The behavior of Listeria monocytogenes (L. monocytogenes) throughout the shelf life of ready-to-eat foodstuffs represents a major concern in relation to human diet and human health. The aim of the study was to evaluate the behavior of L. monocytogenes in Wiener sausage, as an RTE meat product, throughout 15 days of storage (0-7 °C) under the action of fermented juice from parsley (Petroselinum crispum var. tuberosum) roots and common hawthorn (Crataegus monogyna) berry phenolics, compared with the effect of the food additives sodium nitrite and sodium ascorbate used in the standard formulation. For this purpose, one experimental formulation (F1) and one standard formulation (F2) of Wiener sausages were designed using the following preservatives and antioxidants: 50 ppm fermented parsley root juice (as a nitrite source) and 50 ppm hawthorn berry phenolics were used in F1, and 50 ppm sodium nitrite (as food additive E 250) and 50 ppm sodium ascorbate (as food additive E 301) were used in F2. The ability to support L. monocytogenes growth was assessed by a challenge test throughout the 15 days of storage. Based on the results of the assessment, the natural ingredients fermented parsley root juice and hawthorn berry phenolics could act as preservatives that ensure microbiological safety during the shelf life of the product. The nitrite and phenolic compounds of these natural ingredients showed antimicrobial activity against foodborne pathogens, including L. monocytogenes.

We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document} in the transient dimensions d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3$$\\end{document}. We prove that the vacant set of Brownian interlacements contains at most one infinite connected component almost surely. For finite ensembles of Wiener sausages, we provide sharp polynomial bounds on the probability that their vacant set contains at least 2 connected components in microscopic balls. The main proof ingredient is a sharp polynomial bound on the probability that several Brownian motions visit jointly all hemiballs of the unit ball while avoiding a slightly smaller ball.

Upper bounds are obtained for the Newtonian capacity of compact sets in [Formula: see text] in terms of the perimeter of the [Formula: see text]-parallel neighborhood of [Formula: see text]. For compact, convex sets in [Formula: see text] with a [Formula: see text] boundary the Newtonian capacity is bounded from above by [Formula: see text], where [Formula: see text] is the integral of the mean curvature over the boundary of [Formula: see text] with equality if [Formula: see text] is a ball. For compact, convex sets in [Formula: see text] with non-empty interior the Newtonian capacity is bounded from above by [Formula: see text] with equality if [Formula: see text] is a ball. Here, [Formula: see text] is the perimeter of [Formula: see text] and [Formula: see text] is its measure. A quantitative refinement of the latter inequality in terms of the Fraenkel asymmetry is also obtained. An upper bound is obtained for expected Newtonian capacity of the Wiener sausage in [Formula: see text] with radius [Formula: see text] and time length [Formula: see text].

The statistics of the first-encounter time of diffusing particles changes drastically when they are placed under confinement. In the present work, we make use of Monte Carlo simulations to study the behavior of a two-particle system in two- and three-dimensional domains with reflecting boundaries. Based on the outcome of the simulations, we give a comprehensive overview of the behavior of the survival probability S(t) and the associated first-encounter time probability density H(t) over a broad time range spanning several decades. In addition, we provide numerical estimates and empirical formulas for the mean first-encounter time 〈T〉, as well as for the decay time T characterizing the monoexponential long-time decay of the survival probability. Based on the distance between the boundary and the center of mass of two particles, we obtain an empirical lower bound t_{B} for the time at which S(t) starts to significantly deviate from its counterpart for the no boundary case. Surprisingly, for small-sized particles, the dominant contribution to T depends only on the total diffusivity D=D_{1}+D_{2}, in sharp contrast to the one-dimensional case. This contribution can be related to the Wiener sausage generated by a fictitious Brownian particle with diffusivity D. In two dimensions, the first subleading contribution to T is found to depend weakly on the ratio D_{1}/D_{2}. We also investigate the slow-diffusion limit when D_{2}≪D_{1}, and we discuss the transition to the limit when one particle is a fixed target. Finally, we give some indications to anticipate when T can be expected to be a good approximation for 〈T〉.

