You must have chaos within you to give birth to a dancing star.

Friedrich Nietzsche

The first time when I heard the term Brownian motion is from a physics class and it was introduced as the random collisions of particles. In the field of Stochastic Differential Equations, it represents a stochastic process.

## Brownian Motion:

**Definition 1.** A real-valued stochastic process is called a * Brownian motion *or

*if*

**Wiener process**- almost surely.
- is for all .
- For all time , the random variables are independent (independent increments).

Note that the construction of Brownian motion is really careful, and it is important to demonstrate the existence of a Brownian motion. One way is to use an orthonormal

basis of and prove the convergence of a resulting series. One can consult the reference for further study, and here I will present the final result.

**Theorem 1 (Existence of one-dimensional Brownian motion). **Let be a probability space on which countably many , independent random variables are defined. Then there exists a 1-dimensional Crownian motion defined for , .

The existence of Brownian motion in can then be extended from the one-dimensional case.

Now we will take a look at the properties of the sample path of Brownian motions. First, recall the definition of Hölder continuity.

**Definition 2. **Let . A function is called ** Hölder continuous** with exponent if there exists a constant such that

We say is ** uniformly Hölder continuous** with exponent if there exists a constant such that

The continuity of sample paths can be established by a theorem of Kolmogorov

**Theorem 2. **Let be a stochastic process with continuous sample paths a.s., such that

for constants , and for all .

Then for each , , and almost every , there exists a constant such that

Hence the sample path is uniformly Hölder continuous with exponent on .

And we have the following result

**Theorem 3. **For almost every and any , the sample path is uniformly Hölder continuous on for each exponent . For each and almost every , the sample path is nowhere Hölder continuous with exponent . In particular, for almost every , the sample path is nowhere differentiable and is of infinite variation on each subinterval.

These properties are important because they also guarantee the continuity of indefinite Ito’s integral and the continuity of some other applications like the solutions of stochastic differential equations.

Another property of Brownian motion is the Markov property, as represented below.

**Definition 3. **If is a stochastic process, the -algebra

is called the * history* of the process up to and including time .

**Definition 4. **If is a -algebra, , then

is a random variable and defined as the * conditional probability *of given .

**Definition 5. **An -valued stochastic process is called a ** Markov process **if

for all and all Borel subset of .

And now we have the following theorem:

**Theorem 4. **Let be an -dimensional Brownian motion. Then is a Markov process, and

for all , and Borel sets .

Note that one can also prove the strong Markov property for Brownian motion. It helps to define the harmonic measure of an Ito diffusion and further find the solution to the generalized Dirichlet problem.

*Reference*

[1] Øksendal, Bernt K. (2013). *Stochastic differential equations: An introduction with applications*. Heidelberg: Springer.

[2] Evans, Lawrence C. *An Introduction to Stochastic Differential Equations. *Department of Mathematics, UC Berkeley.

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