Backward doubly stochastic differential equations and systems of quasilinear SPDEs

Summary We introduce a new class of backward stochastic differential equations, which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDE's.The research of this author was partially supported by DRET under contract 901636/A000/DRET/DS/SRThe research of this author was supported by a grant from the French Ministère de la Recherche et de la Technologie, which is gratefully acknowledged     

A least-squares-based method for a class of nonsmooth minimization problems with applications in plasticity

This paper introduces a globally convergent algorithm for solving a class of nonsmooth optimization problems, involving square roots of quadratic forms. The class includes in particular limit analysis problems in plasticity. The algorithm combines smoothing with successive approximation. The main computational effort in each iteration is solving a linear weighted least-squares problem. The convergence of the algorithm is proved and ana priori error estimate is obtained. Numerical results are presented for two limit analysis problems.The work of the first author was partially supported by NSF Grant DDM-89-96112. Parts of the work was done during his stay at the University of Bayreuth as a guest of the DFG. The work of the second author was supported in part by the Air Force Office of Scientific Research under contract AFOSR-88-0218 and by a National Science Foundation Grant ECS-8802239 at the University of Maryland, Baltimore County Campus.     

Using the refinement equation for the construction of pre-wavelets III: Elliptic splines

The purpose of this paper is to provide multiresolution analysis, stationary subdivision and pre-wavelet decomposition onL ²(R ^d ) based on a general class of functions which includes polyharmonic B-splines.The work of this author has been partially supported by a DARPA grant.The work of this author has been partially supported by Fondo Nacional de Ciencia y Technologia under Grant 880/89.     

Characterizations of Overtaking Optimality for Controlled Diffusion Processes

In this paper we give conditions for (the existence and) several characterizations of overtaking optimal policies for a general class of controlled diffusion processes. Our characterization results are of a lexicographical type; namely, first we identify the class of so-called canonical policies, and then within this class we search for policies with some special feature—for instance, canonical policies that in addition maximize the bias. This research was partially supported by CONACyT grant 45693-F. The research of the first author (HJF) was also supported by a CONACyT scholarship.     

Normal generation of line bundles of high degrees on smooth algebraic curves

We give necessary and sufficient conditions for non-special line bundles of degree2g — 2 and 2g — 3 being not normally generated. We also provide criteria for special line bundles of degreed > 2g — 6 being normally generated. This work was done under JSPS-KOSEF joint research program 1997. The first named author was partially supported by Grant-in-Aid for Scientific Research, the Ministry of Education, #10440051. The second named author was partially supported by BSRI(1998-015-D00023) and SNU(99-5-l-031). During the period when this paper was prepared for publication, the second named author was enjoying the hospitality of ICTP. The third named author was partially supported by Grant-in-Aid for Scientific Research, the Ministry of Education, #09640043.     

Nonmonotone curvilinear line search methods for unconstrained optimization

We present a new algorithmic framework for solving unconstrained minimization problems that incorporates a curvilinear linesearch. The search direction used in our framework is a combination of an approximate Newton direction and a direction of negative curvature. Global convergence to a stationary point where the Hessian matrix is positive semidefinite is exhibited for this class of algorithms by means of a nonmonotone stabilization strategy. An implementation using the Bunch-Parlett decomposition is shown to outperform several other techniques on a large class of test problems.The work of this author was based on research supported by the National Science Foundation Grant CCR-9157632, the Air Force Office of Scientific Research Grant F49620-94-1-0036 and the Department of Energy Grant DE-FG03-94ER61915.These authors were partially supported by Agenzia Spaziale Italiana, Roma, Italy.     

Potential Theory of Geometric Stable Processes

In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(|x|^d log ²|x|), while the Lévy density behaves like 1/|x|^d. We also study the asymptotic behaviors of the Green function and Lévy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes. The research of this author is supported in part by MZT grant 0037118 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167. The research of this author is supported in part by a joint US-Croatia grant INT 0302167. The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.     

Credibility of Incentive Equilibrium Strategies in Linear-State Differential Games

We characterize in this paper the credibility of incentive equilibrium strategies for the class of linear-state differential games. We derive a general condition for credibility and illustrate its use on two differential games taken from the literature of environmental economics and knowledge accumulation. We show that the proposed linear incentive strategies are not always credible. Further, we provide alternative nonlinear credible strategies which suggest that we should not stick only to linear incentive strategies, even in a simple class of differential games such as the linear-state one.This research was completed when the first author was visiting professor at GERAD, HEC, Montréal. The first author’s research was partially supported by MCYT under project BEC2002-02361 and by JCYL under project VA051/03, confinanced by FEDER funds. The second author’s research was supported by NSERC, Canada.     

