---
tag: QD02B3DEMR
kind: concept
name: Martin boundary
msc: [31C35, 31B25, 60J50]
sources:
  - url: https://en.wikipedia.org/wiki/Poisson_boundary
---

## Summary

The Martin boundary of a domain is an ideal boundary.
It is built from the limits of the [[Martin kernel|1DBC9FYF7M]] as the pole tends to the edge of the domain.
Each of its points is a positive harmonic function, and every positive harmonic function on the domain is an integral of these over the boundary.
R. S. Martin introduced it in 1941 to extend the Poisson representation from the ball to arbitrary domains.
For smooth or Lipschitz domains it coincides with the Euclidean boundary, but for domains with slits or cusps it can be quite different.
In probability it is the boundary seen by Brownian motion and by Markov chains through Doob's $h$-transforms.

## Explanations

### L1 human

The geometric boundary of a domain may not reflect how harmonic functions see it.
For a disc with a slit cut along a radius, the two sides of the slit behave like different boundary points.
The Martin boundary separates such points by asking which positive harmonic functions arise as limits of normalised Green's functions.
For the unit ball, the Martin boundary is just the unit sphere.