On the occupation times of cones by Brownian motion

Summary We study some features concerning the occupation timeA _t of a d-dimensional coneC by Brownian motion. In particular, in the case whereC is convex, we investigate the asymptotic behaviour ofP(A₁u0, when the Brownian motion starts at the vertex ofC. We also give the precise integral test, which decides whether a.s., lim inf _t A _t/(tf(t))=0 or for a decreasing functionf.     

Sojourns and future infima of planar Brownian motion

Summary Let {W(t); 0t1} be a two-dimensional Wiener process starting from 0. We are interested in the almost sure asymptotic behaviour, asr tends to 0, of the processesX(r) andY(r), whereX(r) denotes the total time spent byW in the ball centered at 0 with radiusr andY(r) the distance between 0 and the curve {W(t);rt1}. While a characterization of the lower functions ofY was previously established by Spitzer [S], we characterize via integral tests its upper functions as well as the upper and lower functions ofX.     

On maximum increase and decrease of Brownian motion

The joint distribution of maximum increase and decrease for Brownian motion up to an independent exponential time is computed. This is achieved by decomposing the Brownian path at the hitting times of the infimum and the supremum before the exponential time. It is seen that an important element in our formula is the distribution of the maximum decrease for the three-dimensional Bessel process with drift started from 0 and stopped at the first hitting of a given level. From the joint distribution of the maximum increase and decrease it is possible to calculate the correlation coefficient between these at a fixed time and this is seen to be .     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.     

Traps for reflected Brownian motion

Consider an open set , d ≥ 2, and a closed ball . Let denote the expectation of the hitting time of B for reflected Brownian motion in D starting from x ∈ D. We say that D is a trap domain if . A domain D is not a trap domain if and only if the reflecting Brownian motion in D is uniformly ergodic. We fully characterize the simply connected planar trap domains using a geometric condition and give a number of (less complete) results for d > 2. Research partially supported by NSF grant DMS-0303310. Research partially supported by NSF grant DMS-0303310. Research partially supported by NSF grant DMS-0201435.     

An excursion approach to maxima of the Brownian bridge

Distributions of functionals of Brownian bridge arise as limiting distributions in non-parametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. The idea of rescaling and conditioning on the local time has been used widely in the literature. In this paper it is used to give a unified derivation of a number of known distributions, and a few new ones. Particular cases of calculations include the distribution of the Kolmogorov–Smirnov statistic and the Kuiper statistic.     

MOMENT OF LEvY MEASURE OF OPERATOR-STABLE LAW

In 1979, Jurek gave a characterization of the moment of a full operator-stable μ by eigenvalues of exponent matrix of μ. Here, a characterization of the moment of Lévy measure (restricted on a neighbor of 0) of a full operator-stable μ by eigenvalues of exponent matrix of μ is given.     

Reflected Brownian motion in a cone: Semimartingale property

Summary In this paper, the object of study is reflected Brownian motion in a cone ind-dimensions (d3) with nonconstant oblique reflection on each radial line emanating from the vertex of the cone. The basic question considered here is When is this process a semimartingale?. Conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by Kwon, which is resolved in terms of a real parameter depending on the cone and the direction of reflection. It is shown that starting from any point of the cone, the process is a semimartingale if < 1, + 0 and not a semimartingale if < < 2.This research is supported by KOSEF grant 941-0100-011-1     

Points of increase of the Brownian sheet

Summary. It is well-known that Brownian motion has no points of increase. We show that an analogous statement for the Brownian sheet is false. More precisely, for the standard Brownian sheet in the positive quadrant, we prove that there exist monotone curves along which the sheet has a point of increase. Received: 7 December 1994 / In revised form: 6 August 1996     

Reflecting Brownian motions and a deletion result for Sobolev spaces of order (1, 2)

