We consider the Lévy flow of diffeomorphisms of a manifold obtained by solving a stochastic differential equation driven by an n-dimensional Lévy process and n complete smooth vector fields (which generate a finite-dimensional Lie algebra L) and show that a necessary and sufficient condition for a large class of such flows to have an invariant measure is that each of the vector fields is divergence-free. We investigate the existence and uniqueness of such measures and examine their invariance with respect to the action of the associated Markov semigroup. In particular, we prove that there is an invariant probability measure on M if and only if the transformation Lie group G whose Lie algebra is L is compact. In the unbounded case we show that if there is a unique G-invariant measure, then it is also the unique invariant measure for the Markov semigroup provided the flow is recurrent.

The Fourier method in control systems reduces the study of controllability/observability to the study of related exponential families. In this paper we present examples of such systems, specifically those for which we can prove that the related exponential families form a Riesz basis in corresponding appropriately defined Sobolev spaces. This makes it possible to choose ‘natural’ pairs of spaces: the state space observability space and the control space state space, depending on whether an observation or a control problem is studied, respectively, so that the observation and control operators are isomorphisms.

We improve the known results on interpolation of strictly singular operators and strictly co-singular operators in several directions. Applications are given to embeddings between symmetric spaces.

We consider Hill's equation on the interval [0, a] with periodic and semi-periodic boundary condition. We derive the asymptotic form of the eigenvalues without smoothness conditions on q.

Bifurcation of periodic and chaotic solutions is investigated for coupled perturbed autonomous ordinary differential equations (ODEs) when the first unperturbed ODE has a homoclinic solution and the second unperturbed ODE possesses a Hopf singularity.

We give a structure theorem for simplicial affine semigroups. From this result we deduce characterizations of some properties of semigroup rings of simplicial affine semigroups. We also compute an upper bound for the cardinality of a minimal presentation of a simplicial affine semigroup.

We study the Cauchy problem for the nonlinear Schrödinger equation with dissipation where L is a linear pseudodifferential operator with dissipative symbol ReL(ξ) ≥ C1|ξ|2/(1 + ξ2) and |L′(ξ)| ≤ C2(|ξ|+ |ξ|n) for all ξ ∈ R. Here, C1, C2 > 0, n ≥ 1. Moreover, we assume that L(ξ) = αξ2 + O(|ξ|2+γ) for all |ξ| < 1, where γ > 0, Re α > 0, Im α ≥ 0. When L(ξ) = αξ2, equation (A) is the nonlinear Schrödinger equation with dissipation ut − αuxx + i|u|2u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate under the conditions that u0 ∈ Hn,0 ∩ H0,1 have the mean value and the norm ‖u0‖Hn,0 + ‖u0‖H0,1 = ε is sufficiently small, where σ = 1 if Im α > 0 and σ = 2 if Im α = 0, and Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation ut − αuxx + i|u|p−1u = 0, with p > 3 have the same time decay estimate ‖u‖L∞ = O(t−½) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.

This paper provides a topological and ergodic analysis of random linear Hamiltonian systems. We consider a class of Hamiltonian equations presenting absolutely continuous dynamics and prove the existence of the radial limits of the Weyl M-functions in the L1-topology. The proof is based on previous ergodic relations obtained for the Floquet coefficient. The second part of the paper is devoted to the qualitative description of disconjugate linear Hamiltonian equations. We show that the principal solutions at ±∞ define singular ergodic measures, and determine an invariant region in the Lagrange bundle which concentrates the essential dynamical information. We apply this theory to the study of the n-dimensional Schrödinger equation at the first point of the spectrum.

In this paper we prove, on one hand, extrapolation from infinity for the one-sided classes , and on the other hand, the BMO-boundedness for one-sided singular integrals. We also provide several applications of our results.

We study the stability of periodic solutions of the scalar delay differential equation where f(t) is a periodic forcing term and δ,p∈R. We study stability in the first approximation showing that the non-smooth equation (*) can be linearized along some ‘non-singular’ periodic solutions. Then the corresponding variational equation together with the Krasnosel'skij index are used to prove the existence of multiple periodic solutions to (*). Finally, we apply a generalization of Halanay's inequality to establish conditions for global attractivity in equations with maxima.

The inverse problem of the determination of boundary defects in a planar conductor by a finite number of electrostatic measurements on the boundary is considered. Uniqueness results and stability estimates are proved under essentially minimal regularity assumptions on the data. Finally, Lipschitz estimates for the determination of surface linear cracks are developed.

Using fixed point theorems we discuss continuous solutions of Γ-equivariance for a polynomial-like iterative equation on the real line, where Γ is a closed subgroup of the general linear group GL(R). Our main results guarantee the existence of solutions with certain kinds of symmetry. We show that, under restrictive hypotheses, similar results can be proved in a higher-dimensional case, where the symmetry group is a topologically finitely generated subgroup of the group generated by rotations and dilations in N-dimensional Euclidean space.

The important theory of invariant regions in reaction-diffusion equations has only restricted applications because of its strict requirements on both the reaction terms and the regions. The concept of weakly invariant regions was introduced by us to admit wider reaction-diffusion systems. In this paper we first extend the L∞ estimate technique of semilinear parabolic equations of Rothe to the more general case with convection terms, and then propose more precise criteria for the bounded weakly invariant regions. We illustrate, by three model examples, that they are very convenient for establishing the global existence of solutions for reaction-diffusion systems, especially those from ecology and chemical processes.