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.     

On the expected surface area of the Wiener sausage

For parallel neighborhoods of the paths of the d ‐dimensional Brownian motion, so‐called Wiener sausages, formulae for the expected surface area are given for any dimension d ≥ 2. It is shown by means of geometric arguments that the expected surface area is equal to the first derivative of the mean volume of the Wiener sausage with respect to its radius (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong approximations of three-dimensional Wiener sausages

We prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall [10]șs fine L ²-norm estimates between the Wiener sausage and the Brownian intersection local times. Research supported by the Hungarian National Foundation for Scientific Research, Grants T 037886, T 043037 and K 61052.     

高维Wiener sausage的强逼近

本文研究了四维及四维以上的Wiener sausage 的体积, 得到它们可以由一维Brown 运动强逼近. 作为应用, 推出了弱收敛和重对数率.     

一维Wiener sausage长度的中偏差和重对数律

本文研究一维Wiener sausage.利用布朗运动的相关性质和收缩原理,得到p个Wiener sausage相交部分长度的中偏差和重对数律.     

三维Wiener Sausage体积的中偏差

令{β(s),s≥0}表示R~3空间中的标准Brown运动,|W_r(t)|表示由{β(s),s≥0}产生的观察至时间t且以r为半径的Wiener sausage的体积.由中心极限定理可知,(|W_r(t)|-E|W_r(t)|)/(?)弱收敛至正态分布.本文研究这种情况下的中偏差.     

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 | W_r (t) | - \mathbbE| W_r (t) |\left| {W_r (t)} \right| - \mathbb{E}\left| {W_r (t)} \right| in this case.     

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 ⊂ ℝ ^d 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 ℝ ^d , d ⩾ 3.     

Large Deviations for Intersections of Random Walks

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.     

Expected Volume of Intersection of Pinned Wiener Sausages and Heat Kernel Norms on Compact Riemannian Manifolds with Boundary

Estimates are obtained for the expected volume of intersection of independent pinned Wiener sausages in Euclidean space in the limit of small pinning time.      

On the Convergence of the P-Algorithm for One-Dimensional Global Optimization of Smooth Functions

The Wiener process is a widely used statistical model for stochastic global optimization. One of the first optimization algorithms based on a statistical model, the so-called P-algorithm, was based on the Wiener process. Despite many advantages, this process does not give a realistic model for many optimization problems, particularly from the point of view of local behavior. In the present paper, a version of the P-algorithm is constructed based on a stochastic process with smooth sampling functions. It is shown that, in such a case, the algorithm has a better convergence rate than in the case of the Wiener process. A similar convergence rate is proved for a combination of the Wiener model-based P-algorithm with quadratic fit-based local search.     

Absolute continuity of the law of an infinite dimensional Wiener functional with respect to the Wiener probability

Summary In this paper we study conditions ensuring that the law of aC([0, 1])-valued functional defined on an abstract Wiener space is absolutely continuous with respect to the Wiener measure onC([0,1]). These conditions extend those established byP. Malliavin [12, 13] for finite-dimensional Wiener functionals, and those of [15] for Hilbert-valued functionals.     

Various topologies in the Wiener space and Lévy's stochastic area

Summary In this paper, we observe how Lévy's stochastic area looks when we see it through various topologies in the Wiener space. Our theorem implies that it is quite natural from the viewpoint of topology to define a distinct skeleton of Lévy's stochastic areaS(w) for each distinct topology in the Wiener space, or equivalently, for each distinct abstract Wiener space on which the Wiener measure andS(w) are realized. Thus we cannot determine its intrinsic skeleton in the theory of abstract Wiener spaces.     

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.     

Evaluation formulas for conditional abstract wiener integrals

In this paper we consider abstract Wiener space version of conditional Wiener integrals and establish formulas for evaluating conditional abstract Wiener integrals for various classes of functions on an abstract Wiener space. We then apply our formulas to evaluate certain Wiener integrals and conditional Wiener and Yeh-Wiener integrals     

How big are the Cs(o)rg(o)-Révész increments of two-parameter Wiener processes?

In this paper, we prove a theorem on the set of limit points of the increments of a two-parameter Wiener process via establishing a large deviation principle on the increments of the two-parameter Wiener process.     

Wiener Algebra for the Quaternions

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener–Lévy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener–Hopf operators.     

Dimension-Independent Estimates on the Densities of Wiener Functionals via the Log-Sobolev Inequality

Using the log-Sobolev inequality, we shall present in this note some estimates on the density of finite dimensional non-degenerate Wiener functionals which are independent on the dimension. We shall take the Gaussian measure as the reference measure, contrary to the customary choice of Lebesgue measure in the literature. As an application, we show that the limit in probability of a uniformly bounded sequence of non-degenerate Wiener functionals has a density with respect to the Gaussian measure.     

Stochastic integration,trace and the skeleton of wiener functionals

It is shown that the notion of trace induced by a given complete orthonormal system relates the Skorohod integral with a corresponding Ogawa‐type integral evaluated with respect to the same orthonormal systems. Similarly the multiple Wiener‐Ito integral is shown to be related to a multiple Ogawa‐type integral induced by a complete orthonormal system via the Hu‐Meyer formula with suitably defined multiple traces. The notion of skeleton of a Wiener functional relative to a given orthonormal system is defined and yields what seems to be a “natural” extension of Wiener functionals to the Cameron Martin space and the Wiener processes with a different scale.