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Stability of resonances under singular perturbations

  • Author(s): Drouot, Alexis
  • Advisor(s): Zworski, Maciej R
  • et al.
Abstract

We investigate the stability of complex numbers called resonances in certain open chaotic systems. In the context of waves scattered by a potential, scattering resonances are complex numbers that quantize exponential decay rates of the local energy. In the context of

hyperbolic dynamical systems, Pollicott--Ruelle resonances quantize exponential decay rates of statistical correlations. We will show that resonances occurring in these two situations are stochastically stable. This theoretical result supports the possibility of observing resonances in experimental physics.

This dissertation consists of two independent chapters, based on two papers of the author.

Chapter 1 focuses on scattering resonances. We give a simple model for waves propagating through a localized disordered crystal with small typical scale of heterogeneity. Roughly speaking, our results show that resonances (hence propagating waves) are only weakly perturbed by the crystal. This chapter is organized as follows:

- We describe our model and state our theorems in S1.1. We relate these results to previous study in the idealized case of deterministic highly oscillatory perturbations, whose understanding is very important for the proofs.

- In S1.2, we give an overview of the theory of scattering resonances. We define them using the stationary Schrodinger resolvent and we relate them to local energy decay of waves. This makes Chapter 1 essentially self-contained.

- S1.3 studies the general perturbation theory of scattering resonances. We work with a parameter that typically quantifies the degree of oscillations of a random potential. When this parameter is small enough, we characterize locally scattering resonances as

the zero set of a holomorphic function.

- In S1.4, we give a version of the Hanson--Wright inequality. This is a large deviation estimate for a quadratic form evaluated at random vectors with many entries. It is crucial in the rest of the proof. We also derive a modified version of the central limit

theorem, based on Lindeberg's theorem.

- S1.5 is the core of the chapter. The tools of S1.4 show that the general theory developed in x1.3 applies to random highly oscillatory potentials. In particular, we can write their resonances as the zeroes of a random holomorphic function. We show that this random

function can (roughly speaking) be written as a rescaled Gaussian -- modulo negligeable terms. The variance of the Gaussian arises from large deviation effects, while the average of the Gaussian comes from constructive interference among oscillatory terms

(an effect that was thoroughly studied in the context of deterministic highly oscillatory potentials).

- In S1.6, we prove the stochastic stability theorems. These are valid with high probability. We conclude by exhibiting an example (that appears with small probability) where the conclusions of the theorems do not hold.

Chapter 2 is a study of hyperbolic dynamical systems perturbed by a white noise. These form stochastic processes called kinetic Brownian motion. We use a recent microlocal approach to define Pollicott--Ruelle resonances via kinetic Brownian motion. This is a form of stochastic stability. The presentation is as follows:

- In S2.1, we state our stochastic stability result: the eigenvalues of the generator of the kinetic Brownian motion approach Pollicott--Ruelle resonances in the small white noise regime. We give an overview of recent results for random perturbations of the geodesic equation, and we describe related facts about the hypoelliptic Laplacian of Bismut.

- In S2.2, we recall a modern perspective on Pollicott--Ruelle resonances. It consists of seeing them as spectral quantities rather than dynamical one. This requires the semiclassical construction of anisotropic Sobolev spaces, which improve regularity in the contracting direction of the flow and lower it in the expanding direction. We give an axiomatic introduction to microlocal and semiclassical analysis.

- S2.3 presents the kinetic Brownian motion as a perturbation of the geodesic equation. We also describe its lift to the orthonormal frame bundle (a step required in S2.4).

- In S2.4, we prove a subelliptic estimate for the generator of the kinetic Brownian motion. An informal probabilistic statement is as follows: the white noise perturbation is not too large compared to the kinetic Brownian motion itself.

- S2.5 reformulates the subelliptic estimate in the context of anisotropic Sobolev spaces that are needed to define Pollicott--Ruelle resonances.

- In S2.6, we show that the subelliptic estimate allows to control high frequencies of the white noise perturbation; and that the low frequencies of the white noise perturbations can be treated as an absorbing potential. This enables us to prove the main theorem.

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