Starting from this question which is basically exactly what I was wanting to ask initially, and then its very-well written top answer, I want to make sure everything is clear to me.
Consider the Brownian motion $B_t$.
Do we agree :
that, for any given, specific value of $t$, $dB_t$ follows a normal distribution ?
the covariance of which is infinite (multiple of a Dirac) ?
Or does that not even make sense, thinking of these objects as "functions" of $t$ ?