A general non-convex large deviation result with applications to stochastic equations

We prove an abstract large deviation result for a sequence of random elements of a vector space satisfying an “abstract exponential martingale condition”. The framework naturally generates non-convex rate functions. We apply the result to solutions of It? stochastic equations in R ^d driven by Brownian motion and a Poisson random measure. Received: 23 June 1999 / Revised version: 17 February 2000 / Published online: 22 November 2000     

Ray Hölder-continuity for fractional Sobolev spaces in infinite dimensions and applications

We prove H?lder-continuity on rays in the direction of vectors in the (generalized) Cameron-Martin space for functions in Sobolev spaces in L ^p of fractional order α∈ (, 1) over infinite dimensional linear spaces. The underlying measures are required to satisfy some easy standard structural assumptions only. Apart from Wiener measure they include Gibbs measures on a lattice and Euclidean interacting quantum fields in infinite volume. A number of applications, e.g., to the two-dimensional polymer measure, are presented. In particular, irreducibility of the Dirichlet form associated with the latter measure is proved without restrictions on the coupling constant. Received: 9 November 1998 / Published online: 30 March 2000     

Almost sure asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem ∂ _t u = κΔu + ξ(x)u on ℝ₊×ℝ ^d with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞. Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000     

On logarithmic Sobolev inequalities for continuous time random walks on graphs

We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ ^d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals such as the graph distance in this setting. Received: 6 April 1998 / Revised version: 15 March 1999 / Published on line: 14 February 2000     

Stable windings on hyperbolic surfaces

Let ℳ be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law under the Patterson-Sullivan measure on T ¹ℳ of the windings of the geodesics of ℳ around the cusps. This limit law is stable with parameter 2δ− 1, where δ is the Hausdorff dimension of the limit set of the subgroup Γ of M?bius isometries associated with ℳ. The normalization is t ^{−1/(2δ−1)}, for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan measure mentioned above by measures that are regular along the stable leaves. Received: 8 October 1999 / Revised version: 2 June 2000 / Published online: 21 December 2000     

On the Hopfield model at the critical temperature

We study the Hopfield model at temperature 1, when thenumber M(N) of patterns grows a bit slower than N. We reach a goodunderstanding of the model whenever M(N)≤N/(log N)¹¹. For example, we show that if M(N)→∞, for two typical configurations σ ¹, σ ², (∑ _{i ≤ N} σ¹ _i σ² _i )² is close to NM(N). Received: 15 December 1999 / Revised version: 8 December 2000 / Published online: 23 August 2001     

Self-adjointness of some infinite-dimensional elliptic operators and application to stochastic quantization

We consider an operator K˚ϕ = Lϕ−: <CDU(x), Dϕ> in a Hilbert space H, where L is an Ornstein–Uhlenbeck operator, U∈W ^1,4(H, μ) and μ is the invariant measure associated with L. We show that K˚ is essentially self-adjoint in the space L ²(H, ν) where ν is the “Gibbs” measure ν(dx) = Z ^−:1 e ^−:2U(x) dx. An application to Stochastic quantization is given. Received: 13 August 1998 / Revised version: 20 September 1999 / Published online: 8 August 2000     

Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold

We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on ℤ ^d at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap of the generator of the dyamics in a box of side L centered at the origin scales like L ⁻². Such an estimate is then used to prove a decay to equilibrium for local functions of the form where ε is positive and arbitrarily small and α = ? for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition when β crosses the critical value β _c of the pure system. Received: 10 April 2000 / Revised version: 23 October 2000 / Published online: 5 June 2001     

Transversal fluctuations for increasing subsequences on the plane

Consider a realization of a Poisson process in ℝ² with intensity 1 and take a maximal up/right path from the origin to (N, N) consisting of line segments between the points, where maximal means that it contains as many points as possible. The number of points in such a path has fluctuations of order N ^χ, where χ = 1/3, [BDJ]. Here we show that typical deviations of a maximal path from the diagonal x = y is of order N ^ξ with ξ = 2/3. This is consistent with the scaling identity χ = 2ξ− 1 which is believed to hold in many random growth models. Received: 16 April 1999 / Revised version: 5 July 1999 / Published online: 14 February 2000     

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc Δ ⁿ ⊂ℂ ⁿ in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut (M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut (M) and Aut (Δ ⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δ ⁿ .      

Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3

We consider an asymmetric exclusion process in dimension d≥ 3 under diffusive rescaling starting from the Bernoulli product measure with density 0 < α < 1. We prove that the density fluctuation field Y ^N _t converges to a generalized Ornstein–Uhlenbeck process, which is formally the solution of the stochastic differential equatin dY _t = ?Y _t dt + dB ^∇ _t , where ? is a second order differential operator and B ^∇ _t is a mean zero Gaussian field with known covariances. Received: 31 May 1999 / Revised version: 15 June 2000 / Published online: 24 January 2001     

Supporting-points processes and some of their applications

We introduce a stochastic point process of S-supporting points and prove that upon rescaling it converges to a Gaussian field. The notion of S-supporting points specializes (for adequately chosen S) to Pareto (or, more generally, cone) extremal points or to vertices of convex hulls or to centers of generalized Voronoi tessellations in the models of large scale structure of the Universe based on Burgers equation. The central limit theorems proven here imply i.a. the asymptotic normality for the number of convex hull vertices in large Poisson sample from a simple polyhedra or for the number of Pareto (vector extremal) points in Poisson samples with independent coordinates. Received: 20 July 1997 / Revised version: 18 August 1999 /?Published online: 11 April 2000     

Invariant manifolds for weak solutions to stochastic equations

Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C ² submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: any weak solution, which is viable in a finite dimensional C ² submanifold, is a strong solution. These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves. Received: 15 April 1999 / Revised version: 4 February 2000 / Published online: 18 September 2000     

On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S¹ → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20     

Semitoric integrable systems on symplectic 4-manifolds

Let (M,ω) be a symplectic 4-manifold. A semitoric integrable system on (M,ω) is a pair of smooth functions J,H∈C ^∞(M,ℝ) for which J generates a Hamiltonian S ¹-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce. A. Pelayo was partially supported by an NSF Postdoctoral Fellowship.     

The Markov chain associated to a Pick function

To each function ϕ˜(ω) mapping the upper complex half plane ?⁺ into itself such that the coefficient of ω in the Nevanlinna integral representation is one, we associate the kernel p(y, dx) of a Markov chain on ℝ by

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G = GL _N or SL _N as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2  ^N : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H ^*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H ⁱ (G, M) are Noetherian A ^G -modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.     

The maximum maximum of a martingale constrained by an intermediate law

Let (M _t ) be any martingale with M ₀≡ 0, an intermediate law M ₁∼μ₁, and terminal law M ₂∼μ₂, and let Mˉ ₂≡ sup_{0≤ t ≤2} M _t . In this paper we prove that there exists an upper bound, with respect to stochastic ordering of probability measures, on the law of Mˉ ₂. We construct, using excursion theory, a martingale which attains this maximum. Finally we apply this result to the robust hedging of a lookback option. Received: 26 December 1998 / Revised version: 20 April 2000 /?Published online: 15 February 2001     

Sharp and Meager Elements in Orthocomplete Homogeneous Effect Algebras

We prove that every orthocomplete homogeneous effect algebra is sharply dominating. Let us denote the greatest sharp element below x by x ^↓. For every element x of an orthocomplete homogeneous effect algebra and for every block B with x ∈ B, the interval [x ^↓,x] is a subset of B. For every meager element (that means, an element x with x ^↓ = 0), the interval [0,x] is a complete MV-effect algebra. As a consequence, the set of all meager elements of an orthocomplete homogeneous effect algebra forms a commutative BCK-algebra with the relative cancellation property. We prove that a complete lattice ordered effect algebra E is completely determined by the complete orthomodular lattice S(E) of sharp elements, the BCK-algebra M(E) of meager elements and a mapping h:S(E)→2 ^M(E) given by h(a) = [0,a] ∩ M(E).