Let $X(x):= \exp(Y(x))$ for $x\in D\subseteq \mathbb{R}^d$ for a domain $D$, where $Y$ is a Gaussian random field with some smooth covariance function $c(\cdot,\cdot)\colon D\times D\to \mathbb{R}$. Can I say anything about the law of $$ \int_D X\,dx? $$ For the case of $\int_D Y\,dx$, it is easy to show that it is a normal random variable.
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$\begingroup$ See literature in "Gaussian multiplicative chaos" for results for singular covariance function c. $\endgroup$– Thomas KojarMar 13, 2019 at 19:52
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$\begingroup$ For smooth covariance and bounded domain you can show that the Laplace transform is analytic around an interval and thus uniquely determined by its moments. $\endgroup$– Thomas KojarMar 13, 2019 at 19:55
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You can express the moments $$\mathbb E\left(\int_D X(s)\ ds\right)^n=\int_{D^n}\mathbb E\exp\left(Y(s_1)+\cdots+Y(s_n)\right)\ ds_1\cdots ds_n$$ using the fact that, for a centered Gaussian r.v. $Z$, $\mathbb E e^Z=e^{{\frac12}\mathbb EZ^2}$, and expressing the variance of $Y(s_1)+\cdots+Y(s_n)$ as $\sum_{i,j}c(s_i,s_j)$. The law may not be characterized by its moments, though.