A survey of Markov decision models for control of networks of queues

We review models for the optimal control of networks of queues. Our main emphasis is on models based on Markov decision theory and the characterization of the structure of optimal control policies.This research was partially supported by the National Science Foundation under Grant No. DDM-8719825. The Government has certain rights in this material. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The research was also partially supported by the C.I.E.S. (France), while the author was on leave at INRIA, Sophia-Antipolis, 1991–92.     

On Discretization Schemes for Stochastic Evolution Equations

Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved. Mathematics Subject Classifications (2000) Primary: 60H15; Secondary: 65M60.István Gyöngy: This paper was written while the first named author was visiting the University of Paris X. The research of this author is partially supported by EU Network HARP.Annie Millet: The research of the second named author is partially supported by the research project BMF2003-01345.     

On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.     

Convergence rates for intermediate problems

Convergence rate estimates are derived for a variant of Aronszajn-type intermediate problems that is both computationally feasible and known to be convergent for problems with nontrivial essential spectrum. Implementation of these derived bounds is discussed in general and illustrated on differential eigenvalue problems. Convergence rates are derived for the commonly used method of simple truncation.The work of the first author was partially supported by AFOSR Grant 84-0326. The work of the second author was partially supported by NSF Grant MCS-8301402     

Specialising Aronszajn trees by countable approximations

We show that there are proper forcings based upon countable trees of creatures that specialise a given Aronszajn tree.The first author was partially supported by a Minerva fellowship. The second author's research was partially supported by the ``Israel Science Foundation', founded by the Israel Academy of Science and Humanities. This is the second author's work number 778. Mathematics Subject Classification (2000): 03E15, 03E17, 03E35, 03D65     

The structure of a Brownian bubble

Summary We examine local geometric properties of level sets of the Brownian sheet, and in particular, we identify the asymptotic distribution of the area of sets which correspond to excursions of the sheet high above a given level in the neighborhood of a particular random point. It is equal to the area of certain individual connected components of the random set {(s, t):B(t)>b(s)}, whereB is a standard Brownian motion andb is (essentially) a Bessel process of dimension 3. This limit distribution is studied and, in particular, explicit formulas are given for the probability that a point belongs to a specific connected component, and for the expected area of a component given the height of the excursion ofB(t)-b(s) in this component. These formulas are evaluated numerically and compared with the results from direct simulations ofB andb.The research of this author was partially supported by grants DMS-9103962 from the National Science Foundation and DAAL03-92-6-0323 from the Army Research Office     

The blow up analysis of solutions of the elliptic sinh-Gordon equation

In this paper, using a geometric method we show that the blow-up values of the elliptic sinh-Gordon equation are multiples of 8π. The fourth named author supported partially by NSFC of China (No. 10301020).     

A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two

We prove a new regularity result for systems of nonlinear elliptic equations with quadratic Jacobian type nonlinearity in dimension two. Our proof is based on an adaptation of John Lewis’ method which has not been used for such systems so far. Parts of this work have been done while the second and the third author had been enjoying the hospitality of the Department of Mathematics of the University of Pittsburgh. P.H. was supported by NSF grant DMS-0500966. P.S. was partially supported by the MNiSzW grant no 1 PO 3A 005 29. X.Z. was supported by the Academy of Finland, project 207288.     

An L-like model containing very large cardinals

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with a strong form of diamond and a version of square consistent with supercompactness. This generalises a result due to the first author. There are no restrictions in our model on the structure of the class of supercompact cardinals. A. W. Apter’s research was partially supported by PSC-CUNY Grants and CUNY Collaborative Incentive Grants. J. Cummings’s research was partially supported by NSF Grant DMS-0400982.     

Gently killing S-spaces

We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2^ℵ ₀<2^ℵ ₁. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size ℵ₁, answering a question of the second author [9]. The first author would like to thank The Hebrew University of Jerusalem for their support while the research in this paper was being carried out. The research of the second author was partially supported by NSF Grant DMS-9322613. The research of the third author was partially supported by NSF grant DMS-9704477 and the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. This is publication number 690 in the list of the third author.     

Inertia-preserving secant updates

A class of rank-two, inertia-preserving updates for symmetric matricesH _c is studied. To ensure that inertia is preserved, the updates are chosen to be of the formH ₊=FH _c F ^t, whereF=I+qr ^t, withq andr selected so that the secant equation is satisfied. A characterization is given for all such updates. Using a parameterization of this family of updates, the connection between them and the Broyden class of updates is established. Also, parameter selection criteria that can be used to choose the optimally conditioned update or the update closest to the SR1 update are discussed.The work of the first author was partially supported by AFOSR Grant 84-0326. The work of the second author was partially supported by NSF Grant EAR-82-18743.