I've downloaded a whole book about Brownian Motion (BM)!! Can someone please explain to me
How many types of Brownian motions do we have?
And When somebody says "Brownian motion generated by a random walk", which one of those types of Brownian motion it is?
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
and $X_t $ has homogeneous stable independent increment.
$ 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.