Yeah if you were to discretize it, then you'd get random walks with the only difference being the variance of movement from the last point. Wish I could find this one source that talked about continuous time noise processes - it was pretty interesting. If you assign a random variable to every point t on a continuum, it's not a real process (in terms of the definitions set forth in Measure Theory), so you have to use other methods to define them, like limits of discrete processes.