3
$\begingroup$

Most treatments of Poisson point processes I have encountered define them on $\mathbb{R}^d$, with the key property that the number of points in a bounded region $R$ will have a Poisson distribution with parameter equal to the integral of the intensity function over the region $R$, independent of the number of points in any bounded region disjoint from $R$.

Can we define a Poisson point process on a Riemannian manifold $(M,g)$? If so, are there any constraints on the kinds of manifolds for which we can define Poisson point processes (orientability, etc)? A text to reference would be very helpful.

$\endgroup$
5
$\begingroup$

If $(E,\mathcal E,\mu)$ is a non-atomic $\sigma$-finite measure space such that the diagonal in $E\times E$ is $\mathcal E\otimes \mathcal E$-measurable, then there is a Poisson point process on $E$ with intensity measure $\mu$. See section 2.5 in Poisson Processes by J.F.C. Kingman. In particular, a Riemannian manifold supports a Poisson point process with intensity equal to the Riemannian "volume" measure.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.