Asymptotic probability techniques in monochromatic waves and fuid mechanics

Zusammenfassung

This thesis addresses different question concerning probability, partial differential equations and some aspects of economic theory. We try to answer whether some events in these fields are “typical”, how different probability settings modify their likelihood and how some probability techniques can give us information about expected values of important magnitudes or help us to construct deterministic realizations. The thesis is divided into two parts. In the first part we study monochromatic random waves. First, in Chapter 2 we study monochromatic random waves on the Euclidean space defined by Gaussian variables whose variances tend to zero sufficiently fast. This has the effect that the Fourier transform of the monochromatic wave is an absolutely continuous measure on the sphere with a suitably smooth density, which connects the problem with the scattering regime of monochromatic waves. In this setting, we compute the asymptotic distribution of the nodal components of random monochromatic waves showing that the behavior changes dramatically with respect to the standard theory. Second, in Chapter 2 we consider Gaussian random monochromatic waves u on the plane depending on a real parameter s that is directly related to the regularity of its Fourier transform. Specifically, the Fourier transform of u is f\,d\sigma, where d\sigma is the Hausdorff measure on the unit circle and the density f is a function on the circle that, roughly speaking, has exactly s-\frac12 derivatives in L^2 almost surely. When s=0, one recovers the standard setting for random waves with a translation-invariant covariance-kernel. The main thrust of this chapter is to explore the connection between the regularity parameter s and the asymptotic behavior of the number N(\nabla u,R) of critical points that are contained in the disk of radius R\gg1. Finally, in Chapter 4 we construct deterministic solutions to the Helmholtz equation in Rm which behave accordingly to the Random Wave Model. We then find the number of their nodal domains, their nodal volume (Yau’s conjecture) and the topologies and nesting trees of their nodal set in growing balls around the origin. The proof of the pseudo-random behavior of the functions under consideration hinges on a de-randomization technique pioneered by Bourgain and proceeds via computing their L^p-norms. The study of their nodal set relies on its stability properties and on the evaluation of their doubling index, in an average sense. In the second part of this thesis we study the probability techniques applied to two different fields: fluid mechanics and economic theory. First, in Chapter 5 we show that, with probability 1, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold’s 1965 speculation that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros, and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov–Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including KAM theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on R^3 and of high-frequency Beltrami fields on the 3-torus. The second chapter in this part deals with social choice theory, a branch of theoretical economics. The Condorcet Jury Theorem or the Miracle of Aggregation are frequently invoked to ensure the competence of some aggregate decision-making processes. In Chapter 6 we explore an estimation of the prior probability of the thesis predicted by the theorem (if there are enough voters, majority rule is a competent decision procedure). We use tools from measure theory to conclude that, prima facie, it will fail almost surely. To update this prior either more evidence in favor of competence would be needed or a modification of the decision rule. Following the latter, we investigate how to obtain an almost sure competent information aggregation mechanism for almost any evidence on voter competence (including the less favorable ones). To do so, we substitute simple majority rule by weighted majority rule based on some weights correlated with epistemic rationality such that every voter is guaranteed a minimal weight equal to one.