Let $p, q \in \mathbb{N}$ and $Z$ a gaussian process over $D \subset \mathbb{R}^{p+q}$ : $Z(\cdot) \sim \mathcal{GP}(\mu(\cdot), k(\cdot,\cdot))$.

For $(x_1, \ldots, x_p, y_1, \ldots, y_q)^T \in D$, we define an associated process $W$ as

\begin{align} & W(x_1, \ldots, x_p, y_1, \ldots, y_q) \\[10pt] = {} & \frac{\exp[Z(x_1, \ldots , x_p, y_1, \ldots, y_q)]}{\int_{u_1}\cdots\int_{u_q} \exp[Z(x_1, \ldots, x_p, u_1, \ldots, u_q)] \,du_1 \cdots \,du_q}. \end{align}

How to find the distribution of $W$ ?

I considered the following lead : if $Z$ has continuous paths, then we can see the integral as the limit of a Riemann sum. However, I found that there was no closed-form formula for the distribution of a sum of dependant log-normal variables...

I would be grateful if some of you could give me some hints, other leads I can follow or for any related answer :)

Thanks in advance !