of randomness.”

These equations play an important role in control and planning: shaping uncertainty, formulating probabilistic objectives, coordinating swarms, and designing ergodic exploration strategies. We will examine these applications in later posts.

In this post we introduce the classical Fokker–Planck equation. Later we will discuss **nonholonomic variants** that arise when randomness enters only through certain controlled directions.

---

## Microscopic Dynamics: The Overdamped Langevin Equation

Let $U : \mathbb{R}^d \to \mathbb{R}$ be a potential, and consider the stochastic dynamics

$$
\mathrm{d}X = -\nabla U(X)\, \mathrm{d}t + \sqrt{2}\, \mathrm{d}W + \mathrm{d}Z,
$$

where
- $W(t)$ is a $d$-dimensional Brownian motion,
- $Z(t)$ is a reflection or constraint term keeping $X(t)$ inside a domain $\Omega$.

This is the **overdamped Langevin equation**.

|  |
|--------|
| A particle evolving under overdamped Langevin dynamics in a quadratic potential. Drift pulls toward the minimum of $U$, while Brownian motion spreads the trajectory. |

Let
$$
\mathbb{P}(X(t) \in \mathrm{d}x) = p_t(x)\, \mathrm{d}x.
$$

Then $p_t$ satisfies the **Fokker–Planck equation**

$$
\partial_t p = \Delta p - \nabla \cdot (\nabla U\, p),
$$

together with the **zero-flux boundary condition**

$$
\vec{n}(x)\cdot (\nabla p_t - \nabla U\, p_t) = 0
\quad \text{on } \partial\Omega,
$$

ensuring mass conservation $\int_\Omega p_t = 1$.

|  |
|--------|
| Evolution of the Fokker–Planck equation with a quadratic potential. |

---

## Convergence to Equilibrium

We ask:

> **Does $p_t$ converge to a stationary distribution $p_\infty$ as $t\to\infty$?**

We begin with a simple case.

---

## Case 1: $U \equiv 0$ — The Heat Equation

When $U=0$, the PDE becomes:

$$
\partial_t p = \Delta p.
$$

On a bounded domain with Neumann boundary conditions, the solution converges to the uniform distribution

$$
p_\infty = \frac{1}{|\Omega|}.
$$

A key tool is the **Poincaré inequality**:

$$
\boxed{
\|f - f_\Omega\|_2^2 \le C \|\nabla f\|_2^2,
\qquad
f_\Omega = \frac{1}{|\Omega|} \int_\Omega f
}
$$

### Claim
If $p_0 \ge 0$ and $\int_\Omega p_0 = 1$, then

$$
\|p_t - p_\infty\|_2 \le e^{-t/C}\, \|p_0 - p_\infty\|_2.
$$

### Informal Proof

Define the Lyapunov functional
$$
\mathcal{F}(p_t) = \|p_t - p_\infty\|_2^2.
$$

Differentiate using $\partial_t p_t = \Delta p_t$:

$$
\frac{d}{dt} \mathcal{F}
= 2 \int_\Omega (\Delta p_t)(p_t - p_\infty)\, dx.
$$

Integrate by parts (zero-flux BC):

$$
\frac{d}{dt} \mathcal{F}
= -2 \int_\Omega |\nabla p_t|^2 dx.
$$

Using Poincaré:

$$
\mathcal{F}(p_t) \le C \int_\Omega |\nabla p_t|^2 dx,
$$

so

$$
\frac{d}{dt} \mathcal{F}(p_t) \le -\frac{2}{C}\mathcal{F}(p_t).
$$

Grönwall’s inequality gives exponential convergence.

---

## Case 2: General Potential $U(x)$

The stationary distribution is the Gibbs measure

$$
p_\infty(x) = \frac{1}{Z}\, e^{-U(x)}, \qquad Z = \int_\Omega e^{-U(x)} dx.
$$

The plain $L^2$ norm is no longer adapted. Instead use the **weighted $L^2$ norm**

$$
\|f\|_{L^2(p_\infty^{-1})}
= \left( \int_\Omega \frac{|f(x)|^2}{p_\infty(x)}\, dx \right)^{1/2}.
$$

If $U$ is smooth, then $p_\infty$ is bounded above and below. One can show the **weighted Poincaré inequality**

$$
\boxed{
\| f - f_{\Omega,p_\infty} \|_{L^2(p_\infty^{-1})}
\le C' \|\nabla f\|_{L^2(p_\infty^{-1})}
}
$$

where
$$
f_{\Omega,p_\infty} = \int_\Omega f\, p_\infty\, dx.
$$

---

## Stability via a Lyapunov Functional

Define
$$
f_t = \frac{p_t}{p_\infty}
$$

and

$$
\mathcal{F}(p_t)
= \int_\Omega (f_t - 1)^2\, p_\infty \, dx
= \|p_t - p_\infty\|_{L^2(p_\infty^{-1})}^2.
$$

Rewrite the Fokker–Planck equation:

$$
\partial_t p_t = \nabla\cdot (p_\infty \nabla f_t),
$$

$$
\partial_t f_t = \frac{1}{p_\infty} \nabla\cdot(p_\infty \nabla f_t).
$$

Differentiate $\mathcal{F}$:

$$
\frac{d}{dt}\mathcal{F}(p_t)
= -2 \int_\Omega p_\infty |\nabla f_t|^2 dx.
$$

Weighted Poincaré gives:

$$
\mathcal{F}(p_t)
\le C' \int_\Omega \frac{|\nabla f_t|^2}{p_\infty} dx.
$$

Thus for some constant $M>0$,

$$
\frac{d}{dt} \mathcal{F}(p_t)
\le -\frac{2M}{C'}\, \mathcal{F}(p_t).
$$

Grönwall yields:

$$
\boxed{
\|p_t - p_\infty\|_{L^2(p_\infty^{-1})}
\le e^{-Mt/C'}\, \|p_0 - p_\infty\|_{L^2(p_\infty^{-1})}.
}
$$

This shows exponential convergence to equilibrium in the weighted $L^2$ metric.

---