---
tag: GDY62GEC1H
kind: definition
name: Stochastic geometry
msc: [60D05, 60G55, 60G57]
defines: 2J9GB49FHA
uses: [0NFPJVJJHX, 0PNRKMRC55, DNDKBVHS3H, 0MTNR28K60, 0XPR1E85W0]
sources:
  - url: https://en.wikipedia.org/wiki/Stochastic_geometry
---

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]], and let $E$ be a space with a geometric structure, usually the [[Euclidean space|0XPR1E85W0]] $\R^d$.
*Stochastic geometry* is the study of [[random elements|0PNRKMRC55]] defined on $(\Omega, \mathcal{F}, P)$ whose values are geometric objects in $E$.
Its basic objects are:

1. [[point processes|DNDKBVHS3H]], that is, random locally finite sets or counting measures in $E$;
2. random closed sets, that is, random elements of the family of closed subsets of $E$;
3. random measures on $E$, and structures built from the above, such as random tessellations, fibre processes and germ-grain models.

The questions concern the laws of these objects, their mean values such as intensities and volume fractions, and their behaviour under motions of $E$.

## Remarks

Example: the [[Poisson point process|0MTNR28K60]] in $\R^d$ is a point process, and the [[Boolean model|NK3CZTVEJD]] built from it is a random closed set; both are central models of the field.

Non-example: a fixed lattice $\Z^d \subset \R^d$ involves no randomness, so on its own it is a question of deterministic geometry.
A randomly shifted lattice $U + \Z^d$, with $U$ uniform on $[0, 1)^d$, is again a point process and belongs to the field.

The term geometric probability is often used for the older part of the subject, which deals with a fixed number of random points, lines or bodies.