On families of Lagrangian submanifolds

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω _t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ _t of diffeomorphisms of X such that ω _t =Φ _t ^*(ω₀). If L⊂X is a Lagrangian submanifold for (X,ω₀), L _t =Φ _t ^-1(L) is thus a Lagrangian submanifold for (X,ω _t ). Here we show that if we simply assume that L is compact and ω _t | _L is exact for every t, a family L _t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure. Received: 29 May 2001/ Revised version: 17 October 2001     

Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t ?1/ω ?1(t ?1) for ${t\in (0,\infty).}Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t −1/ω −1(t −1) for t ? (0,￥).{t\in (0,\infty).} In this paper, the authors introduce the generalized VMO spaces VMOr, L(\mathbb Rn){{\mathop{\rm VMO}_ {\rho, L}({\mathbb R}^n)}} associated with L, and characterize them via tent spaces. As applications, the authors show that (VMOr,L (\mathbb Rn))*=Bw,L*(\mathbb Rn){({\rm VMO}_{\rho,L} ({\mathbb R}^n))^\ast=B_{\omega,L^\ast}({\mathbb R}^n)}, where L * denotes the adjoint operator of L in L2(\mathbb Rn){L^2({\mathbb R}^n)} and Bw,L*(\mathbb Rn){B_{\omega,L^\ast}({\mathbb R}^n)} the Banach completion of the Orlicz–Hardy space Hw,L*(\mathbb Rn){H_{\omega,L^\ast}({\mathbb R}^n)}. Notice that ω(t) = t p for all t ? (0,￥){t\in (0,\infty)} and p ? (0,1]{p\in (0,1]} is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and (VMO1, L(\mathbb Rn))*=HL*1(\mathbb Rn){({\mathop{\rm VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)}, where HL*1(\mathbb Rn){H_{L^\ast}^1({\mathbb R}^n)} was the Hardy space introduced by Hofmann and Mayboroda.     

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Large deviations and continuum limit in the 2D Ising model

Summary. We study the 2D Ising model in a rectangular box Λ _L of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑ _t∈ΛL σ(t) when L→∞ for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m ^* is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function <σ(0)σ(t)>, as |t|→∞, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature representation of the model. These results are valid for dimensions D≥2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation. We then study the Gibbs measure conditioned by {|∑ _t∈ΛL σ(t) −m|Λ _L ||≤|Λ _L |L ^{− c} }, with 0<c<1/4 and −m ^*<m<m ^*. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric type. Received: 17 October 1996 / In revised form: 7 March 1997     

<Emphasis Type="Italic">BV</Emphasis> functions and parabolic initial boundary value problems on domains

Given a uniformly elliptic second order operator on a possibly unbounded domain , let (T(t)) _t≥0 be the semigroup generated by in L ¹(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L ¹-norm of the spatial gradient D _x T(t)u ₀ tends to the total variation of the initial datum u ₀, and in particular is finite if and only if u ₀ belongs to BV(Ω). This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of (T(t)) _t≥0 is also given.      

Approximation by de la Vallée-Poussin operators on the classes of functions locally summable on the real axis

For the least upper bounds of deviations of the de la Vallée-Poussin operators on the classes [^(L)]_b^y \hat{L}_\beta^\psi of rapidly vanishing functions ψ in the metric of the spaces [^(L)]_p {\hat{L}_p} , 1 ≤ p ≤ ∞, we establish upper estimates that are exact on some subsets of functions from [^(L)]_p {\hat{L}_p} .     

Dérivé convexe d’une loi de probabilité et lois stables. Application a un problème d’isomorphisme

The “convex derived set” of a symmetric probability lawF on the real line is defined as the set of limits of laws ∗ _j−1/k ⁿ F(t _j ⁿ η), inf 1≤j≤k _n t _j ⁿ →∞ ifn→∞ and the stable laws it contains are exhibited. A new criterion of stochastic compacity of the set of the powers of a probability law is established. Finally, an isomorphism theorem between somel ^p andL ⁰ spaces is given.

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Laboratoire associé au C.N.R.S. n^o 224 “Processus stochastiques et applications”.     

