Paths of Brownian motion
For all omegas (or almost surely every) it holds that the path is continous, undifferentiable... and yes, one particular omega corresponds to one particular path. So if you generate BM in R, one generation corresponds in this case to one omega.
E.g. this code gives 10 different paths, i.e. ten different $B_t(\omega_i), i=1, ..., 10$.
\begin{verbatim}
n = 1000
t = 100
No.Ex = 10
steps = seq(0,t,length=n+1)
A = replicate(No.Ex, {
bm <- c(0, cumsum(rnorm(n,0,sqrt(t/n))))
})
cols = rainbow(No.Ex)
matplot(A, type = "l", col = cols, lty = 1)
\end{verbatim}