**Exercise 1.** *The Hermite polynomial of order is defined as
*

- Compute .
- Show that if is a Brownian motion, then the process is a martingale.
- Show that

**Exercise 2.** *(Probabilistic proof of Liouville theorem) By using martingale methods, prove that if is a bounded harmonic function, then is constant.*

**Exercise 3.** *Show that if is a local martingale of a Brownian filtration , then there is a unique progressively measurable process such that for every , and *

**Exercise 4 ***[Skew-product decomposition]*

Let be a complex Brownian motion started at .

- Show that for ,

- Show that there exists a complex Brownian motion such that

where . - Show that the process is independent from the Brownian motion .
- We denote which can be interpreted as a winding number around 0 of the complex Brownian motion paths. For , we consider the stopping time

- Compute for every , the distribution of the random variable

- Prove Spitzer theorem: In distribution, we have the following convergence

where is a Cauchy random variable with parameter 1 that is a random variable with density .