Weak KAM Theory topics in the stationary ergodic setting

We perform a qualitative analysis of the critical equation associated with a stationary ergodic Hamiltonian through a stochastic version of the metric method, where the notion of closed random stationary set, issued from stochastic geometry, plays a major role. Our purpose is to give an appropriate notion of random Aubry set, to single out characterizing conditions for the existence of exact or approximate correctors, and write down representation formulae for them. For the last task, we make use of a Lax-type formula, adapted to the stochastic environment. This material can be regarded as a first step of a long-term project to develop a random analog of Weak KAM Theory, generalizing what done in the periodic case or, more generally, when the underlying space is a compact manifold.     

On the HSL-flow

We introduce a natural geometric fourth order flow associated to Hamiltonian stationary submanifolds in Kähler–Einstein manifolds. Afterwards we study some of its properties and show short-time existence. In case of Hamiltonian stationary submanifolds with bounded second fundamental form evolving in flat space we obtain an existence time estimate (Theorem 3.7).     

Hamiltonian stationary tori in K?hler manifolds

A Hamiltonian stationary Lagrangian submanifold of a K?hler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a K?hler manifold of real dimension four to guarantee the existence of a family of small Hamiltonian stationary Lagrangian tori.     

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

The existence of Hamiltonian stationary Lagrangian tori in K?hler manifolds of any dimension

Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They are natural generalizations of special Lagrangians or Lagrangian and minimal submanifolds. In this paper, we obtain a local condition that gives the existence of a smooth family of Hamiltonian stationary Lagrangian tori in K?hler manifolds. This criterion involves a weighted sum of holomorphic sectional curvatures. It can be considered as a complex analogue of the scalar curvature when the weighting are the same. The problem is also studied by Butscher and Corvino (Hamiltonian stationary tori in Kahler manifolds, 2008).     

On solving the potential equation

We study radial solutions to the generalized Swift-Hohenberg equation on the plane with an additional quadratic term. We find stationary localized radial solutions that decay at infinity and solutions that tend to constants as the radius increases unboundedly (“droplets”). We formulate existence theorems for droplets and sketch the proofs employing the properties of the limit system as r → ∞. This system is a Hamiltonian system corresponding to a spatially one-dimensional stationary Swift-Hohenberg equation. We analyze the properties of this system and also discuss concentric-wave-type solutions. All the results are obtained by combining the methods of the theory of dynamical systems, in particular, the theory of homo-and heteroclinic orbits, and numerical simulation.     

Stability of Riccati's Equation with Random Stationary Coefficients

The purpose of this paper is to study under weak conditions of stabilizability and detectability, the asymptotic behavior of the matrix Riccati equation which arises in stochastic control and filtering with random stationary coefficients. We prove the existence of a stationary solution of this Riccati equation. This solution is attracting, in the sense that if P _t is another solution, then onverges to 0 exponentially fast as t tends to +∞ , at a rate given by the smallest positive Lyapunov exponent of the associated Hamiltonian matrices. Accepted 13 January 1998     

Homogenization of “Viscous” Hamilton–Jacobi Equations in Stationary Ergodic Media

ABSTRACT

We study the homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media. The “viscosity” and the spatial oscillations are assumed to be of the same order. We identify the asymptotic (effective) equation, which is a first-order deterministic Hamilton–Jacobi equation. We also provide examples that show that the associated macroscopic problem does not admit suitable solutions (correctors). Finally, we present as applications results about large deviations of diffusion processes and front propagation (asymptotics of reaction-diffusion equations) in random environments.     

On orientation-preserving automorphisms of Hamiltonian cycles in the <Emphasis Type="Italic">N</Emphasis>-dimensional Boolean cube

We obtain an upper bound for the order of the group of orientation-preserving automorphisms of a Hamiltonian cycle in the Boolean n-cube. We prove that the existence of a Hamiltonian cycle with the order of the group of orientation-preserving automorphisms attaining this upper bound is equivalent to the existence of a Hamiltonian cycle with an additional condition on the graph of orbits of a fixed automorphism of the n-cube.     

