Mitia Duerinckx

Analyticity of Homogenized Coefficients Under Bernoulli Perturbations and the Clausius–Mossotti Formulas

This paper is concerned with the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a Bernoulli perturbation. Introducing a new family of energy estimates that combine probability and physical spaces, we prove the analyticity of the perturbed homogenized coefficients with respect to the Bernoulli parameter. Our approach holds under the minimal assumptions of stationarity and ergodicity, both in the scalar and vector cases, and gives analytical formulas for each derivative that essentially coincide with the so-called cluster expansion used by physicists. In particular, the first term yields the celebrated (electric and elastic) Clausius–Mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium. This work constitutes the first general proof of these formulas in the case of random inclusions.

Maximum likelihood characterization of distributions

A famous characterization theorem due to C. F. Gauss states that the maximum likelihood estimator (MLE) of the parameter in a lo- cation family is the sample mean for all samples of all sample sizes if and only if the family is Gaussian. There exist many extensions of this result in diverse directions, most of them focussing on location and scale families. In this paper we propose a unified treatment of this literature by providing general MLE characterization theorems for one-parameter group families (with particular attention on location and scale parameters). In doing so we provide tools for determining whether or not a given such family is MLE-characterizable, and, in case it is, we define the fundamental concept of minimal necessary sample size at which a given characterization holds. Many of the cornerstone references on this topic are retrieved and discussed in the light of our findings, and several new characterization theorems are provided. Of particular interest is that one part of our work, namely the introduction of so-called equivalence classes for MLE characterizations, is a modernized version of Daniel Bernoulli’s viewpoint on maximum likelihood estimation.

Stochastic noise reduction upon complexification: Positively correlated birth-death type systems.

Cell systems consist of a huge number of various molecules that display specific patterns of interactions, which have a determining influence on the cell׳s functioning. In general, such complexity is seen to increase with the complexity of the organism, with a concomitant increase of the accuracy and specificity of the cellular processes. The question thus arises how the complexification of systems - modeled here by simple interacting birth-death type processes - can lead to a reduction of the noise - described by the variance of the number of molecules. To gain understanding of this issue, we investigated the difference between a single system containing molecules that are produced and degraded, and the same system - with the same average number of molecules - connected to a buffer. We modeled these systems using Itō stochastic differential equations in discrete time, as they allow straightforward analytical developments. In general, when the molecules in the system and the buffer are positively correlated, the variance on the number of molecules in the system is found to decrease compared to the equivalent system without a buffer. Only buffers that are too noisy themselves tend to increase the noise in the main system. We tested this result on two model cases, in which the system and the buffer contain proteins in their active and inactive state, or protein monomers and homodimers. We found that in the second test case, where the interconversion terms are non-linear in the number of molecules, the noise reduction is much more pronounced; it reaches up to 20% reduction of the Fano factor with the parameter values tested in numerical simulations on an unperturbed birth-death model. We extended our analysis to two arbitrary interconnected systems, and found that the sum of the noise levels in the two systems generally decreases upon interconnection if the molecules they contain are positively correlated.

Mean-field limits for some Riesz interaction gradient flows

This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to infinity. Based on a modulated energy method, using regularity and stability properties of the limiting equation, as inspired by the work of Serfaty in the context of the Ginzburg-Landau vortices, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open.

Functions with Constant Mean on Similar Countable Subsets of ℝ 2

Abstract We prove the following generalization of a problem proposed at the 70th William Lowell Putnam Mathematical Competition. Given a nonempty finite set E of n points in ℝ2 and a function f : ℝ2 → ℝd such that the arithmetic mean of the values of f at the n points of every image of E by a direct similarity is equal to a constant, then f is constant on ℝ2. This result is extended to nonempty countable sets, and its validity is discussed in a more general context.

Approximate spectral theory and wave propagation in quasi-periodic media

In this article we make specific in the quasi-periodic setting the general Floquet-Bloch theory we have introduced for stationary ergodic operators together with the associated approximate spectral theory. As an application we consider the long-time behavior of the Schrödinger flow with a quasi-periodic potential (in the regime of small intensity of the discorder), and the long-time behavior of the wave equation with quasi-periodic coefficients (in the homogenization regime).