BSDEs with two reflecting barriers : the general result

In this paper we show the existence of a solution for the BSDE with two reflecting barriers when those latter are completely separated. Neither Mokobodzkis condition nor the regularity of the barriers are supposed. The main tool is the notion of local solution of reflected BSDEs. Applications related to Dynkin games and double obstacle variational inequality are given.Mathematics Subject Classification (2000): 91A15, 60G40, 91A60     

Finite Element Methods for Parabolic Stochastic PDE’s

We study the rate of convergence of some explicit and implicit numerical schemes for the solution of a parabolic stochastic partial differential equation driven by white noise. These include the forward and backward Euler and the Crank–Nicholson schemes. We use the finite element method. We find, as expected, that the rates of convergence are substantially similar to those found for finite difference schemes, at least when the size of the time step k is on the order of the square of the size of the space step h: all the schemes considered converge at a rate on the order of h^1/2+k^1/4, which is known to be optimal. We also consider cases where k is much greater than h², and find that only the backward Euler method always attains the optimal rate; other schemes, even though they are stable, can fail to convergence to the true solution if the time step is too long relative to the space step. The Crank–Nicholson scheme behaves particularly badly in this case, even though it is a higher-order method. Mathematics Subject Classifications (2000) 60H15, 60H35, 65N30, 35R60.     

Renormalization and homogenization of fractional diffusion equations with random data

 This paper presents a renormalization and homogenization theory for fractional-in-space or in-time diffusion equations with singular random initial conditions. The spectral representations for the solutions of these equations are provided. Gaussian and non-Gaussian limiting distributions of the renormalized solutions of these equations are then described in terms of multiple stochastic integral representations. Received: 30 May 2000 / Revised version: 9 November 2001 / Published online: 10 September 2002 Mathematics Subject Classification (2000): Primary 62M40, 62M15; Secondary 60H05, 60G60 Key words or phrases: Fractional diffusion equation – Scaling laws – Renormalised solution – Long-range dependence – Non-Gaussian scenario – Mittag-Leffler function – Stable distributions – Bessel potential – Riesz potential     

Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term

 We study existence and uniqueness of a mild solution in the space of continuous functions and existence of an invariant measure for a class of reaction-diffusion systems on bounded domains of ℝ ^d , perturbed by a multiplicative noise. The reaction term is assumed to have polynomial growth and to be locally Lipschitz-continuous and monotone. The noise is white in space and time if d=1 and coloured in space if d>1; in any case the covariance operator is never assumed to be Hilbert-Schmidt. The multiplication term in front of the noise is assumed to be Lipschitz-continuous and no restrictions are given either on its linear growth or on its degenaracy. Our results apply, in particular, to systems of stochastic Ginzburg-Landau equations with multiplicative noise. Received: 1 November 2001 / Revised version: 17 June 2002 / Published online: 14 November Mathematics Subject Classification (2000): 60H15, 35R60, 47A35     

Propagation and conditional propagation of chaos for pressureless gas equations

We study the existence and uniqueness of a weak solution of a viscous d-dimensional system of pressureless gas equations. We construct a nonlinear diffusion by using the propagation and conditional propagation of chaos. The latter diffusion is associated with the above pressureless gas equations.Mathematics Subject Classification (2000):60H15, 35R60, 60H30     

Symmetric Ornstein-Uhlenbeck semigroups and their generators

 We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure μ. For a reversible process the domain of its generator in L ^p (μ) is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup (R _t ) to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. We show also that the existence of spectral gap implies a smoothing property of R _t and provide an estimate for the (appropriately defined) gradient of R _t φ. Finally, in the Hilbert-Schmidt case, we show tha t for any the function R _t φ is an (almost) classical solution of a version of the Kolmogorov equation. Received: 17 September 2001 / Revised version: 3 June 2002 / Published online: 30 September 2002 This work was partially supported by the Small ARC Grant Scheme. Mathematics Subject Classification (2000): Primary: 60H15, 47F05; Secondary: 60J60, 35R15, 35K15 Key words or phrases: Ornstein-Uhlenbeck operator – Second quantization – Reversibility – Spectral gap – Sobolev spaces – Domain of generator     

A representation of the Belavkin equation via Feynman path integrals

 The Belavkin equation, describing the continuous measurement of the position of a quantum particle, is studied. A rigorous representation of its solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the complex Cameron-Martin space is given. Received: 7 January 2002 / Revised version: 20 June 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 81, 81S40, 60H15 Key words or phrases: Belavkin equation – Continuous measurement – Quantum theory – Oscillatory integrals – Feynman path integrals     

Busy periods in M/M/∞ systems with heterogeneous servers

This note gives a solution for the problem of finding the probability density and probability distribution functions of the N-busy-period length for the M/M/∞ system where the servers are not necessarily the same. A solution in case of the same servers was done in [3]. AMS Subject Classification 60K25 68M20     

Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data

We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u ₀ belonging to the Lorentz space L ^2,∞ and such that curl u ₀ is a finite measure. Giga, Miyakawaand Osada [7] proved that a solution u exists and that u=K* curl u, where K is the Biot-Savartkernel and v = curl u is solution of a nonlinear equation in dimension one, called the vortex equation. In this paper, we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a Monte-Carlo approximation for a solution of the Navier-Stokesequation. That gives in this case a pathwise proofof the vortex algorithm introducedby Chorin and consequently generalizes the works ofMarchioro-Pulvirenti [12] and Méléardv [15] obtained in the case of a vortex equation with bounded density initial data. Received: 6 October 1999 / Revised version: 15 September 2000 / Published online: 9 October 2001     

Singular Monge-Ampère foliations

 This paper generalizes results of Lempert and Sz?ke on the structure of the singular set of a solution of the homogeneous Monge-Ampère equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be C ³ everywhere is replaced by a smoothness condition on the blowup of the singular set. Under these conditions, the singular set is shown to inherit a Finsler metric, which in the real analytic case uniquely determines the solution of the Monge-Ampère equation. These results are proved using techniques from contact geometry. Received: 6 April 2001 / Published online: 2 December 2002 Mathematics Subject Classification (2000): 53C56, 32F, 53C60     

Asymptotic analysis via Mellin transforms for small deviations in L 2-norm of integrated Brownian sheets

We use Mellin transforms to compute a full asymptotic expansion for the tail of the Laplace transform of the squared L²-norm of any multiply-integrated Brownian sheet. Through reversion we obtain corresponding strong small-deviation estimates.Research supported by NSF grant DMS–0104167, and by The Johns Hopkins Universitys Acheson J. Duncan Fund for the Advancement of Research in Statistics.Mathematics Subject Classification (2000):Primary 60G15, 41A60; secondary 60E10, 44A15, 41A27     

On the infinite divisibility of squared Gaussian processes

 We show that fractional Brownian motions with index in (0,1] satisfy a remarkable property: their squares are infinitely divisible. We also prove that a large class of Gaussian processes are sharing this property. This property then allows the construction of two-parameters families of processes having the additivity property of the squared Bessel processes. Received: 1 April 2002 / Revised version: 7 September 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 60E07, 60G15, 60J25, 60J55 Key words or phrases: Gaussian processes – Infinite divisibility – Markov processes     

Applications of Lobatto polynomials to an adaptive finite element method: A posteriori error estimates for HP-Adaptivity and Grid-to-Grid Interpolation

Summary.  Hp-adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions between grids. Lobatto polynomial-based techniques are presented for both. An interpolation error-based error estimation strategy for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution. The estimates involve high-order derivatives of the solution that must be approximated from the computed solution. Differentiating a ``Taylor-like' series for error in the Lobatto interpolant and using the weak form of the equations yields the correct derivative approximations. This leads to a more robust order selection strategy. Interpolation between grids is done over each element using the Lobatto interpolating polynomial. Explicit formulas for the inverse of the resulting Lobatto interpolation matrices are given. Computational results illustrate the theory. Received June 25, 2001 / Revised version received February 12, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65M15,65M20,65M60 This research was partially supported by NSF Grant #DMS-0196108.     

