On page 2 of *"Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm"* by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at [arXiv](https://arxiv.org/abs/1312.6536), they claim the following on the bottom right of the page:



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> "...in the stationary case the intensity of the Cox process is equal to the expectation of Λ(x)"
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My assumption is that by "intensity", they are referring to some extension of $$ \lambda (t)=\lim \_{h\downarrow 0}{\frac {1}{h}}\mathbb {E} [N(t+h)-N(t)|{\mathcal {F}}\_{t}],$$ as taken from wikipedia at <https://en.wikipedia.org/wiki/Intensity_of_counting_processes>.


I mentioned "extension" because in the Diggle–Moraga–Rowlingson–Taylor paper, they define the Cox process on $\mathbb {R}^2$, whereas the intensity definition from Wikipedia is to do with processes defined on $\mathbb{R}$.


My issue is that I'm unable to find a proof of their claim. If someone more experienced in this field could explain to me why the authors' claim is true / guide me to a relevant source, I would really appreciate it.
Thank you.



Question about the intensity of a cox process, Diggle–Moraga–Rowlingson–Taylor (2013)
The [intensity function](https://mathworld.wolfram.com/IntensityFunction.html) of a [point process](https://en.wikipedia.org/wiki/Point_process#Definition) $(N\_B)$ over $\mathbb R^d$ is defined as the density of the intensity measure $\mu$ of $(N\_B)$ relative to the Lebesgue measure over $\mathbb R^d$ (in your case, the Lebesgue measure over $\mathbb R^2$).


In turn, the [intensity measure](https://mathworld.wolfram.com/IntensityMeasure.html) $\mu$ of $(N\_B)$ is defined by the formula $\mu(B):=EN\_B$ for all Borel sets $B$, in your case all Borel subsets $B$ of $\mathbb R^2$.




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Now, conditionally on a nonnegative-valued process $\Lambda=(\Lambda\_x)\_{x\in\mathbb R^2}$, the Cox process $(N\_B)$ over $\mathbb R^2$ is an inhomogeneous Poisson process with intensity $(\Lambda\_x)$. So, for all Borel subsets $B$ of $\mathbb R^2$  

$$E(N\_B|\Lambda)=\int\_B\Lambda\_x\,dx,$$
whence
$$EN\_B=EE(N\_B|\Lambda)=E\int\_B \Lambda\_x\,dx=\int\_B E\Lambda\_x\,dx.$$
Thus, $(E\Lambda\_x)\_{x\in\mathbb R^2}$ is indeed the intensity function of the Cox process; the stationarity assumption is not needed here.


