Lévy's Characterization of Brownian Motion

Lévy's theorem provides a characterization of Brownian motion via the martingale property and quadratic variation.

Statement

Theorem4.1Lévy's characterization

A continuous process $(M_t)_{t \geq 0}$ with $M_0 = 0$ is a Brownian motion if and only if:

$(M_t)$ is a martingale.
$[M]_t = t$ (quadratic variation equals $t$ ).

This theorem is powerful: to verify a process is Brownian, we need only check the martingale property and quadratic variation, not the full Gaussian increment structure.

ExampleTime-changed Brownian motion

If $(B_t)$ is Brownian motion and $(A_t)$ is a continuous increasing process with $A_0 = 0$ , define $M_t = B_{A_t}$ . If $A_t = t$ (the identity), then $M_t$ is Brownian. More generally, $M_t$ is a time-changed Brownian motion.

Application: Donsker's invariance principle

Theorem4.2Donsker's theorem

Let $X_1, X_2, \ldots$ be i.i.d. with mean 0 and variance 1. Define the random walk scaled to continuous time:

$W_n(t) = \frac{1}{\sqrt{n}} \sum_{i=1}^{\lfloor nt \rfloor} X_i.$

Then $W_n \Rightarrow B$ in distribution (in the space of continuous functions), where $B$ is Brownian motion.

Donsker's theorem shows that Brownian motion is the universal scaling limit of random walks.

Summary

Lévy's characterization simplifies verification of Brownian motion and provides a foundation for stochastic calculus, where processes are often constructed as martingales with specified quadratic variation.