Brown运动在Hlder范数下的拟必然Strassen重对数律的收敛速率

本文研究了Brown运动的泛函极限问题.利用Brown运动在Hlder范数下关于容度的大偏差与小偏差,获得了Brown运动在Hlder范数下的Strassen型重对数律的拟必然收敛速率,推广了文[2]中的结果.     

The Mean Perimeter of Some Random Plane Convex Sets Generated by a Brownian Motion

If C ₁ is the convex hull of the curve of a standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex hulls of C ₁ and several rotations of it and compute the mean of the length of their perimeter by elementary calculations. This can be seen geometrically as a study of the exit time by a Brownian motion from certain polytopes having the unit circle as an inscribed one.     

Optimality of AIC in Inference About Brownian Motion

In the usual Gaussian White-Noise model, we consider the problem of estimating the unknown square-integrable drift function of the standard Brownian motion using the partial sums of its Fourier series expansion generated by an orthonormal basis. Using the squared L ₂ distance loss, this problem is known to be the same as estimating the mean of an infinite dimensional random vector with l ₂ loss, where the coordinates are independently normally distributed with the unknown Fourier coefficients as the means and the same variance. In this modified version of the problem, we show that Akaike Information Criterion for model selection, followed by least squares estimation, attains the minimax rate of convergence. An erratum to this article can be found at     

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

<Emphasis Type="Italic">p</Emphasis>-Adic Brownian motion over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a p-adic Brownian motion over ℚ _p . First, we construct directly a p-adic white noise over ℚ _p by using a specific complete orthonormal system of (ℚ _p ). A p-adic Brownian motion over ℚ _p is then constructed by the Paley-Wiener method. Finally, we introduce a p-adic random walk and prove a theorem on the approximation of a p-adic Brownian motion by a p-adic random walk.     

An Extension of Vervaat's Transformation and Its Consequences

Vervaat⁽¹⁸⁾ proved that by exchanging the pre-minimum and post-minimum parts of a Brownian bridge one obtains a normalized Brownian excursion. Let s (0, 1), then we extend this result by determining a random time m _s such that when we exchange the pre-m _s-part and the post-m _s-part of a Brownian bridge, one gets a Brownian bridge conditioned to spend a time equal to s under 0. This transformation leads to some independence relations between some functionals of the Brownian bridge and the time it spends under 0. By splitting the Brownian motion at time m _s in another manner, we get a new path transformation which explains an identity in law on quantiles due to Port. It also yields a pathwise construction of a Brownian bridge conditioned to spend a time equal to s under 0.     

Small Ball Constants and Tight Eigenvalue Asymptotics for Fractional Brownian Motions

In this paper we prove rigorous large n asymptotics for the Karhunen–Loeve eigenvalues of a fractional Brownian motion. From the asymptotics of the eigenvalues the exact constants for small L ₂ ball estimates for fractional Brownian motions follows in a straightforward way.     

The approximation of additive functionals of diffusion processes

Let us consider a diffusion process in R_d . Around each point x one may consider a ring of size ? and a process which counts the crossings over the ring. Integrating with respect to a measure μ(dx) and letting ?→ 0 one gets an additive functional. This is a natural generalization of the approximation theorem of the local time of one dimensional Brownian motion by means of “downcrossings”. For multidimensional Brownian motion the result was established by Bally. The present paper introduces a new method which allows us to handle general diffusions     

Continuity in Law with Respect to the Hurst Parameter of the Local Time of the Fractional Brownian Motion

We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter H converges weakly to that of the local time of , when H tends to H ₀.      

On the Potential Theory of One-dimensional Subordinate Brownian Motions with Continuous Components

Suppose that S is a subordinator with a nonzero drift and W is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion X defined by X _t  = W(S _t ). We give sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and prove a boundary Harnack principle. In the case when S is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of X in (0, ∞ ), and sharp bounds for the Poisson kernel of X in a bounded open interval.     

Etude asymptotique des enlacements du mouvement brownien autour des droites de l'espace

Summary The existence of a joint asymptotic distribution for the windings of a three-dimensional Brownian motion around a finite number of straight lines is obtained. This complements the recent studies, by Pitman- Yor, and the authors, of the joint asymptotic distribution for the windings of planar Brownian motion around a finite number of points.The following principle governs the passage from results in the plane to results in space:Let B be a three-dimensional Brownian motion, and P ₁, ..., P _k, k planes which intersect two by two. Then, the convergences in distribution concerning the planar Brownian motions B ⁱ (1ik), defined respectively as the orthogonal projections of B on P _i (1ik), take place jointly, and the corresponding limit variables are independent.     

An enlargement of filtration for Brownian motion

Let Bt be an Ft Brownian motion and Gt be an enlargement of filtration of Ft from some Gaussian random variables. We obtain equations for ht such that Bt ht is a Gt-Brownian motion.     

多参数双分数布朗运动相遇局部时的存在性和联合连续性

本文研究了两个相互独立的(N,d)双分数布朗运动BH1,K1和BH2,K2的相遇局部时的问题.利用Fourier分析,获得了相遇局部时的存在性和联合连续性的结果,推广了分数布朗运动相遇局部时的相关结果.     

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H _n , n > 2. For ν > 0, the Brownian bridge B ^(ν) of length ν on H is the process B _t , 0 ≤t≤ν, conditioned by B ₀ = B _ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ³). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999     

The most visited sites of symmetric stable processes

Let X be a symmetric stable process of index α∈ (1,2] and let L ^x _t denote the local time at time t and position x. Let V(t) be such that L _t ^V(t) = sup _{x∈ ℝ} L _t ^x . We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim _{t →∞} |V(t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension of the Ray–Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and further to the winding problem for planar Brownian motion. Received: 14 October 1998 / Revised version: 8 June 1999 / Published online: 7 February 2000     

Attractiveness of Brownian queues in tandem

Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one-sided reflection.

     

Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion

Let E be a real Banach space with property (α) and let W _Γ be an E-valued Brownian motion with distribution Γ. We show that a function is stochastically integrable with respect to W _Γ if and only if Γ-almost all orbits Ψx are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence of Γ-measurable linear extensions of γ-radonifying operators with values in spaces of γ-radonifying operators. As an application we obtain a necessary and sufficient condition for solvability of stochastic evolution equations driven by an E-valued Brownian motion. The first named author gratefully acknowledges the support by a ‘VIDI subsidie’ in the ‘Vernieuwingsimpuls’ programme of The Netherlands Organization for Scientific Research (NWO) and the Research Training Network HPRN-CT-2002–00281. The second named author was supported by grants from the Volkswagenstiftung (I/78593) and the Deutsche Forschungsgemeinschaft (We 2847/1–1).     

Normal limit theorems for symmetric random matrices

Using the machinery of zonal polynomials, we examine the limiting behavior of random symmetric matrices invariant under conjugation by orthogonal matrices as the dimension tends to infinity. In particular, we give sufficient conditions for the distribution of a fixed submatrix to tend to a normal distribution. We also consider the problem of when the sequence of partial sums of the diagonal elements tends to a Brownian motion. Using these results, we show that if O _n is a uniform random n×n orthogonal matrix, then for any fixed k>0, the sequence of partial sums of the diagonal of O ^k _n tends to a Brownian motion as n→∞. Received: 3 February 1998 / Revised version: 11 June 1998