I am learning about stochastic processes, in particular Wiener and related processes. Wiener process $W(t)$ has the property that $W(0) = 0$. Does this mean that in fact for $t=0$ Wiener process is not stochastic? In that at $t=0$ its value is not a random variable?
Further. It is stated that for any $t <T$ the random variable $W(T) - W(t)$ follows normal distribution $N(0, T-t)$. Since $W(0) =0$, does it mean then that $W(t)$ itself follows $N(0,t)$ for any $t>0$?
Although $W(0) = 0$, it can still be thought of as a random variable, just a trivial one: $W_0(\omega) = 0$ for all $\omega \in \Omega$. You don't typically need to worry about whether a constant is a random variable or not - it can basically be treated like one for every purpose I can think of.
For your second question, yes, $W(t) \sim N(0,t)$ for all $t > 0$.