Wiener measure in nLab

Contents

Contents

Idea

The Wiener measure is a measure on the space of continuous paths in a given manifold. The Lebesgue integral with respect to Wiener’s measure is called the Wiener integral.

The Wiener measure serves to make precise the path integral quantization for the (charged) non-relativistic particle (that of the relativistic particle may be amenable to Wiener measure methods via Wick rotation, i.e. analytic continuation to imaginary time. ).

• Brownian motion

References

Named after Norbert Wiener‘s discussion of Brownian motion:

• Norbert Wiener, The Average of an Analytic Functional and the Brownian Movement, Proceedings of the National Academy of Sciences of the United States of America, 7 10 (1921) 294-298 [jstor:84434]

Introduction:

• Tamas Szabados, An elementary introduction to the Wiener process and stochastic integrals, Studia Scientiarum Mathematicarum Hungarica 31 (1996) 249-297 [arXiv:1008.1510]

History:

• Arthur Genthon, The concept of velocity in the history of Brownian motion – From physics to mathematics and back, Eur. Phys. J. H 45 (2020) 49-105 [arXiv:2006.05399, doi:10.1140/epjh/e2020-10009-8]

See also:

• Enc. Math, Wiener integral

• PlanetMath Wiener measure

A textbook account in the context of path integral quantization:

• Barry Simon, around p. 49 in: Functional integration and quantum physics, AMS Chelsea Publ., Providence, 2005