When studying functionals of a Brownian motion, it may be useful to embed this functional into a larger dimensional Markov process.

Consider the case of the Levy area

where , , is a two dimensional Brownian motion started at 0. We can write

where . Since , we interpret as (two times) the algebraic area swept out in the plane by the Brownian curve up to time . The process is not a Markov process in its own natural filtration. However, if we consider the 3-dimensional process

then is solution of a stochastic differential equation

As a consequence is a Markov process with generator

where are the following vector fields

Observe that the Lie bracket

Thus, for every , is a basis of . From the celebrated Hormander’s theorem, this implies that for every the random variable has a smooth density with respect to the Lebesgue measure of . In particular also has smooth density whenever . We are interested in an expression for this density. The first idea is to reduce the complexity of the random variable by making use of symmetries.

**Lemma: **Let , . Then, the couple

is a Markov process with generator

**Proof:**

From Ito’s formula, we have

Since the two processes

are two independent Brownian motions, the conclusion easily follows.

We are now ready to prove the celebrated Levy’s area formula.

**Theorem: **For and , and

**Proof:**

First, we observe that by rotational symmetry of the Brownian motion , we have

Then, according to the previous lemma,

where is a Brownian motion independent from . We deduce

As we have seen, solves a stochastic differential equation

where is a one-dimensional Brownian motion.

One considers then the new probability

where is the natural filtration of . Observe that

Therefore

In particular, one deduces that

which proves that is a martingale. By using this change of probability, if is a bounded and Borel function, we have

Putting things together, we are thus let with the computation of the distribution of under the probability . From Girsanov’s theorem, the process

is a Brownian motion under the probability . Thus

In law, this is the stochastic differential equation solved by where

We deduce that is distributed as , the norm of a two-dimensional Ornstein Uhlenbeck process with parameter . Since is a Gaussian random variable with mean 0 and variance , the conclusion follows from standard computations about the Gaussian distribution.

This formula is due to Paul Levy who originally used a series expansion of the Brownian motion. The proof we present here is due to Marc Yor.

The Levy’s area formula has several interesting consequences. First, when , we deduce that

This gives a formula for the characteristic function of the algebraic stochastic area within the Brownian loop with length . Inverting this Fourier transform yields

Next, integrating the Levy’s area formula with respect to the distribution of yields the characteristic function of :

Unfortunately, this Fourier transform may not be easily inverted. However, one may deduce from it the following formula (due to Biane-Yor): For ,

where is the Dirichlet function. This provides an unexpected and fascinating connection with the Riemann zeta function.