Diffusion Processes — Mathematics Knowledge System

Overview

A diffusion process is a continuous-time Markov process with continuous sample paths that solves an Ito SDE. Diffusions are characterized by their infinitesimal generator L, which encodes the drift and diffusion coefficients. The Fokker-Planck equation (forward Kolmogorov equation) governs the evolution of the probability density of the process, while the backward Kolmogorov equation is related to the Feynman-Kac formula. Diffusion processes provide the rigorous mathematical framework for modeling heat flow, Brownian motion, population genetics, and financial markets.

Intuition

A diffusion process is a continuous random motion where two forces act: a drift (like a current pushing the particle in a direction) and a diffusion (random jostling in all directions). The generator L tells you how quickly, on average, any observable function of the current state is changing. The Fokker-Planck equation tracks how the probability cloud spreads and shifts over time: particles drift with the drift field and spread out due to diffusion. At stationarity, the probability density satisfies L*p = 0 -- the probability flux balances.

Formal Definition

Definition

A diffusion process X_t with drift mu and diffusion coefficient sigma is defined by its SDE and characterized by its infinitesimal generator.

dX_t = \mu(t, X_t)\,dt + \sigma(t, X_t)\,dW_t

Diffusion SDE

\mathcal{L}f(x) = \mu(x)f'(x) + \tfrac{\sigma^2(x)}{2}f''(x)

Infinitesimal generator (time-homogeneous case)

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

Fokker-Planck (forward Kolmogorov) equation

-\frac{\partial u}{\partial s} = \mathcal{L}_x u, \quad u(T,x) = g(x)

Backward Kolmogorov equation

Notation

Notation	Meaning
$\mathcal{L}$	Infinitesimal generator: Lf = mu f' + (sigma^2/2) f''
$p(t,x)$	Probability density of X_t at position x
$p_s(x)$	Stationary density satisfying the stationary Fokker-Planck equation

Theorems

Theorem 1: Fokker-Planck Equation

\text{If } X_t \text{ solves } dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \text{ and } p(t,x) \text{ is the density of } X_t, \text{ then:} \quad \partial_t p = -\partial_x(\mu p) + \tfrac{1}{2}\partial_{xx}(\sigma^2 p).

Theorem 2: Dynkin's Formula

\text{For } f \in C_c^2 \text{ and stopping time } T \text{ with } \mathbb{E}[T] < \infty{:} \quad \mathbb{E}^x[f(X_T)] = f(x) + \mathbb{E}^x\!\left[\int_0^T \mathcal{L}f(X_s)\,ds\right].

Theorem 3: Stationary Distribution of a Diffusion

\text{For a time-homogeneous diffusion on } \mathbb{R} \text{ with drift } \mu \text{ and diffusion } \sigma, \text{ the stationary density (when it exists) is:} \quad p_s(x) \propto \frac{1}{\sigma^2(x)}\exp\!\left(2\int_0^x \frac{\mu(y)}{\sigma^2(y)}\,dy\right).

Worked Examples

1
Apply the stationary density formula with mu(x) = -alpha x, sigma^2(x) = sigma^2.
$p_s(x) \propto \exp\!\left(2\int_0^x \frac{-\alpha y}{\sigma^2}\,dy\right) = \exp\!\left(-\frac{\alpha x^2}{\sigma^2}\right)$
2
Normalize to get the Gaussian density.
$p_s(x) = \sqrt{\frac{\alpha}{\pi\sigma^2}}\,\exp\!\left(-\frac{\alpha x^2}{\sigma^2}\right) = N\!\left(0,\, \frac{\sigma^2}{2\alpha}\right)$

✓ Answer

The stationary distribution of the OU process is N(0, sigma^2/(2*alpha)): a Gaussian centered at 0 with variance sigma^2/(2*alpha).

Practice Problems

Mediumapplication

Compute the generator L of geometric Brownian motion dS_t = mu S_t dt + sigma S_t dW_t and use Dynkin's formula to find E[S_t^2].

Hardfree response

State the connection between the backward Kolmogorov equation and the Feynman-Kac formula.

Common Mistakes

Common Mistake

The Fokker-Planck and backward Kolmogorov equations are the same

The forward (Fokker-Planck) equation evolves the density in time and space variables (t,x); the backward equation involves the initial condition and runs backward in time.

Common Mistake

The generator L only involves the diffusion coefficient

The generator Lf = mu f' + (sigma^2/2) f'' involves both the drift mu and the diffusion sigma^2; both contribute to how the process evolves.

Quiz

Historical Background

The connection between Brownian motion and the heat equation was observed by Einstein in 1905 and made mathematically precise by Kolmogorov in 1931 through his forward and backward equations. Ito's 1944 theory of stochastic integration provided the analytical foundation for the general theory of diffusions. The Fokker-Planck equation was developed independently by Adriaan Fokker (1914) and Max Planck (1917) in the context of physical diffusion, and unified with the probabilistic theory by Kolmogorov.

1905
Einstein derives the diffusion equation for Brownian motion
Albert Einstein
1914
Fokker derives the forward equation for Brownian motion in an external field
Adriaan Fokker
1931
Kolmogorov derives both the forward (Fokker-Planck) and backward equations
Andrei Kolmogorov
1944
Ito's stochastic integral provides the foundation for the general theory of diffusions
Kiyosi Ito

Summary

A diffusion process solves an Ito SDE; it is a continuous-path Markov process characterized by drift mu and diffusion sigma.
The infinitesimal generator Lf = mu f' + (sigma^2/2) f'' encodes how the process evolves; Dynkin's formula uses L to compute expected values.
The Fokker-Planck (forward Kolmogorov) equation governs the evolution of the probability density: partial_t p = -partial_x(mu p) + (1/2) partial_{xx}(sigma^2 p).
The Feynman-Kac formula: E[g(X_T)|X_t=x] solves the backward Kolmogorov PDE, connecting SDEs to PDEs.

References

BookKaratzas, I. and Shreve, S. -- Brownian Motion and Stochastic Calculus
BookOksendal, B. -- Stochastic Differential Equations
WebsiteWikipedia -- Fokker-Planck equation