I find the two very confusing as some seem to use them interchangeably and some don't seem to. Wiki says they're both the same "…is often called the standard Brownian motion" it says in the "Wiener Process" page.
I understand $B_t$ a Brownian motion is normally distributed $N(0,t)$. But is the "standard" Brownian motion distributed as $N(0,1)$? The name analogous to the standard normal distribution?
Best Answer
Standard Brownian motion is the process you describe: a continuous Gaussian process $B_t$ whose distribution at time $t$ is normal with mean zero and variance $t$ (or in higher dimensions, mean zero and covariance $tI$).
Some authors also use the term "Brownian motion" to refer to translations or scalings of this process; e.g. $B_t + x_0$, which at time $t$ has distribution $N(x_0,t)$, or $c B_t$, which has distribution $N(0, c^2 t)$. So adding the word "standard" serves to clarify when we are not doing that; when we really do mean the process with $N(0,t)$ distribution at time $t$ (i.e. $x_0 = 0$ and $c=1$).
There is no process analogous to Brownian motion that has distribution $N(0,1)$ at time $t$. For one thing, it would have to have either a random starting point, or a jump immediately after time 0, which are typically things we don't want to have in our definition of Brownian motion. (However, you might like to look up the stationary Ornstein-Uhlenbeck process, which is a nontrivial continuous process with random starting point whose distribution at every time $t$ is indeed $N(0,1)$.)
In other contexts, the term "Brownian motion" can refer to other processes that are somehow analogous to standard Euclidean Brownian motion. For instance, on a Riemannian manifold $M$, one can define a stochastic process whose generator is the Laplace-Beltrami operator; this process is often called "Brownian motion on $M$" because it plays that role.