Théorème central limite pour l'intersection de deux saucisses de Wiener indépendantes

Summary J.F. Le Gall [4] proved thatn ² times the volume of the intersection of two independent Wiener sausages in ³, with radius 1/n, converges inL ², asn, towards a multiple of the intersection local time at 0, for the underlying Brownian motions.We complete this result by proving a corresponding central limit theorem.

     

The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in R d

Summary LetL(x, T),xR ^d ,TR ₊ ^N , be the local time of theN-parameter Wiener processW taking values inR ^d . Even in the distribution valued casedd2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|0 and/orT and of related occupation integrals asT. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.X _T (f), and of the second order, i.e. normalized convergence laws forL(x, T)–E(L(x, T)) resp.X _T (f)–E(X _T (f)).This research was made during a stay at the LMU in München supported by DAAD     

Applications of the degree theorem to absolute continuity on Wiener space

Summary Let (,H, P) be an abstract Wiener space and define a shift on byT()=+F() whereF is anH-valued random variable. We study the absolute continuity of the measuresPºT ^–1and ( _F P)ºT ¹ with respect toP using the techniques of the degree theory of Wiener maps, where _F =det₂(1+F) × Exp{–F–1/2|F|²}.The work of the second author was supported by the fund for promotion of research at the Technion     

Diffusions with singular drift related to wave functions

Summary Schrödinger equations are equivalent to pairs of mutually time-reversed non-linear diffusion equations. Here the associated diffusion processes with singular drift are constructed under assumptions adopted from the theory of Schrödinger operators, expressed in terms of a local space-time Sobolev space.By means of Nagasawa's multiplicative functionalN _s ^t , a Radon-Nikodym derivative on the space of continuous paths, a transformed process is obtained from Wiener measure. Its singular drift is identified by Maruyama's drift transformation. For this a version of Itô's formula for continuous space-time functions with first and second order derivatives in the sense of distributions satisfying local integrability conditions has to be derived.The equivalence is shown between weak solutions of a diffusion equation with singular creation and killing term and the solutions of a Feynman-Kac integral equation with a locally integrable potential function.     

Random rotations of the Wiener path

Summary Let (W, H, ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyhH. It is shown that (R(w)h) is again a Gaussian (0, |h| _H ² )-random variable. Consequently, if (e _i ,i)W ^* is a complete, orthonormal basis ofH, then defines a measure preserving transformation, a rotation, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, (R(w+k)h) is (0, |h| _H ² )-Gaussian for allk, hH, thenR is an isometry and Rh is quasi-nilpotent for allHH. The relation between the stochastic calculi for these Wiener pathsw and , as well as the conditions of the inverbibility of the map are discussed and the problem of the absolute continuity of the image of the Wiener measure under Euclidean motion on the Wiener space (i.e. composed with a shift) is studied.The research of the second author was supported by the Fund for the Promotion of Research at the TechnionDedicated to the memory of Albert Badrikian     

Randomizable point systems

This paper gives a survey of the theory of point-set randomizations with respect to a group. It is shown that only the so-calledk-lattices are randomizable with respect to the parallel translations and all motions of theR ⁿ space. It is shown how the randomizability problems are connected with questions of the theory of discrete Lie groups and regular point systems. The results are applied to the construction of stationary random line processes.     

Asymptotics for the Euclidean TSP with power weighted edges

Summary LetU ₁,...,U_n denote i.i.d. random variables with the uniform distribution on [0, 1]², and letT ²T²(U₁,...,U_n) denote the shortest tour throughU ₁,...,U_n with square-weighted edges. By drawing on the quasi-additive structure ofT ² and the boundary rooted dual process, it is shown that lim _n E T ²(U ₁,...,U_n)= for some finite constant .This work was supported in part by NSF Grant DMS-9200656, Swiss National Foundation Grant 21-298333.90, and the US Army Research Office through the Mathematical Sciences Institute of Cornell University, whose assistance is gratefully acknowledged     

A new approach to the single point catalytic super-Brownian motion

Summary A new approach is provided to the super-Brownian motionX with a single point-catalyst _c as branching rate. We start from a superprocessU with constant branching rate and spatial motion given by the 1/2-stable subordinator. We prove that the occupation density measure ^c ofX at the catalystc is distributed as the total occupation time measure ofU. Furthermore, we show thatX _t is determined from ^c by an explicit representation formula. Heuristically, a mass ^c (ds) of particles leaves the catalyst at times and then evolves according to Itô's Brownian excursion measure. As a consequence of our representation formula, the density fieldx ofX satisfies the heat equation outside ofc, with a noisy boundary condition atc given by the singularly continuous random measure ^c . In particular,x isC outside the catalyst. We also provide a new derivation of the singularity of the measure ^c .     

