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     

Mappings of finite distortion: Hausdorff measure of zero sets

We prove that a for a mapping f of finite distortion , the -Hausdorff measure of any point preimage is zero provided is integrable, with , and the multiplicity function of f is essentially bounded. As a consequence for we obtain that the mapping is then open and discrete. Received: 18 June 2001 / Revised version: 31 January 2002 / Published online: 27 June 2002     

Stochastic integration with respect to q Brownian motion

 We develop a stochastic integration with respect to a q-Brownian motion (for ), i.e. a non commutative process where the operator a _t and its adjoint fulfill the q commutation relation ; under the vacuum state expectation. We show that this process enjoys a predictable representation type property. Received: 15 February 2002 / Revised version: 25 May 2002 / Published online: 30 September 2002 Mathematics Subject Classification (2000): 60H05, 46L50, 81S25     

The mathematics of playing golf, or: a new class of difficult non-linear mixed integer programs

We consider a class of non-linear mixed integer programs with n integer variables and k continuous variables. Solving instances from this class to optimality is an NP-hard problem. We show that for the cases with k=1 and k=2, every optimal solution is integral. In contrast to this, for every k≥3 there exist instances where every optimal solution takes non-integral values. Received: August 2001 / Accepted: January 2002?Published online March 27, 2002     

On uniqueness for semilinear wave equations

 We prove that uniqueness holds in for solutions of for suitable values of s, p. This includes notably the critical case for n=4,5,6. Received: 4 February 2002 ; in final form: 9 August 2002 / Published online: 1 April 2003     

Onsager-Machlup functional for the fractional Brownian motion

 Let X be the solution of the stochastic differential equation where B ^H is a fractional Brownian motion with Hurst parameter H. In this paper we compute the Onsager-Machlup functional of X for the supremum norm and H?lder norms of order β with in the case and for H?lder norms of order β with when . Received: 16 July 2001 / Revised version: 12 March 2002 / Published online: 10 September 2002     

Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations

In this paper we are concerned with multi-lump bound states of the nonlinear Schr?dinger equation for sufficiently small , where for and for . V is bounded on . For any finite collection of nondegenerate critical points of V, we show the uniqueness of solutions of the form for , where u is positive on and is a small perturbation of a sum of one-lump solutions concentrated near , respectively for sufficiently small . Received: 30 October 2001; in final form: 10 June 2002 /Published online: 2 December 2002 RID="*" ID="*" Research supported by Alexander von Humboldt Foundation in Germany and NSFC in China     

Inversion of adjunction for non-degenerate hypersurfaces

 We prove a precise inversion of adjunction formula for the log variety (ℂ ^{d +1},X), where X is a non-degenerate hypersurface. As a corollary, the minimal log discrepancies of non-degenerate normal hypersurface singularities are bounded by dimension. Received: 17 September 2002 / Revised version: 22 November 2002 Published online: 14 February 2003 Current address: DPMMS, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England. e-mail: f.ambro@dpmms.cam.ac.uk Mathematics Subject Classification (2000): Primary 14B05; Secondary 14M25, 52B20     

Girth of sparse graphs

For each fixed k ≥ 0, we give an upper bound for the girth of a graph of order n and size n + k. This bound is likely to be essentially best possible as n → ∞. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 194–200, 2002; DOI 10.1002/jgt.10023     

Energy quantization for triholomorphic maps

Let be weakly convergent stationary triholomorphic maps from a hyperkähler manifold M to another hyperkähler manifold N. We establish an energy quantization for the density function of the defect measure on the concentration set.Received: 10 July 2002, Accepted: 30 September 2002, Published online: 17 December 2002     

The two phase stochastic Stefan problem

 This work is concerned with existence for a stochastic free boundary problem arising in phase transition (the Stefan two phase problem). The existence of an invariant ergodic measure associated with the corresponding transition semigroup P(t) is also proved, together with an integration by parts formula for the generator of P(t). Received: 20 June 2001 / Revised version: 17 June 2002 / Published online: 24 October 2002 Mathematics Subject Classification (2000): 35K35, 35R15, 60H15 Key words or phrases: Stochastic Stefan problem – Invariant measures – Wiener process – Transition semigroup – Kolmogorov operator     

Backward SDEs and Cauchy problem for semilinear equations in divergence form

 We extend the definition of solutions of backward stochastic differential equations to the case where the driving process is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic partial differential equations in Sobolev spaces. Received: 22 January 2002 / Revised version: 10 September 2002 / Published online: 19 December 2002 Research supported by KBN Grant 0253 P03 2000 19. Mathematics Subject Classification (2002): Primary 60H30; Secondary 35K55 Key words or phrases: Backward stochastic differential equation – Semilinear partial differential equation – Divergence form operator – Weak solution     

Vers un algorithme pour la réduction stable des revêtements <Emphasis Type="Italic">p</Emphasis>-cycliques de la droite projective sur un corps <Emphasis Type="Italic">p</Emphasis>-adique

 In his Ph. D. thesis, C. Lehr offers an algorithm which gives the stable model for p-cyclic covers of the projective line over a p-adic field under the conditions that the branch locus whose cardinal is m+1 has the so called equidistant geometry and m<p. In this note we give an algorithm also in the equidistant geometry case but without condition on m. In particular we are able to study the reduction at 2 of hyperelliptic curves with equidistant branch locus.

---

Vers un algorithme pour la réduction stable des revêtements p-cycliques de la droite projective sur un corps p-adique

---

Received: 11 February 2002 / Revised version: 8 May 2002 / Published online: 2 December 2002

---

Mathematics Subject Classification (2000): 11G20, 14H30, 14Q05     

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     

Non-existence of phase transition of oriented percolation on Sierpinski carpet lattices

 A percolation problem on Sierpinski carpet lattices is considered. It is obtained that the critical probability of oriented percolation is equal to 1. In contrast it was already shown that the critical probability p _c of percolation is strictly less than 1 in Kumagai [9]. This result shows a difference between fractal-like lattice and ℤ ^d lattice. Received: 15 May 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003 Mathematics Subject Classification (2000): Primary: 60K35, 82B43; Secondary: 82B26     

Vanishing of local homology

 We study the vanishing properties of local homology of complexes of modules without assuming that its homology is artinian. Using vanishing results for local homology and cohomology we prove new vanishing results for Ext- and Tor-modules. Received: 1 August 2002; in final form: 23 September 2002 / Published online: 16 May 2003 Mathematics Subject Classification (1991): 13C12, 13D07, 13D45.     

The central limit theorem for Markov chains started at a point

 The aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic Markov chain on a general state space, with respect to the law of the chain started at a point. No irreducibility assumption nor mixing conditions are imposed; the only assumption bears on the growth of the L ² -norms of the ergodic sums for the function generating the additive functional, which must be with . The result holds almost surely with respect to the invariant probability of the chain. Received: 17 October 2001 / Revised version: 5 April 2002 / Published Online: 24 October 2002 Mathematics Subject Classification (2000): 60F05, 60J05