Wiener sausage

: "For the food sometimes called a Wiener (sausage), see hot dog or Vienna sausage."In the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time "t", given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after Norbert Wiener by harvs|txt=yes|authorlink=M. D. Donsker |author2-link=S. R. Srinivasa Varadhan|first=M. D.|last= Donsker |first2=S. R. Srinivasa|last2= Varadhan|year=1975 because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" means "Viennese" in German.

The Wiener sausage is one of the simplest non-Markovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction. It was first described by harvs|txt=yes|authorlink=Frank Spitzer|first=Frank|last= Spitzer|year=1964, and it was used by harvs|txt=yes|authorlink=Mark Kac |author2-link=Joaquin Mazdak Luttinger |first=Mark|last= Kac |first2=Joaquin Mazdak|last2= Luttinger|year=1973|year2=1974 to explain results of a Bose–Einstein condensate, with proofs published by harvs|txt=yes|authorlink=M. D. Donsker |author2-link=S. R. Srinivasa Varadhan|first=M. D.|last= Donsker |first2=S. R. Srinivasa|last2= Varadhan|year=1975.

Definitions

The Wiener sausage "W"_δ("t") of radius δ and length "t" is the set-valued random variable on Brownian paths b (in some Euclidean space) defined by:$W\_delta(t)(\{\; b\})$ is the set of points within a distance δ of some point b("x") of the path b with 0≤"x"≤"t".

The volume of the Wiener sausage

There has been a lot of work on the behavior of the volume (Lebesgue measure) |"W"_δ("t")| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long ("t"→∞).

harvtxt|Spitzer|1964 showed that in 3 dimensions the expected value of the volume of the sausage is:$E(|W\_delta(t)|)\; =\; 2pidelta\; t\; +\; 4delta^2sqrt\{2pi\; t\}\; +4pidelta^3/3.$In dimension "d" at least 3 the volume of the Wiener sausage is asymptotic to :$delta^\{d-2\}\; pi^\{d/2\}2t/Gamma((d-2)/2)$as "t" tends to infinity. In dimensions 1 and 2 this formula gets replaced by $sqrt\{8t/pi\}$ and $2pi\; t/log(t)$ respectively.

harvtxt|Whitman|1964, a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compacts sets than balls.

References

*citation|first= M. D.|last= Donsker |first2=S. R. S. |last2=Varadhan|title=Asymptotics for the Wiener sausage|journal=Communications in Pure and Applied Mathematics| volume =28 |year=1975|pages= 525–565
doi=10.1002/cpa.3160280406
*springer
title=Wiener sausage
id=W/w130100
first=F. den |last=Hollander
*citation|id=MR|0342114|last= Kac|first= M.|last2= Luttinger|first2= J. M. |title=Bose-Einstein condensation in the presence of impurities. |journal=J. Mathematical Phys.|volume= 14 |year=1973|pages= 1626--1628|doi=10.1063/1.1666234
*citation|id=MR|0342115|last= Kac|first= M.|last2= Luttinger|first2= J. M. |title=Bose-Einstein condensation in the presence of impurities. II. |journal=J. Mathematical Phys.|volume= 15 |year=1974|pages= 183--186
doi=10.1063/1.1666617
*citation|id=MR|2105995|last= Simon|first= Barry|title= Functional integration and quantum physics|publisher= AMS Chelsea Publishing|place= Providence, RI|year= 2005|isbn= 0-8218-3582-3|ISBN status=May be invalid - please double check Especially chapter 22.
*citation|first=F.|last= Spitzer|title=Electrostatic capacity, heat flow and Brownian motion|journal= Probability Theory and Related Fields |volume= 3 |year=1964|pages= 110–121|doi=10.1007/BF00535970
*citation|last=Spitzer|title= Principles of random walks|id=MR|0171290
first= Frank
series=Graduate Texts in Mathematics|volume= 34|publisher= Springer-Verlag|place= New York-Heidelberg, |year=1976|page=40 (Reprint of 1964 edition)
*citation|id=MR|1717054
last=Sznitman|first= Alain-Sol
title=Brownian motion, obstacles and random media
series=Springer Monographs in Mathematics|publisher= Springer-Verlag|place= Berlin|year= 1998|isbn= 3-540-64554-3|ISBN status=May be invalid - please double check An advanced monograph covering the Wiener sausage.
*citation|last=Whitman|first=Walter William|title=Some Strong Laws for Random Walks and Brownian Motion|series= PhD Thesis|year=1964|publisher=Cornell U.