Ergodicity in infinite Hamiltonian systems with conservative noise

Summary. We study the stationary measures of an infinite Hamiltonian system of interacting particles in ℝ ³ subject to a stochastic local perturbation conserving energy and momentum. We prove that the translation invariant measures that are stationary for the deterministic Hamiltonian dynamics, reversible for the stochastic dynamics, and with finite entropy density, are convex combination of “Gibbs” states. This result implies hydrodynamic behavior for the systems under consideration. Received: 17 December 1994/In revised form: 12 April 1996     

Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions

We derive a Weierstrass-type formula for conformal Lagrangian immersions in Euclidean 4-space, and show that the data satisfies an equation similar to Dirac equation with complex potential. Alternatively this representation has a simple formulation using quaternions. We apply it to the Hamiltonian stationary case and construct all possible tori, thus obtaining a first approach to a moduli space in terms of a simple algebraic-geometric problem on the plane. We also classify Hamiltonian stationary Klein bottles and show they self-intersect. Received: January 25, 2000.     

Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

Abstract

We prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.     

Stationary focusing mean-field games

We consider stationary viscous mean-field games (MFG) systems in the case of local, decreasing and unbounded coupling. These systems arise in ergodic MFG theory and describe Nash equilibria of games with a large number of agents aiming at aggregation. We show how the dimension of the state space, the behavior of the coupling, and the Hamiltonian at infinity affect the existence and nonexistence of regular solutions. Our approach relies on the study of Sobolev regularity of the invariant measure and a blow-up procedure that is calibrated on the scaling properties of the system. In very special cases, we observe uniqueness of solutions. Finally, we apply our methods to obtain new existence results for MFG systems with competition, namely, when the coupling is local and increasing.     

Metric techniques for convex stationary ergodic Hamiltonians

We adapt the metric approach to the study of stationary ergodic Hamilton?CJacobi equations, for which a notion of admissible random (sub)solution is defined. For any level of the Hamiltonian greater than or equal to a distinguished critical value, we define an intrinsic random semidistance and prove that an asymptotic norm does exist. Taking as source region a suitable class of closed random sets, we show that the Lax formula provides admissible subsolutions. This enables us to relate the degeneracies of the critical stable norm to the existence/nonexistence of exact or approximate critical admissible solutions.     

An isoperimetric inequality for Hamiltonian stationary Lagrangian tori in {\mathbb C}^2 related to Oh's conjecture

We compute loops integrals on Hamiltonian stationary Lagrangian tori in which are symplectic invariants, then we show an isoperimetric inequality involving these invariants and the area. Finally, we show that the flat torus has least area among Hamiltonian stationary Lagrangian tori of its isotopy class. Received: 4 December 2000; in final form: 18 January 2002 / Published online: 5 September 2002     

Time Delay and Short-Range Scattering in Quantum Waveguides

Although many physical arguments account for using a modified definition of time delay in multichannel-type scattering processes, one can hardly find rigorous results on that issue in the literature. We try to fill in this gap by showing, both in an abstract setting and in a short-range case, the identity of the modified time delay and the Eisenbud-Wigner time delay in waveguides. In the short-range case we also obtain limiting absorption principles, state spectral properties of the total Hamiltonian, prove the existence of the wave operators and show an explicit formula for the S-matrix. The proofs rely on stationary and commutator methods. Communicated by Yosi Avron submitted 12/04/05, accepted 13/05/05     

On Hamiltonian stationary Lagrangian spheres in non-Einstein K?hler surfaces

Hamiltonian stationary Lagrangian spheres in K?hler-Einstein surfaces are minimal. We prove that in the family of non-Einstein K?hler surfaces given by the product Σ₁?×?Σ₂ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example, defined when the surfaces Σ₁ and Σ₂ are spheres, is unstable.     

Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems

We study some monotonicity and iteration inequality of the Maslov-type index i-1of linear Hamiltonian systems.As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity.This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption.We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity which was proved by Fei,Kim and Wang in 2001.     

Nonlinear evolution equations and their stationary reductions

In recent years there have been many papers on stationary flows of integrable nonlinear evolution equations and their Hamiltonian properties. In particular there have been some results concerning the reversal of the roles of x and t, resulting in PDEs which are Hamiltonian and give the usual stationary Poisson brackets in the reduced case. To date the results have been rather ad hoc and disparate. In this brief report we give a systematic construction of these x−t reversed equations and their Hamiltonian properties, using their isospectral properties. We illustrate our approach with examples from the KdV hierarchy. Bibliography: 5 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1996, pp. 245–159.     

Multiple periodic solutions for non-canonical Hamiltonian systems with application to differential delay equations

We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system. Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory. As an application of the results, we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.