Let D be an open set in ^d and E be a relatively closed subset of D having zero Lebesgue measure. A necessary and sufficient integral condition is given for the Sobolev spaces W ^1,2 (D) and W ^1,2(D\E) to be the same. The latter is equivalent to (normally) reflecting Brownian motion (RBM) on being indistinguishable (in distribution) from RBM on . This integral condition is satisfied, for example, when E has zero (d–1)-dimensional Hausdorff measure. Therefore it is possible to delete from D a relatively closed subset E having positive capacity but nevertheless the RBM on is indistinguishable from the RBM on , or equivalently, W ^1,2(D\E)=W^1,2(D). An example of such kind is: D=² and E is the Cantor set. In the proof of above mentioned results, a detailed study of RBMs on general open sets is given. In particular, a semimartingale decomposition and approximation result previously proved in [3] for RBMs on bounded open sets is extended to the case of unbounded open sets.Research supported in part by NSF Grant DMS 86-57483.     

Estimates of transition densities for Brownian motion on nested fractals

Summary We obtain upper and lower bounds for the transition densities of Brownian motion on nested fractals. Compared with the estimate on the Sierpinski gasket, the results require the introduction of a new exponent,d _J, related to the shortest path metric and chemical exponent on nested fractals. Further, Hölder order of the resolvent densities, sample paths and local times are obtained. The results are obtained using the theory of multi-type branching processes.     

Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant

Summary This work is concerned with the existence and uniqueness of a class of semimartingale reflecting Brownian motions which live in the non-negative orthant of ^d . Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the orthant the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the (d-1)-dimensional faces that form the boundary of the orthant, the bounded variation part of the process increases in a given direction (constant for any particular face) so as to confine the process to the orthant. For historical reasons, this pushing at the boundary is called instantaneous reflection. In 1988, Reiman and Williams proved that a necessary condition for the existence of such a semimartingale reflecting Brownian motion (SRBM) is that the reflection matrix formed by the directions of reflection be completely-L. In this work we prove that condition is sufficient for the existence of an SRBM and that the SRBM is unique in law. It follows from the uniqueness that an SRBM defines a strong Markov process. Our results have potential application to the study of diffusions arising as approximations tomulti-class queueing networks.Research supported in part by NSF Grants DMS 8657483, 8722351 and 9023335, and a grant from AT&T Bell Labs. In addition, R.J. Williams was supported in part during the period of this research by an Alfred P. Sloan Research Fellowship     

On almost sure limit inferior for B-valued stochastic processes and applications

Summary A criterion on almost sure limit inferior for the increments of B-valued stochastic processes is presented. Applications to processes of independent increments and to Gaussian processes with stationary increments are given. In particular, an exact limit inferior bound is established for increments of infinite series of independent Ornstein-Uhlenbeck processes.Work supported by an NSERC Canada grant at Carleton UniversityWork supported by the Fok Yingtung Education Foundation of China     