Bivariate Bonferroni inequalities

Summary LetA ₁,A ₂, ,A _m ,C ₁,C ₂, ,C _n be events on a given probability speace. LetV _m andU _n , respectively, be the numbers among theA _i 's andC _j 's which occur.Upper and lower bounds ofP(V _m 1, U _n 1) are obtained by means of the bivariate binomial moments. These extend recent univariate optimal Bonferroni-type inequalities.     

Two-parameter diffusion processes and martingales

We introduce a class of two-parameter processes which are diffusions on each coordinate and satisfy a particular Markov property related to the partial ordering in R²₊. These processes can be expressed as solutions of some stochastic integral equations driven by a two-parameter Wiener process and two families of ordinary Brownian motions. This result is based on a characterization of two-parameter martingales with orthogonal increments.     

On the prediction of vector-valued random fields and the spectral distribution of their evanescent component

Let (X_m,n)_(m,n)∈Z² be a C^p-valued wide sense stationary process. We study the prediction theory of such processes according to different total orders on Z². In the case of a “rational order”, we give the spectral distribution of the resulting evanescent component and prove that for two different rational orders, the resulting evanescent components are mutually orthogonal.     

Hölder continuity for spatial and path processes via spectral analysis

For ν(dθ), a σ-finite Borel measure on R ^d , we consider L ²(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ ^t ₀ e ^{−λ(θ)( t − s )} dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ ^t ₀ g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process _t (τ)≗Y(τt∧t). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001     

Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.     

Onm-dependence and Edgeworth expansions

This paper contains two results. The first establishes, under mild assumptions, the validity of an Edgeworth expansion with remaindero(N ^–1/2) for aU-statistic with a kernel of degree two using observations from anm-dependent shift. The second result gives a necessary and sufficient condition for the distribution of a sum ofm-dependent random variables to possess an Edgeworth expansion. This generalizes a result of Bickel and Robinson from the i.i.d. case to them-dependent case.This research was supported in part by National Science Foundation, Grant DMS 89-23071.     

M-structure in dual Banach spaces

A result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach spaces. Let ℒ be the Banach algebra of operators generated by theL-projections of a real Banach spaceU, and let ℳ (U ^* be the bounded operators on the dual spaceU ^* with adjoint in ℒ(U ^**. Then the adjoint operation maps ℒ (U) onto ℳ (U ^*). In particular, anyM-projection ofU ^* is weak^* continuous. Supported in part by the National Science Foundation.     

Path and semimartingale properties of chaos processes

The present paper characterizes various properties of chaos processes which in particular include processes where all time variables admit a Wiener chaos expansion of a fixed finite order. The main focus is on the semimartingale property, p$p$-variation and continuity. The general results obtained are finally used to characterize when a moving average is a semimartingale.     

Large deviations and continuum limit in the 2D Ising model

Summary. We study the 2D Ising model in a rectangular box Λ _L of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑ _t∈ΛL σ(t) when L→∞ for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m ^* is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function <σ(0)σ(t)>, as |t|→∞, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature representation of the model. These results are valid for dimensions D≥2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation. We then study the Gibbs measure conditioned by {|∑ _t∈ΛL σ(t) −m|Λ _L ||≤|Λ _L |L ^{− c} }, with 0<c<1/4 and −m ^*<m<m ^*. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric type. Received: 17 October 1996 / In revised form: 7 March 1997     

Conditioned super-Brownian motion

Summary We investigate classes of conditioned super-Brownian motions, namely H-transformsP ^H with non-negative finitely-based space-time harmonic functionsH(t, ). We prove thatH ^H is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching Brownian motions. We identify the limit behaviour of H-transforms with functionsH(t, )=h(t, (1)) depending only on the total mass (1). Using the Palm measures of the super-Brownian motion we describe for an additive spacetime harmonic functionH(t, )=h(t, x) (dx) theH-transformP ^H as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.     

Comparison inequalities on Wiener space

We define a covariance-type operator on Wiener space: for F$F$ and G$G$ two random variables in the Gross–Sobolev space D^1,2${\mathbb{D}}^{1,2}$ of random variables with a square-integrable Malliavin derivative, we let _ΓF,G?〈DF,−DL⁻¹G〉${\Gamma }_{F,G}?〈{DF,-DL}^{-1}G〉$, where D$D$ is the Malliavin derivative operator and L⁻¹$L^{-1}$ is the pseudo-inverse of the generator of the Ornstein–Uhlenbeck semigroup. We use Γ$\Gamma$ to extend the notion of covariance and canonical metric for vectors and random fields on Wiener space, and prove corresponding non-Gaussian comparison inequalities on Wiener space, which extend the Sudakov–Fernique result on comparison of expected suprema of Gaussian fields, and the Slepian inequality for functionals of Gaussian vectors. These results are proved using a so-called smart-path method on Wiener space, and are illustrated via various examples. We also illustrate the use of the same method by proving a Sherrington–Kirkpatrick universality result for spin systems in correlated and non-stationary non-Gaussian random media.