---
tag: 46436KE7NT
kind: concept
name: Excessive function
msc: [60J45, 31D05]
sources:
  - url: https://en.wikipedia.org/wiki/Hunt_process
---

## Summary

An excessive function of a Markov process is a nonnegative function that decreases on average along the paths of the process.
It is also recovered as the limit of its averages at small times.
Excessive functions are the probabilistic counterpart of nonnegative superharmonic functions.
For Brownian motion the two classes coincide.
For general Markov processes, excessive functions play the role of superharmonic functions in an abstract potential theory.

## Explanations

### L1 human

Imagine a quantity attached to each position, and a particle moving at random.
The function is excessive if its expected value at the particle's later position never exceeds its value at the starting position.
Constants are excessive, and so is the expected total time that the particle will spend in a given region.
For Brownian motion in space, $1/\abs{x}$ is excessive, matching the fact that it is superharmonic.