We review properties of the Beurling correlation function related to differentiability and functional equations. The relevance of this function is due to the fact that some properties of the Riemann zeta function can be expressed interms of it. / Se repasan algunas propiedades de la función de correlación de Beurling, que sirven para expresar ciertas propiedades de la función zeta de Riemann.

We consider the incompressible 2D Navier-Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and extremely degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a uniquely ergodic and exponentially mixing quasi-periodic invariant measure. The result is true for any value of the viscosity ν > 0. By utilizing this quasi-periodic invariant measure, we show the strong law of large numbers and central limit theorem for the continuous time inhomogeneous solution processes. Estimates of the corresponding rate of convergence are also obtained, which is the same as in the time homogeneous case for the strong law of large numbers, while the convergence rate in the central limit theorem depends on the Diophantine approximation property on the quasi-periodic frequency and the mixing rate of the quasi-periodic invariant measure. We also prove the existence of a stable quasi-periodic solution in the laminar case (when the viscosity is large). The scheme of analyzing the statistical behavior of the time inhomogeneous solution process by the quasi-periodic invariant measure could be extended to other inhomogeneous Markov processes.

Made available in DSpace on 2016-06-02T20:28:25Z (GMT). No. of bitstreams: 1 3207.pdf: 939340 bytes, checksum: b708a600566bb7e50aa91c249a665893 (MD5) Previous issue date: 2010-06-21 / In this work, we will see that if the transpose operator of a smooth real vector field L defined on the N-dimensional torus, regarded as a linear differential operator with coefficients in C1(TN), is globally hypoelliptic, then there exists a vector field with constant coefficients L0 such that L and L0 are C1-conjugated, with such constants satisfying a condition called Diofantina (*). We will also show the converse of this fact, that is, if there is a coordinate system such that in this new system L has constant coefficients with such constant satisfying the Diophantine condition (*) then its transpose L* is globally hypoelliptic. We will see that the Diophantine condition implies that the flow generated by the field, regarded as a Dynamical system is minimal. / Neste trabalho, veremos que se o operador transposto de um campo vetorial real suave L definido no toro N-dimensional, visto como um operador diferencial linear com coeficientes em C1(TN), for globalmente hipoelíptico, então existe um campo vetorial com coeficientes constantes L0 tal que L e L0 são C1- conjugados, com tais constantes satisfazendo uma condição chamada de Diofantina (*). Mostraremos também a recíproca deste fato, isto é, se existir um sistema de coordenadas tal que, neste novo sitema L possui coeficientes constantes com tais constantes satisfazendo a condição Diofantina (*) então, seu transposto L* é globalmente hipoelíptico. Veremos que a condição Diofantina implica que, os fluxos gerados pelo campo, vistos como um sistema dinânico, são minimais.