Intensity function $\lambda(u)$ of non-stationary MatérnI hard-core point process?

Question

MatérnI description

In a MatérnI hard-core process, a stationary PPP $\Phi$ defined at $\mathbb{R}^d$ with intensity $\lambda$ is generated. Then the points are removed if there exists others lying inside the same ball of radius $r$, that is, a thinning is applied with deletion probability $f(x,x')=\mathbb{1}(\|x-x'\| < r)$.

It is known that the intensity function of the MatérnI process is: $$ \lambda_{MatérnI} = \lambda\ exp\left\{-\lambda\ d\ b_d\int_o^\infty f(r)\ r^{d-1}\ dr\right\} $$ where $b_d$ is the volume of the unit ball in $\mathbb{R}^d$.

Question

What is the intensity function when $\Phi$ is not stationary, i.e., when $\lambda: \mathbb{R}^d \mapsto \mathbb{R}^+$ is not a constant?

Answer 1

Dr. Møller kindly answered my doubt saying the following:

This intensity function of the Matern hard core process is

$$ \rho(x) = exp(-\nu(B(x,r))) \lambda(x)$$

where $r$ is the hard core parameter and $\lambda$ is the intensity function and $\nu$ is the intensity measure of the PPP.

So it seems like the expression I reached was ok.