Intersections of random walks and Wiener sausages in four dimensions

Let {S ₁ (n)} _n0and {S ₂ (n)} _n0be independent simple random walks in Z ⁴ starting at the origin, and let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaqGPbaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiaadggacaGGSaGaamOyaiab-LcaPiabg2da9i% ab-Tha7Hqbciab+Hha4jabgIGiolab+PfaAnaaCaaaleqabaGaaGin% aaaakiaacQdaieGacaqFtbWaaSbaaSqaaiaabMgaaeqaaOGae8hkaG% Iaa0xBaiab-LcaPiabg2da9iab+Hha4baa!5761!\[\Pi _{\rm{i}} (a,b) = \{ x \in Z^4 :S_{\rm{i}} (m) = x\]for the some % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciaa-1gacqGHiiIZtCvAUfKttLearyqr1ngBPrgaiuaacqGF% OaakcaWGHbGaaiilaiaadkgacqGFPaqkcqGF9bqFaaa!4936!\[m \in (a,b)\} \]. Let two integervalued sequences {a _n}_n0and {b _n}_n0be given, such that the limit lim_n a _nexists and lim _n b _n=+. In this paper, it is shown that the probability of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaaIXaaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiab-bdaWiab-XcaSiabg6HiLkab-LcaPiabgM% Iihlabfc6aqnaaBaaaleaacaaIYaaabeaakiab-HcaOiaadggadaWg% aaWcbaGaamOBaiaacYcaaeqaaOGaamyyamaaBaaaleaacaWGUbaabe% aakiabgUcaRiaadkgadaWgaaWcbaGaamOBaaqabaGccqWFPaqkcqGH% GjsUieaacaGFydaaaa!5904!\[\Pi _1 (0,\infty ) \cap \Pi _2 (a_{n,} a_n + b_n ) \ne \O \] is asymptotic to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdaaaGaciiBaiaac+gacaGGNbWe% xLMBb50ujbqeguuDJXwAKbacfaGae8hkaGIae8xmaeJae83kaSIaam% OyamaaBaaaleaacaWGUbaabeaakiaac+cacaWGHbWaaSbaaSqaaiaa% d6gaaeqaaOGae8xkaKIae83la8IaciiBaiaac+gacaGGNbGaamOyam% aaBaaaleaacaWGUbaabeaaaaa!5364!\[\frac{1}{2}\log (1 + b_n /a_n )/\log b_n \] if it tends to zero as n, and the probability of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaaIXaaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiab-bdaWiab-XcaSiabg6HiLkab-LcaPiabgM% Iihlabfc6aqnaaBaaaleaacaaIYaaabeaakiab-HcaOiaadggadaWg% aaWcbaGaamOBaaqabaGccaGGSaGaamyyamaaBaaaleaacaWGUbaabe% aakiabgUcaRiaadkgadaWgaaWcbaGaamOBaaqabaGccqWFPaqkcqWF% 9aqpieaacaGFydaaaa!583C!\[\Pi _1 (0,\infty ) \cap \Pi _2 (a_n ,a_n + b_n ) = \O \]is asymptotic to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% abaeqabaaabaGaam4yaiaacUfaciGGSbGaai4BaiaacEgatCvAUfKt% tLearyqr1ngBPrgaiuaacqWFOaakcaWGHbWaaSbaaSqaaiaad6gaae% qaaOGaey4kaSIaamOyamaaBaaaleaacaWGUbaabeaakiab-LcaPiab% -9caViab-XgaSjab-9gaVjab-DgaNjab-HcaOiaadggadaWgaaWcba% GaamOBaaqabaGccqGHRaWkcaaIYaGae8xkaKIae8xxa01aaWbaaSqa% beaacqWFTaqlcqWFXaqmcqWFVaWlcqWFYaGmaaaaaaa!5BAC!\[\begin{array}{l} \Pi _1 (0,\infty ) \cap \Pi _2 (a_n ,a_n + b_n ) = \O \\ c[\log (a_n + b_n )/log(a_n + 2)]^{ - 1/2} \\ \end{array}\], for some constant c, if it tends to a finite constant (1) as n. These results extend some results obtained by G. F. Lawler about the intersection properties of simple random walks in Z ⁴. By using similar arguments, we also get corresponding results for the intersections of Wiener sausages in four dimensions. In particular, a conjecture suggested by M. Aizenman, which describes nonintersection of independent Wiener sausages in R ⁴, is proven.Partly supported by AvH Foundation.     

Tree structured independence for exponential Brownian functionals

The product of GIG and gamma distributions is preserved under the transformation (x,y)?((x+y)⁻¹,x⁻¹−(x+y)⁻¹)$(x,y)?({(x+y)}^{-1},x^{-1}-{(x+y)}^{-1})$. It is also known that this independence property may be reformulated and extended to an analogous property on trees. The purpose of this article is to show the independence property on trees, which was originally derived outside the framework of stochastic processes, in terms of a family of exponential Brownian functionals.     

Multiple points of trajectories of multiparameter fractional Brownian motion

Consider 0<α<1 and the Gaussian process Y(t) on ℝ ^N with covariance E(Y(s)Y(t))=|t|^2α+|s|^2α−|t−s|^2α, where |t| is the Euclidean norm of t. Consider independent copies X ¹,…,X ^d of Y and␣the process X(t)=(X ¹(t),…,X ^d (t)) valued in ℝ ^d . When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^{k N /α−( k −1) d} (loglog(1/ɛ)) ^k . If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^d (log(1/ɛ) logloglog 1/ɛ) ^k . (This includes the case k=1.) Received: 20 May 1997 / Revised version: 15 May 1998