Math 2.0

Probability theory and stochastic processes Definition

Stochastic geometry

Builds on Probability spaceRandom variablePoint processPoisson point processEuclidean space

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space, and let EE be a space with a geometric structure, usually the Euclidean space ℝd\R^d. Stochastic geometry is the study of random elements defined on (Ω,ℱ,P)(\Omega, \mathcal{F}, P) whose values are geometric objects in EE. Its basic objects are:

  1. point processes, that is, random locally finite sets or counting measures in EE;
  2. random closed sets, that is, random elements of the family of closed subsets of EE;
  3. random measures on EE, 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 EE.

Remarks

Example: the Poisson point process in ℝd\R^d is a point process, and the Boolean model built from it is a random closed set; both are central models of the field.

Non-example: a fixed lattice ℤd⊂ℝd\Z^d \subset \R^d involves no randomness, so on its own it is a question of deterministic geometry. A randomly shifted lattice U+ℤdU + \Z^d, with UU uniform on [0,1)d[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.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectPoint process, concept, no evidence levelPoint processconceptPoisson point process, concept, no evidence levelPoisson point processconceptProbability space, concept, no evidence levelProbability spaceconceptRandom variable, concept, no evidence levelRandom variableconceptStochastic geometry, definition, no evidence levelStochastic geometrydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Point process concept
  3. Poisson point process concept
  4. Probability space concept
  5. Random variable concept
  6. Stochastic geometry definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stochastic_geometry
About this page
TagGDY62GEC1H
URIm2://GDY62GEC1H
Kinddefinition
WritingW0 stub
MSC60D05, 60G55, 60G57
DefinesStochastic geometry
UsesProbability space, Random variable, Point process, Poisson point process, Euclidean space
Changed2 November 2026

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