Wiener Sausage Research Articles

ASYMPTOTIC EXPANSION OF THE EXPECTED VOLUME OF THE WIENER SAUSAGE IN EVEN DIMENSIONS

We consider the Wiener sausage for a Brownian motion up to time $t$ associated with a closed ball in even dimensional cases. We obtain the asymptotic expansion of the expected volume of the Wiener sausage for large $t$. The result says that the expansion has many $\log$ terms, which do not appear in odd dimensional cases.

On the speed of the one-dimensional polymer in the large range regime

We consider a Hamiltonian involving the range of the simple random walk and the Wiener sausage so that the walk tends to stretch itself. This Hamiltonian can be easily extended to the multidimensional cases, since the Wiener sausage is well-defined in any dimension. In dimension one, we give a formula for the speed and the spread of the endpoint of the polymer path. Also, we provide the CLT. It can be easily showed that if the self-repelling strength is stronger, the end point is going away faster. This strict monotonicity of speed has not been proven in the literature for the one-dimensional case.

Sensory evaluation of various lamb meat foods by judging consumers

Four kinds of lamb meat foods (Wiener sausage, ham, roasted meat, and roasted leg) were developed based on 8 different genotypes to improve lamb meat quality and quantity, and to study the differences among them. The stringiness, flavour, and odour of the products were judged in food sensory evaluation by a total of 265 (60% male and 40% female) randomly selected consumers. In stringiness and flavour five (from 1 to 5), and concerning odour three (from 1 to 3) categories were created according to decreasing quality. The consumers represented different age groups: below 20 years, between 21–30, 31–40, 41–50, 51–60, 61–70, and above 70 years. The groups of consumers were involved in sheep farming, other animal farming, other agriculture sector, industry, catering, education, other service; administrative department of the state, were students, and others. Analyses of variance were calculated to estimate the effects of the genotype of lamb, the gender, age, and occupation of judging consumers on the sensory ...

The Rate of Decay of the Wiener Sausage in Local Dirichlet Space

In the context of a heat kernel diffusion which admits a Gaussian type estimate with parameter \(\beta \) on a local Dirichlet space, we consider the log asymptotic behavior of the negative exponential moments of the Wiener sausage. We show that the log asymptotic behavior up to time \(t^{\beta }V(x,t)\) is \(-V(x,t)\), which is analogous to the Euclidean result. Here, \(V(x,t)\) represents the mass of the ball of radius \(t\) about a point \(x\) of the local Dirichlet space. The proof of the upper asymptotic uses a known coarse graining technique which must be adapted to the non-transitive setting. This result provides the first such asymptotics for several other contexts, including diffusions on complete Riemannian manifolds with nonnegative Ricci curvature.

On an Upper Bound of the Euler Characteristic of the Wiener Sausage

We study the asymptotic number of the connected components of the complement of a Wiener sausage in the plane. We prove the statement on the limit behaviour of the number of the connected components of the complement of a Wiener sausage with dependance on its radius. As the corollary we obtain the upper bound of the Euler characteristic of the Wiener sausage in the plane.

Moderate deviations for the mutual intersection time of Wiener sausages in subcritical dimensions

In this paper, we study the mutual intersection time of Wiener sausages in subcritical dimensions. Using the new high moment method and the Feynman-Kac semigroup, we obtain the corresponding moderate deviations.

The spectral shift function for planar obstacle scattering at low energy

AbstractLet H signify the free non‐negative Laplacian on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^2$\end{document} and HY the non‐negative Dirichlet Laplacian on the complement Y of a nonpolar compact subset K in the plane. We derive the low‐energy expansion for the Krein spectral shift function (scattering phase) for the obstacle scattering system \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\left\lbrace \,H_Y\hspace*{-3.0pt},\,H\right\rbrace$\end{document} including detailed expressions for the first three coefficients. We use this to investigate the large time behaviour of the expected volume of the pinned Wiener sausage associated to K.

Symmetric rearrangements around infinity with applications to Lévy processes

We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger (Arch Ration Mech 61:35–44, 1976) and can be interpreted as involving symmetric rearrangements of domains around $$\infty $$ . As applications, we prove two comparison results for general Levy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Levy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Levy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi (Geom Funct Anal 22(4):1000–1014, 2012) for the Wiener sausage. In the second application, we show that the $$q$$ -capacity of a Borel measurable set for a Levy process can only decrease if the set and the Levy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe (Z Wahrsch Verw Gebiete 63:487–499, 1983) for symmetric Levy processes.

