---
tag: CPA34DVD8J
kind: concept
name: Abstract Wiener space
msc: [28C20, 60B11, 46G12]
sources:
  - url: https://en.wikipedia.org/wiki/Abstract_Wiener_space
same_as:
  wikidata: Q4669933
  wikipedia: Abstract Wiener space
---

## Summary

An abstract Wiener space is a [[Hilbert space|HM2HHE3J51]] sitting densely inside a larger [[Banach space|06GZ956657]].
The standard Gaussian [[cylinder set measure|G95YDNFXQN]] of the Hilbert space must become a true [[Gaussian measure|P0DV2TCH1N]] on the Banach space.
The notion was introduced by Leonard Gross.
It is the general frame for Wiener measure, Malliavin calculus and Gaussian free fields.

## Explanations

### L1 human

There is no standard Gaussian measure on an infinite-dimensional Hilbert space: a typical point would have infinite length.
The fix is to enlarge the space, with a weaker norm, until the typical points fit.
The classical example is Brownian motion.
The Hilbert space is the space of paths with square-integrable derivative, and the larger space is the space of all continuous paths, where Brownian paths actually live.