Wiener sausage

For the food sometimes called a Wiener (sausage), see hot dog or Vienna sausage.

A long, thin Wiener sausage in 3 dimensions

A short, fat Wiener sausage in 2 dimensions

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 M. D. Donsker and S. R. Srinivasa Varadhan (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 Frank Spitzer (1964), and it was used by Mark Kac and Joaquin Mazdak Luttinger (1973, 1974) to explain results of a Bose–Einstein condensate, with proofs published by M. D. Donsker and S. R. Srinivasa Varadhan (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→∞).

Spitzer (1964) showed that in 3 dimensions the expected value of the volume of the sausage is

E(|W_\delta(t)|) = 2\pi\delta t - 4\delta^2\sqrt{2\pi t} +4\pi\delta^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 $2{\pi}t/\log(t)$ respectively. Whitman (1964), a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compact sets than balls.

References

Donsker, M. D.; Varadhan, S. R. S. (1975), "Asymptotics for the Wiener sausage", Communications in Pure and Applied Mathematics 28 (4): 525–565, doi:10.1002/cpa.3160280406
Hollander, F. den (2001), "Wiener sausage", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Kac, M.; Luttinger, J. M. (1973), "Bose-Einstein condensation in the presence of impurities", J. Mathematical Phys. 14 (11): 1626–1628, doi:10.1063/1.1666234, MR 0342114
Kac, M.; Luttinger, J. M. (1974), "Bose-Einstein condensation in the presence of impurities. II", J. Mathematical Phys. 15 (2): 183–186, doi:10.1063/1.1666617, MR 0342115
Simon, Barry (2005), Functional integration and quantum physics, Providence, RI: AMS Chelsea Publishing, ISBN 0-8218-3582-3, MR 2105995 Especially chapter 22.
Spitzer, F. (1964), "Electrostatic capacity, heat flow and Brownian motion", Probability Theory and Related Fields 3 (2): 110–121, doi:10.1007/BF00535970
Spitzer, Frank (1976), Principles of random walks, Graduate Texts in Mathematics 34, New York-Heidelberg,: Springer-Verlag, p. 40, MR 0171290 (Reprint of 1964 edition)
Sznitman, Alain-Sol (1998), Brownian motion, obstacles and random media, Springer Monographs in Mathematics, Berlin: Springer-Verlag, ISBN 3-540-64554-3, MR 1717054 An advanced monograph covering the Wiener sausage.
Whitman, Walter William (1964), Some Strong Laws for Random Walks and Brownian Motion, PhD Thesis, Cornell U.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

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