Abstract We prove a large deviations principle for the number of intersections of two independent infinite‐time ranges in dimension 5 and greater, improving upon the moment bounds of Khanin, Mazel, Shlosman, and Sinaï [9]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen, and den Hollander [15], who analyzed this question for the Wiener sausage in the finite‐time horizon. The proof builds on their result (which was adapted in the discrete setting by Phetpradap [12]), and combines it with a series of tools that were developed in recent works of the authors [2, 3, 5]. Moreover, we show that most of the intersection occurs in a single box where both walks realize an occupation density of order 1. © 2022 Wiley Periodicals, Inc.

Perspectives on Sarcopenia and Protein Intake in Aged and Diabetic Patients

We prove an upper large deviation bound on the scale of the mean for a symmetric random walk in the plane satisfying certain moment conditions. This result complements the study by Phetpradap for the random walk range, which is restricted to dimension three and higher, and of van den Berg, Bolthausen and den Hollander, for the volume of the Wiener sausage.

The branching Brownian sausage in $\mathbb{R} ^{d}$ was defined in [4] similarly to the classical Wiener sausage, as the random subset of $\mathbb{R} ^{d}$ scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated branching Brownian sausage (BBM-sausage) with radius exponentially shrinking in time. Using a previous result on the density of the support of BBM, and some well-known results on the classical Wiener sausage and Brownian hitting probabilities, we obtain almost sure limit theorems as time tends to infinity on the volume of the shrinking BBM-sausage in all dimensions.

We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements [Comets, Popov, Vachkovskaia, 2016]. At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of [Comets, Popov, Vachkovskaia, 2016; Comets, Popov, 2016], as well as the results specific to the continuous case.

In the present paper, we consider long time behaviors of the volume of the Wiener sausage on Dirichlet spaces. We focus on the volume of the Wiener sausage for diffusion processes on metric measure spaces other than the Euclid space equipped with the Lebesgue measure. We obtain the growth rate of the expectations and almost sure behaviors of the volumes of the Wiener sausages on metric measure Dirichlet spaces satisfying Ahlfors regularity and sub-Gaussian heat kernel estimates. We show that the growth rate of the expectations on a "bounded" modification of the Euclidian space is identical with the one on the Euclidian space equipped with the Lebesgue measure. We give an example of a metric measure Dirichlet space on which a scaled of the means fluctuates.

We prove a strong law of large numbers for the Newtonian capacity of a Wiener sausage in the critical dimension four, where a logarithmic correction appears in the scaling. The main step of the proof is to obtain precise asymptotics for the expected value of the capacity. This requires a delicate analysis of intersection probabilities between two independent Wiener sausages.

In order to valorize tainted meat from entire male pigs, two options exist in the production of different meat products: either to dilute tainted meat or to mask off-odors in the final product. Processing steps may also reduce the concentrations of boar taint compounds. This study investigated the impact of different processing methods on the concentrations of boar taint compounds skatole and androstenone in meat products. Three different types of German sausages, such as raw fermented (salami), boiled (wiener), and cooked meat sausages (liver sausage), were produced from boar meat and fat with either “high” or “low” taint level. Heating during processing, especially the production of wiener and liver sausage, reduced androstenone concentrations between 44 and 87%. In salami, androstenone concentrations were not reduced during production process. In contrast, skatole reductions of up to 26% were observed for salami and up to 44% for wiener, whereas liver sausage was not affected. Practical applications The risk of offensive boar taint is one of the main disadvantages of pork production with entire males. If the detection of tainted carcasses at the slaughter line can be improved, a valid strategy to valorize such carcasses is crucial. The study revealed that open heating during processing has the potential to reduce androstenone, whereas smoking of the products seems to reduce skatole concentrations in the final product.

The space-time distribution, $Q_A(x,dt d\xi)$ say, of Brownian hitting of a bounded Borel set $A$ of the $d$-dimensional Euclidian space is studied. We derive the asymptotic form of the leading term of the time-derivative $Q_A(x, dtd\xi)/dt$ for each $d =2, 3, ...$, valid uniformly with respect to the starting point $x$ of the Brownian motion, which result extends significantly the classical results for $Q_A(x, dt d\xi)$ itself by Hunt ($d=2$), Joffe and Spitzer ($d= 3, 4,...$). The results are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joining the origin to a distant point.