Weak Convergence of a Nonlinear Functional of a Stationary Gaussian Process. Application to the Local Time

Let {X _t : 0 ≦ t ≦ 1} be a centered stationary Gaussian process, with correlation function satisfying the condition ρ(t) = 1 − t ^β L(t), 0 < β < 2, and let L be a slowly varying function at zero. Observing the process at points i/N, i = 0,1,..., N and considering |X _i/N − X _(i-1)/N | ^p with p > 0, we study the properties of the Donsker line associated with p-th order variations . We also study the relationship between the number of crossings of a regularization of the initial process and the local time of the initial process. The results depend on the values of β. This revised version was published online in June 2006 with corrections to the Cover Date.     

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

Geodesics On The Symplectomorphism Group

Let M be a compact manifold with a symplectic form ω and consider the group D_w{\mathcal{D}_\omega} consisting of diffeomorphisms that preserve ω. We introduce a Riemannian metric on M which is compatible with ω and use it to define an L ²-inner product on vector fields on M. Extending by right invariance we get a weak Riemannian metric on D_w{\mathcal{D}_\omega} . We show that this metric has geodesics which come from integral curves of a smooth vector field on the tangent bundle of D_w{\mathcal{D}_\omega} . Then, estimating the growth of such geodesics, we show that they extend globally.     

Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

In [Do], Doi proved that the ${L^{2}_{t}H^{1/2}_{x}}In [Do], Doi proved that the L²_tH^1/2_x{L^{2}_{t}H^{1/2}_{x}} local smoothing effect for Schr?dinger equations on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and L ¹ → L ^∞ dispersive estimates still hold without loss for e ^itΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.     

The geometry of whips

In this article, we study geometric aspects of the space of arcs parameterized by unit speed in the L ² metric. Physically, this corresponds to the motion of a whip, and it also arises in studying shape recognition. The geodesic equation is the nonlinear, nonlocal wave equation η _tt = ∂ _s (σ η _s ), with \lvert h_s\rvert o 1{\lvert \eta_{s}\rvert\equiv 1} and σ given by s_ss- \lvert h_ss\rvert² s = -\lvert h_st\rvert²{\sigma_{ss}- \lvert \eta_{ss}\rvert^2 \sigma = -\lvert \eta_{st}\rvert^2}, with boundary conditions σ(t, 1) = σ(t, −1) = 0 and η(t, 0) = 0. We prove that the space of arcs is a submanifold of the space of all curves, that the orthogonal projection exists but is not smooth, and as a consequence we get a Riemannian exponential map that is continuous and even differentiable but not C ¹. This is related to the fact that the curvature is positive but unbounded above, so that there are conjugate points at arbitrarily short times along any geodesic.     

Probabilistic Representations of Solutions of the Forward Equations

In this paper we prove a stochastic representation for solutions of the evolution equation

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.     

Two examples of local ergodic divergence

This note contains the first example of a 1-parameter semigroup {T _pt≧0} of linear contractions inL _p (1<p<∞) for which the assertion of the local ergodic theorem (t ¹∝₀′T _sfds conv. a.e. ast → 0+0 for allf ∈L _p) fails to be true. The first example is a continuous semigroup of unitary operators inL ₂, the second a power-bounded continuous semigroup of positive operators inL ₁. This answers problems of Kubokawa, Fong and Sucheston. In memory of our friend and colleague Shlomo Horowitz     

Precise asymptotics in Chung’s law of the iterated logarithm

Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.     

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(X ^ε (t)): ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψ_ε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X ε (t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψ_ε(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of Ψ_ε(u). AMS Subject Classifications 60F10, 91B30 This work has been partially supported by Murst Project “Metodi Stocastici in Finanza Matematica”     

Asymptotic closeness to limiting shapes for expanding embedded plane curves

We show that for embedded or convex plane curves expansion, the difference u(x,t)-r(t) in support functions between the expanding curves γ_t and some expanding circles C_t (with radius r(t)) has its asymptotic shape as t→∞. Moreover the isoperimetric difference L²-4πA is decreasing and it converges to a constant if the expansion speed is asymptotically a constant and the initial curve is not a circle. For convex initial curves, if the expansion speed is asymptotically infinite, then L²-4πA decreases to and there exists an asymptotic center of expansion for γ_t. Mathematics Subject Classification (2000) 35K15, 35K55     

On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ _t μ = L ^* μ for bounded Borel measures on ℝ ^d  × (0, T), where L is a second order elliptic operator, for example, Lu = D_xu + ( b,?_xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity