The problem of the most visited site in random environment

We prove that the process of the most visited site of Sinai's simple random walk in random environment is transient. The rate of escape is characterized via an integral criterion. Our method also applies to a class of recurrent diffusion processes with random potentials. It is interesting to note that the corresponding problem for the usual symmetric Bernoulli walk or for Brownian motion remains open. Received: 17 April 1998     

On the length of the homotopic Brownian word in the thrice punctured sphere

We consider the word associated to the homotopic class of the Brownian path (properly closed) in the thrice punctured sphere. We prove that its length has almost surely the same behaviour as a totally asymmetric Cauchy process on the line. More precisely, the liminf has the same normalization in t log(t) and the limsup can be described by the same integral test. They are the Brownian motion counterparts of some Lévy and Khintchine results on continued fraction expansions. Received: 17 December 1996 / Revised version: 23 February 1998     

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     

Exact capacity results for stable processes

Exact results are proved for the capacity of pullbacks of analytic sets by stable processes. Received: 25 May 1988 / Revised version: 15 September 1997     

Poisson point processes,excursions and stable processes in two-dimensional structures

Itô’s contributions lie at the root of stochastic calculus and of the theory of excursions. These ideas are also very useful in the study of conformally invariant two-dimensional structures, via conformal loop ensembles, excursions of Schramm–Loewner evolutions and Poisson point processes of Brownian loops.     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

On the convergence of a dual-primal substructuring method

Summary.   In the Dual-Primal FETI method, introduced by Farhat et al. [5], the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated. The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the Reissner-Mindlin plate model. Received January 20, 2000 / Revised version received April 25, 2000 / Published online December 19, 2000     

Lower bounds for three-dimensional nonoverlapping domain decomposition algorithms

Summary. Lower bounds for the condition numbers of the preconditioned systems are obtained for the wire basket preconditioner and the Neumann-Neumann preconditioner in three dimensions. They show that the known upper bounds are sharp. Received January 28, 2001 / Revised version received September 3, 2001 / Published online January 30, 2002 This work was supported in part by the National Science Foundation under Grant Nos. DMS-9600133 and DMS-0074246     

On the uniqueness of stable ultimate shapes for the anisotropic curve-shortening problem

The uniqueness of stable ultimate shapes for the generalized curve-shortening problem is established for a class of anisotropic factors. Received: 24 June 1999     

On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations

Summary. For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle (being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev. A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented. Received August 29, 1994 / Revised version received September 19, 1995     

Increment sizes of the principal value of Brownian local time

Let W be a standard Brownian motion, and define Y(t)= ∫₀ ^t ds/W(s) as Cauchy's principal value related to local time. We determine: (a) the modulus of continuity of Y in the sense of P. Lévy; (b) the large increments of Y. Received: 1 April 1999 / Revised version: 27 September 1999 / Published online: 14 June 2000     

A super-Brownian motion with a locally infinite catalytic mass

Summary. A super-Brownian motion in with “hyperbolic” branching rate , is constructed, which symbolically could be described by the formal stochastic equation (with a space-time white noise ). Starting at this superprocess will never hit the catalytic center: There is an increasing sequence of Brownian stopping times strictly smaller than the hitting time of such that with probability one Dynkin's stopped measures vanish except for finitely many Received: 27 November 1995 / In revised form: 24 July 1996     

Analysis of a stabilized three-fields domain decomposition method

Summary. In this paper we prove that, for suitable choices of the bilinear form involved in the stabilization procedure, the stabilized three fields domain decomposition method proposed in [8] is stable and convergent uniformly in the number of subdomains and with respect to their sizes under quite general assumptions on the decomposition and on the discretization spaces. The same is proven to hold for the resulting discrete Steklov-Poincaré operator. Received April 4, 2000 / Revised version received January 9, 2001 / Published online June 17, 2002     

On the locking phenomenon for a class of elliptic problems

Summary. In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not. Revised version received October 25, 1993     

Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

Asymmetric positive solutions for a symmetric nonlinear problem in {mathbb R}^N

We study a symmetric semilinear elliptic problem in all and we prove existence of an asymmetric positive solution by using variational arguments. The corresponding problem in dimension N=2, which provides the motivation of this work, arises in Nonlinear Optics from the study of the behaviour of optical cylindrical waveguides. Received September 28, 1999/ Accepted January 14, 2000 / Published online June 28, 2000     

Excision in entire cyclic cohomology

We prove that entire and periodic cyclic cohomology satisfy excision for extensions of bornological algebras with a bounded linear section. That is, for such an extension we obtain a six term exact sequence in cohomology. Received December 16, 1999 / final version received April 18, 2001?Published online May 29, 2001     

Asymptotic behaviour of disconnection and non-intersection exponents

Summary. We study the asymptotic behaviour of disconnection and non-intersection exponents for planar Brownian motionwhen the number of considered paths tends to infinity. In particular, if η _n (respectively ξ (n, p)) denotes the disconnection exponent for n paths (respectively the non-intersection exponent for n paths versus p paths), then we show that lim _{n →∞} η _n /n = 1 2 and that for a > 0 and b > 0,lim _{n →∞} ξ ([na],[nb])/n = (√ a + √ b) ² /2. Received: 28 February 1996 / In revised form: 3 September 1996