If $f(x)$ is Gaussian process, i.e.,
$$ f(x) = GP [m(x), \kappa(x, x')] $$
can we say anything about the distribution of the integral of the exponential of this, i.e.,
$$ F = \int e^{f(x)} dx $$
or, if it's more helpful, the logarithm $ln[F]$?
This previous post uses the Riemann sum to show the distribution of the integral of a Gaussian process is Gaussian, though I've had no luck applying the same idea to the above problem. This post discusses a similar problem but for the complex case. A paper linked in that post discusses the distribution of $\int_0^T exp(u X(t)) dt$ where $X(t)$ is Brownian motion, though I'm not sure I understand the connection between Gaussian processes and Brownian motion enough for this to help.
