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?



Properties of the time integral of Wiener process
As an integral of a zero-mean Gaussian process, your $X\_T$ is a zero-mean Gaussian process as well. Its covariance function can be calculated via


$$ c(s, t) = \int^s\_0 \int^t\_0 \min(u, v) \; du \; dv \; , $$


which yields


$$ c(s, t) = \frac{\min(s, t)^2}{6} \left( 3 \max(s, t) - \min(s, t) \right) . $$


In terms of sample functions, your expectation is correct. Being an integral of a random process with (almost surely) continuous sample paths, it indeed has (almost surely) differentiable sample paths.


It does not have independent increments, as discussed [here](https://math.stackexchange.com/questions/221032/independent-increments).


