Brown motion process

I've downloaded a whole book about Brownian Motion (BM)!! Can someone please explain to me

1. How many types of Brownian motions do we have?

2. And When somebody says "Brownian motion generated by a random walk", which one of those types of Brownian motion it is?

3. How do we judge if a stochastic process is a Brownian motion? IS it that any process which has the characteristics mentioned below is a BM process?

My current knowledge is this: If $X_t$ be a BM process then

1. $X_0=0$

2. and $X_t$ has homogeneous stable independent increment.

3. $X_t$ is normally distributed $N(0,c^2t)$

If $c=1$ then we have a Standard BM and we show it by $B_t$.

BM is not a stationary process, but its increment is stationary.

• What counts as "a type" of Brownian motion for you? – Henning Makholm Dec 25 '16 at 9:53
• Only if in knew ... We have this question in the book that is asking us to compare two Brownian motion. I think (not sure) maybe it means like: Standard BM, BM with drift, Geometrical BM, Brownian bridge, 2-D BM? maybe? I have no idea...just guessing – Tayebe Dec 25 '16 at 10:18