It is not clear to me exactly what the Wiener process is. I know the theory, I just don't know if $dW(t) = \epsilon \sqrt(dt)$ where $\epsilon \sim N(0,1)$ and so $dW(t) \sim N(0, d t)$ is the Wiener process or $W(t) = \int_0^t dW(\tau)$ is the Wiener process.
My understanding is that $W(t)$ is the Wiener process and $d W(t)$ is called the differential of the Wiener process.