Let $W_t$ be a Wiener process and consider the time integral $$ X_T:= \int_0^T W_t dt $$

It is often mentionend in literature that $X_T$ is a Gaussian with mean 0 and variance $T^3/6$.

I am interested in learning more about the process $X_T$ for $T>0$. Except for the description of the individual random variable $X_T$ I have not found much.

But there must be more to it. It is if one stopped talking about a Wiener process after mentioning that $W_t$ is a Gaussian with mean 0 and variance $t$.

As an example, it would like to have sample paths discussed. I wonder, since as being defined by an integral over a continuous function it should be differentiable. On the other hand it feels wrong, since I have never encountered a non-degenerate stochastic process that is (a.s.) differentiable.

Do you know were this process is discussed in depth?