**Definition:*** Let be a probability space. A continuous real-valued process is called a standard Brownian motion if it is a Gaussian process with mean function*

and covariance function

It is seen that is a covariance function, because it is symmetric and for and ,

The distribution probability of a standard Brownian motion is called the Wiener measure. It is probability measure on the space of continuous functions (See Lecture 3).

Similarly, a -dimensional stochastic process is called a standard Brownian motion if

where the processes are independent standard Brownian motions.

Of course, the definition of Brownian motion is worth only because such an object exists.

**Theorem.**

There exist a probability space and a stochastic process on it which is a standard Brownian motion.

**Proof**

From the Daniell-Kolmogorov existence theorem, there exists a probability space and a Gaussian process on it, whose mean function is and covariance function is

We have for and :

Therefore, by using the Kolmogorov continuity theorem, there exists a modification of whose paths are locally -Hölder if

From the previous proof, we also easily deduce that the paths of a standard Brownian motion are locally -Hölder for every . It will later be shown that they are not -Hölder (It is a consequence of the law of iterated logarithm).

The following exercises give some first basic properties of Brownian motion. In these exercises, is a standard one-dimensional Brownian motion.

**Exercise.**

Show the following properties:

- a.s.;
- For any , the process is a standard Brownian motion;
- For any , the random variable is independent of the -algebra .

**Exercise.***(Symmetry property of the Brownian motion)*

- Show that the process is a standard Brownian motion.
- More generally, show that if
is a -dimensional Brownian motion and if is an orthogonal matrix, then is a standard Brownian motion.

**Exercise.***(Scaling property of the Brownian motion)*

*Show that for every , the process has the same distribution as the process . *

**Exercise.***(Time inversion property of Brownian motion)*

- Show that almost surely, .
- Deduce that the process has the same law as the process .

**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.