Solution of Maxwell equation in axisymmetric geometry by Fourier series decompostion and by use of H (rot) conforming finite element

Summary. This study deals with the mathematical and numerical solution of time-harmonic Maxwell equation in axisymmetric geometry. Using Fourier decomposition, we define weighted Sobolev spaces of solution and we prove expected regularity results. A practical contribution of this paper is the construction of a class of finite element conforming with the H (rot) space equipped with the weighted measure rdrdz. It appears as an extension of the well-known cartesian mixed finite element of Raviart-Thomas-Nédélec [11]–[15]. These elements are built from classical lagrangian and mixed finite element, therefore no special approximations functions are needed. Finally, following works of Mercier and Raugel [10], we perform an interpolation error estimate for the simplest proposed element. Received March 15, 1996 / Revised version received November 30, 1998 / Published online December 6, 1999     

Cost operator algorithms for the transportation problem

A primal transportation algorithm is devised via post-optimization on the costs of a modified problem. The procedure involves altering the costs corresponding to the basic cells of the initial (primal feasible) solution so that it is dual feasible as well. The altered costs are then successively restored to their true values with appropriate changes in the optimal solution by the application of cell or area cost operators discussed elsewhere. The cell cost operator algorithm converges to optimum within (2T – 1) steps for primal nondegenerate transportation problems and [(2T + 1) min (m, n)] – 1 steps for primal degenerate transportation problems, whereT is the sum of the (integer) warehouse availabilities (also the sum of the (integer) market requirements) andm andn denote the number of warehouses and markets respectively. For the area cost operator algorithm the corresponding bounds on the number of steps areT and (T + 1) min (m, n) respectively.This report was prepared as part of the activities of the Management Sciences Research Group, Carnegie—Mellon University, under Contract N00014-67-A-0314-0007 NR 047-048 with the U.S. Office of Naval Research.     

Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux

Résumé Nous considérons le système des équations d'Euler isentropiques en dimension deux; pour des données initiales invariantes par rotation et perturbations de taille d'un état de repos, on établit un équivalent du temps de vieT de la solution classique (lim ² T = _* ² ).De plus, on donne, pour une estimation de la vraie solution, en calculant la taille de son écart à une solution approchée construite dans un précédent travail.

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Summary We consider the 2D isentropic Euler equations; for rotationnally invariant data which are a perturbation of size of a rest state, we establish the first term asymptotic of the life spanT of the classical solution (lim ² T = _* ² ).Moreover, we give, for an estimate of the true solution, by computing the size of its difference with an approximate solution obtained in a previous work.

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Oblatum 2-XII-1991 & 24-IX-1992     

Green function estimate for censored stable processes

 Sharp two-sided estimates for Green functions of censored α-stable process Y in a bounded C ^1,1 open set D are obtained, where α  (1, 2). It is shown that the Martin boundary and minimal Martin boundary of Y can all be identified with the Euclidean boundary of D. Sharp two-sided estimates for the Martin kernel of Y are also derived. Received: 27 January 2002 / Revised version: 10 June 2002 / Published online: 24 October 2002 This research is supported in part by NSF Grant DMS-0071486. Mathematics Subject Classification (2002): Primary: 60J45, 31C35; Secondary: 60G52, 31C15 Keywords or phrases: Censored stable process – Green function – Capacity – Martin boundary – Martin kernel – Harmonic function     

Some existence results for solutions to SU(3) Toda system

We study SU(3) Toda system in non-abelian relativistic self-dual gauge theory. In the range of parameters where the corresponding Trudinger-Moser inequality fails, we show the existence of the solution by a different variational formulation from Lucia-Nolasco's [15]. This work was supported by a grant of the Japan-Korea Scientific Cooperation Program - Joint Research “Mathematical analysis and mathematical science for self-interacting particles.” The second author was partially supported by Grant-in-Aid for Scientific Research (No. 16740103), Japan Society for the Promotion of Science. Mathematics Subject Classification (2000) 35B40 - 35J50 - 35J60 - 49Q99 - 58E15 - 58J05 - 70S15     

Error analysis of a mixed finite element method for the Cahn-Hilliard equation

Summary. We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation u_t + (u–^–1f(u)) = 0, where > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on only in some lower polynomial order for small . The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as 0 in [29].Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10Acknowledgments. The first author would like to thank Nicholas Alikakos for explaining all the fascinating properties of the Allen-Cahn and Cahn-Hilliard equations to him. He would also like to thank Nicholas Alikakos and Xinfu Chen for answering his questions regarding the spectrum estimate in Proposition 1. The second author gratefully acknowledges financial support by the DFG.