---
tag: 4HS3C12472
kind: concept
name: Feller process
msc: [60J25, 60J35, 47D07]
specializes: [0QH5ENKZM1]
sources:
  - url: https://en.wikipedia.org/wiki/Feller_process
same_as:
  wikidata: Q5442452
  wikipedia: Feller process
---

## Summary

A Feller process is a Markov process whose transition semigroup acts nicely on continuous functions that vanish at infinity.
The semigroup maps such functions to such functions and is continuous in time at $0$.
These two conditions give a version with right-continuous paths that have left limits, the strong Markov property, and an [[infinitesimal generator|VPA41PWRR1]].
Brownian motion, [[Lévy processes|8NT5CFG2ZR]] and many [[diffusions|9HDQ094ADQ]] are Feller processes.

## Explanations

### L1 human

A Markov process forgets its past: only the present position matters for the future.
For a Feller process, the future also depends continuously on the starting point.
If two particles start very close, their laws at a later time are close.
Also, after a very short time the process has barely moved in law.
Brownian motion on the line is the main example.