Suppose that $\Phi\_1$ and $\Phi\_2$ represents two independent Poisson point processes respectively with intensity $\lambda\_1$ and $\lambda\_2$ (therefore). We know very well different operations on Poisson processes preserving the Poisson law like superposition, dilation and thinning.


I would like to know if there is any work on the law of Minkowski sum of two Poisson processes. By Minkowski sum of $\Phi\_1=\sum \delta\_{x^1\_i}$ and $\Phi\_2=\sum \delta\_{x^2\_i}$, I mean a new Point processes like: $\Phi\_3=\displaystyle\sum\_{i,j} \delta\_{x^1\_i+x^2\_j}$.


I could not find any related literature on this topic, except works on germ-grain model, so any help is appreciated.



On Minkowski sum of two independent Poisson point processes
Indeed the process is not locally finite if your processes are homogeneous on $R^d$ (that can be deduced from the computations below).


If on the other hand your intensity measures $\lambda\_1$ and $\lambda\_2$ are finite, then you have a point process that is not Poisson.


Let $K$ be some compact set. Then $$\Phi\_3(K)=\sum\_{x\in \Phi\_1}\Phi\_2(K-x).$$ Using Mecke formula yields that 
$$
E \Phi\_3(K)=\int \lambda\_1(dx) E(\Phi\_2(K-x))=\int \lambda\_2(K-x)\lambda\_1(dx),$$
and therefore for $K,L$ disjoint
$$
E\Phi\_3(K)E\Phi\_3(L)=\int \int \lambda\_2(K-x)\lambda\_2(L-y)\lambda\_1(dx)\lambda\_1(dy)$$
Similar computations yield
$$E\Phi\_3(K)\Phi\_3(L)=\int\int E \Phi\_2(K-x)\Phi\_2(L-y) \lambda\_1(dx)\lambda\_1(dy)$$
For any $K,L,x,y,$ writing $$\Phi\_2(K-x)=\Phi\_2((K-x)\cap(L-y))+\Phi\_2((K-x)\setminus(L-y))$$, idem for $\Phi\_2(L-y)$, and developing the product yields that in non trivial cases for well chosen $L,K$, $$E\Phi\_3(K)\Phi\_3(L)>E\Phi\_3(K)E\Phi\_3(L)$$ 