The expected volume and surface area of the Wiener sausage in odd dimensions

We consider the Wiener sausage up to time $t$ associated with a closed ball. A formula for the expected volume of the Wiener sausage is obtained in odd dimensions. In these cases, we also find that the formula leads to the asymptotic expansion for large $t$ and each coefficient is represented by zeros of a modified Bessel function of the second kind. Moreover we obtain a formula for the expected surface area of the Wiener sausage.

The expected area of the Wiener sausage swept by a disc

The expected areas of the Wiener sausages swept by a disc attached to the two-dimensional Brownian Bridge joining the origin to a point x over a time interval [0,t] are computed. It is proved that the leading term of the expectation is given by Ramanujan’s function if |x|=O(t). The second term is also given explicitly when |x|=o(t). The corresponding result for unconditioned process is also obtained.

An isoperimetric inequality for the Wiener sausage

Let (ξ(s)) s ≥ 0 be a standard Brownian motion in d ≥ 1 dimensions and let (D s ) s ≥ 0 be a collection of open sets in \({\mathbb{R}^d}\). For each s, let B s be a ball centered at 0 with vol(B s ) = vol(D s ). We show that \({\mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + D_s))] \geq \mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + B_s))]}\), for all t. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion.

Almost sure asymptotic behaviour of the r-neighbourhood surface area of Brownian paths

We show that whenever the q-dimensional Minkowski content of a subset A ⊂ ℝd exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in ℝd, d ⩾ 3.

Surface area of the domain visited by a spherical Brownian particle

A spatial domain swept out by a spherical particle, whose center follows a Wiener trajectory, is referred to as a Wiener sausage. The present study focuses on the surface area of the Wiener sausage. Using intuitive arguments we derive the mean and variance of the surface area, as well as the asymptotic behavior of its probability density in the limits when the area tends to zero and infinity.

Strong approximation of high dimensional Wiener sausage

In this paper, we study the volume of Wiener sausage in R d for d 4. We obtain that it can be strongly approximated by a one-dimensional standard Brownian motion. As an application, we give the weak convergence and laws of the iterated logarithm.

Laws of the iterated logarithm for high-dimensional wiener sausage

Let {β(s), s ≥ 0} be the standard Brownian motion in ℝ d with d ≥ 4 and let |W r (t)| be the volume of the Wiener sausage associated with {β(s), s ≥ 0} observed until time t. From the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for \(\left| {W_r (t)} \right| - \mathbb{E}\left| {W_r (t)} \right|\) in this case.

On the expected volume of the Wiener sausage

We consider the expected volume of the Wiener sausage on the time interval [0,t] associated with a closed ball. Let L(t) be the expected volume minus the volume of the ball. We obtain that L(t) is asymptotically equal to a constant multiple of t1/2 as t tends to 0 and that it is represented as an absolutely convergent power series of t1/2 for any t > 0 in the odd dimensional cases. Moreover, the explicit form of L(t) can be given in five and seven dimensional cases.

Large time volume of the pinned Wiener sausage to second order

AbstractWe investigate the large time behaviour of the expected volume of the pinned Wiener sausage associated to a compact subset K in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ {\mathbb R}^d $\end{document} for d ⩾ 3. The structure of the asymptotic series is known: it depends strongly on whether the dimension d is odd or even; and the leading coefficient is given by the Newtonian capacity of K. In this article, we obtain detailed expressions for the second order coefficients. It is noteworthy that these coefficients feature novel potential‐theoretic terms that do not appear in the unpinned case. The proof exploits a trace formula of Krein. In the course of the argument, we derive a low‐energy expansion to second order for the Krein spectral shift function for an obstacle scattering system. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

On the Volume of the Intersection of Two Independent Wiener Sausages

The expected volume of intersection of two independent Wiener sausages in ℝm, m ≥ 3, up to time t, and associated to non-polar, compact sets K1 and K2 respectively, is obtained in the limit of large t.

On the volume of intersection of three independent Wiener sausages

Soit K un ensemble compact, non-polaire dans ℝm (m≥3) et soit SKi(t)={Bi(s)+y: 0≤s≤t, y∈K} des saucisses de Wiener associées à des processus Browniens indépendants Bi, i=1, 2, 3 initalisés à 0. L’espérance des volumes de ⋂i=13SKi(t) par rapport à la mesure produit est obtenue en termes de la mesure d’équilibre de K lorsque t tend vers l’infini.

Approximations of the Wiener sausage and its curvature measures

A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.