Say $x(t)$ is a Gaussian white noise process with $\sigma^2$ as variance.
Now what is the probabilistic distribution of $Y(t)$?
$$ Y(t) = \int_{0}^t x(s) e^{a(t-s)} ds $$
say $a \in \mathbb{R}$. I want to know the PDF.
What I know
when $a = 0$, $Y(t)$ is Wiener process and I know the PDF.