**Exercise 2.** ** **Solve Exercise 3 in Chapter 1 of the book.

**Exercise 3. ** Solve Exercise 39 in Chapter 1 of the book.

**Exercise 4.*** *The heat kernel on is given by .

- By using the subordination identity show that for ,
- The Bernoulli numbers are defined via the series expansion By using the previous identity show that for , ,

**Exercise 5.*** *Show that the heat kernel on the torus is given by

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- 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 .

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Let . Let be a continuous Gaussian process such that for ,

Show that for every , there is a positive random variable such that , for every and such that for every , \textbf{Hint:} You may use without proof the Garsia-Rodemich-Rumsey inequality: Let and , then there exists a constant such that for any continuous function on , and for all one has:

**Exercise.***(Non-canonical representation of Brownian motion)*

- Show that for , the Riemann integral almost surely exists.
- Show that the process is a standard Brownian motion.

**Exercise. **[Non-differentiability of the Brownian paths]

1) Show that if is differentiable at , then there exist an interval and a constant such that for ,

2) For , let

Show that .

3) Deduce that

**Exercise.**[Fractional Brownian motion] Let . 1) Show that for , the function is square integrable on . 2) Deduce that is a covariance function. 3) A continuous and centered Gaussian process with covariance function is called a fractional Brownian motion with parameter . Show that such process exists and study its Holder sample path regularity. 4) Let be a fractional Brownian motion with parameter . Show that for any , the process is a fractional Brownian motion. 5) Show that for every , the process has the same law as the process

**Exercise. **(Brownian bridge)

Let and .

- Show that the process

is a Gaussian process. Compute its mean function and its covariance function. - Show that is a Brownian motion conditioned to be at time , that is for every , and Borel sets of ,

- Let be two independent sequences of i.i.d. Gaussian random variables with mean 0 and variance 1. By using the Fourier series decomposition of the process , show that the random series

is a Brownian motion on .

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and for , is the restriction of to . Let be a probability measure on such that for every ,

- Show that there exists a right continuous and left limited martingale such that for every ,
- Show that the following properties are equivalent:

1) ;

2) The martingale is uniformly integrable;

3) converges in ;

4) almost surely converges to an integrable and measurable random variable such that

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**Exercise 2. **Let be the open unit ball in . Let in . Show that there exists a smooth vector field on , such that and if is not in .

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**Exercise. **Let be a one-dimensional compact manifold with boundary. Show that is diffeomorphic to a finite union of segments and circles (You may use the appendix in Milnor’s lecture notes).

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Let be a continuous process adapted to a filtration . Let

*where is a closed subset of . Show that is a stopping time of the filtration .*

**Exercise. (Closed martingale)**

*Let be a filtration defined on a probability space and let be an integrable and -measurable random variable. Show that the process is a martingale with respect to the filtration .
*

**Exercise.** *Let be a filtration defined on a probability space and let be a submartingale with respect to the filtration . Show that the function is non-decreasing.
*

**Exercise.** *Let be a filtration defined on a probability space and let be a martingale with respect to the filtration . Let now be a convex function such that for , . Show that the process is a submartingale.*

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