In the next few Lectures we will illustrate through several examples of application the power of the stochastic integration theory.

We start with a study of the multidimensional Brownian motion. As already pointed out, a multidimensional stochastic process , is called a Brownian motion if the processes , , are independent Brownian motions. In the sequel we denote by the Laplace operator on , that is

The following result is an easy consequence of the Itō’s formula.

**Proposition.*** Let be a function that is once continuously differentiable with respect to its first variable and twice continuously differentiable with respect to its second variable and let be a -dimensional Brownian motion. The process
is a local martingale. If moreover is such that
for some continuous function and some constant , then is a martingale.*

In particular, if is a harmonic function, i.e. , and if is a multidimensional Brownian motion, then the process is a local martingale. As we will see it later, this nice fact has many consequences. A first nice application is the study of recurrence or transience of the multidimensional Brownian motion paths. As we have seen before, the Brownian motion recurrent: It reaches any value with probability 1. In higher dimensions, the situation is more subtle.

Let be a -dimensional Brownian motion with . For and , we consider the stopping time

**Proposition.** * For ,
*

**Proof.** For , we consider the function

A straightforward computation shows that . The process is therefore a martingale, which implies . This yields

Since

we deduce that

By letting , we get

**Corolllary.*** For ,
As a consequence, for the Brownian motion is recurrent, that is, for every non empty set ,
*

Though the two-dimensional Brownian motion is recurrent, points are always polar.

**Proposition.**

For every ,

**Proof.** It suffices to prove that for every , , . We have

Since , we get

As we have just seen, the two-dimensional Brownian motion will hit every non empty open set with probability one. The situation is different in dimension higher than 3: Brownian motion paths will eventually leave any bounded set with probability one.

**Proposition.** *Let be a -dimensional Brownian motion. If then almost surely
*

**Proof.** Let us assume . Let where . Since will never hit the point , we can consider the process which is seen to be a positive local martingale from Itō’s formula. A positive local martingale is always a supermartingale. Therefore from the Doob’s convergence theorem, the process converges almost surely when to an integrable and non negative random variable . From Fatou’s lemma, we have . By the scaling property of the Brownian motion, it is clear that . We conclude

**Exercise** *(Probabilistic proof of Liouville theorem) By using martingale methods, prove that if is a bounded harmonic function, then is constant.*