Let \U0001d54bm be the m-dimensional unit torus, m ∈ ℕ. The torsional rigidity of an open set Ω ⊂ \U0001d54bm is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = \U0001d54bm\\β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in \U0001d54b2\\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of \U0001d54b3\\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage Wr(t)[0, t] of radius r(t) = o(t-1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ℝ3 and W1[0, t] in ℝm, m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on \U0001d54bm, which has received a lot of attention in the literature in past years.

We derive formulae for some ratios of the Macdonald functions by using their zeros, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory and one in classical analysis. We show a formula for the Levy measure of the distribution of the first hitting time of a Bessel process and an explicit form for the expected volume of the Wiener sausage for an even dimensional Brownian motion. In addition, we show that the complex zeros of the Macdonald functions are the roots of some algebraic equations with real coefficients.

We prove a comparison theorem for the spatial mass of the solutions of two exterior parabolic problems, one of them having symmetrized geometry, using approximation of the Schwarz symmetrization by polarizations, as it was introduced in Brock and Solynin (Trans Am Math Soc 352(4):1759–1796, 2000). This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the Wiener sausage, which was proved in Peres and Sousi (Geom Funct Anal 22(4):1000–1014, 2012).

In this paper, we establish laws of the iterated logarithm for intersection times and volumes of Wiener sausages in critical dimensions by the high moment method.

Abstract We consider the Wiener sausage for a Brownian motion with a constant drift up to time t associated with a closed ball. In the two or more dimensional cases, we obtain the explicit form of the expected volume of the Wiener sausage. The result says that it can be represented by the sum of the mean volumes of the multi-dimensional Wiener sausages without a drift. In addition, we show that the leading term of the expected volume of the Wiener sausage is written as κ ⁢ t ⁢ ( 1 + o ⁢ [ 1 ] ) ${\kappa t(1+o[1])}$ for large t by a constant κ. The expression for κ is of a complicated form, but it converges to the known constant as the drift tends to 0.

The Wiener Sausage, the volume traced out by a sphere attached to a Brownian particle, is a classical problem in statistics and mathematical physics. Initially motivated by a range of field-theoretic, technical questions, we present a single loop renormalised perturbation theory of a stochastic process closely related to the Wiener Sausage, which, however, proves to be exact for the exponents and some amplitudes. The field-theoretic approach is particularly elegant and very enjoyable to see at work on such a classic problem. While we recover a number of known, classical results, the field-theoretic techniques deployed provide a particularly versatile framework, which allows easy calculation with different boundary conditions even of higher momenta and more complicated correlation functions. At the same time, we provide a highly instructive, non-trivial example for some of the technical particularities of the field-theoretic description of stochastic processes, such as excluded volume, lack of translational invariance and immobile particles. The aim of the present work is not to improve upon the well-established results for the Wiener Sausage, but to provide a field-theoretic approach to it, in order to gain a better understanding of the field-theoretic obstacles to overcome.

We consider a continuum percolation model on $$\mathbb {R}^d$$ , $$d\ge 1$$ . For $$t,\lambda \in (0,\infty )$$ and $$d\in \{1,2,3\}$$ , the occupied set is given by the union of independent Brownian paths running up to time t whose initial points form a Poisson point process with intensity $$\lambda >0$$ . When $$d\ge 4$$ , the Brownian paths are replaced by Wiener sausages with radius $$r>0$$ . We establish that, for $$d=1$$ and all choices of t, no percolation occurs, whereas for $$d\ge 2$$ , there is a non-trivial percolation transition in t, provided $$\lambda $$ and r are chosen properly. The last statement means that $$\lambda $$ has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero (which is infinite when $$d\in \{2,3\}$$ , but finite and dependent on r when $$d\ge 4$$ ). We further show that for all $$d\ge 2$$ , the unbounded cluster in the supercritical phase is unique. Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest. The present paper settles the basic properties of the model and should be viewed as a springboard for finer results.