# Wiener processes and Brownian motion

## Wiener proceses

### Independent increments

The changes in any non-overlapping time increments are independent.

Formally:

$$t_0<t_1<t_2<...<t_m$$

With $$X_t$$

$$X_{t_1}-X_{t_0}$$ is indepentent from $$X_{t_2}-X{t_1}$$ etc.

### Wiener processes

A Wiener process is a process $$W_t$$ with independent increments, which: + Is continuous + Has normally distributed increments.

Can be constructed as limit of random walk. Can also be constructed as integral of Gaussian noise?

## Brownian motion

### Brownian motion

brownian motion in stats. given we start at a, what is chance be end up at b? normal. do 1d then multi d