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
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.
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.