**Exercise 1.**

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 .