Academic Year 2019-2020

Francesco Grotto (SNS, Pisa), Invariant Measures for 2d Incompressible Fluid Dynamics Models

Abstract: The Hamiltonian structure of 2d Euler’s equations and its variants allows the formal derivation of invariant measures from conservation laws. Gaussian and Poissonian invariant measures thus obtained pose nontrivial questions, concerning the singular dynamics they induce and the relations between their very different natures. We will give an overview of classical and more recent results on the topic.

Fei Pu (University of Luxembourg), Spatial limit theorems for stochastic heat equation via Poincare inequality

Abstract: In this talk, I will present spatial limit theorems for the solution to stochastic heat equations, which include ergodicity, central limit theorem and Poisson limit theorem. The tool to study these properties is Malliavin calculus, in particular, the Poincare inequality.

Arturo Jaramillo (University of Luxembourg), Quantitative Erdös-Kac theorem for additive functions, a self-contained probabilistic approach

Abstract: The talk will have as starting point the classical Erdös-Kac theorem, a result of great importance in probabilistic number theory, which states that the fluctuations of the standardized number of distinct primes of a uniformly chosen number between one and n, are asymptotically Gaussian. Naturally, after the publication of this result, a quantitative version of it was explored by many authors. LeVeque conjectured that the optimal rate of convergence (in the topology of Kolmogorov distance) was of the order . This was subsequently proved by Turan and Rényi by means of a very clever manipulation of the associated characteristic function. Unfortunately, up to this day, all of the approaches for solving LeVeque’s conjecture(in its full generality) rely on highly non-trivial complex analysis tools, whereas the purely probabilistic tools have only been successfully applied for obtaining non-optimal assessments of the aforementioned rate.
In this talk, we present a new perspective to estimate the distance to a Gaussian distribution (with respect to Kolmogorov and Wasserstein metric), for general additive functions applied to a uniformly chosen number between one and n . Our approach is probabilistic and does not rely on prior knowledge of the underlying characteristic function. Our main result is an optimal Berry-Esseen type bound in the Kolmogorov distance and the Wasserstein distance. In the special case where the additive function is taken to be the prime factors counting function (with and without accounting of multiplicities), we also show Poisson approximations with optimal error bounds in the total variational distance.

Cecile Durot (Université Paris Nanterre), Divide and Conquer methods in monotone regression

Abstract: The divide and conquer principle in studied in the isotonic regression problem, where rates of convergence are slower than the square-root of the sample size, and limit distributions are non-Gaussian. For a fixed model, the pooled estimator obtained by averaging non-standard estimates across mutually exclusive subsamples, outperforms the non-standard monotonicity-constrained (global) estimator based on the entire sample in the sense of point wise estimation. However, this gain in efficiency under a fixed model comes at a price: the pooled estimator’s performance, in a uniform sense over a class of models worsens as the number of subsamples increases, leading to a version of the super-efficiency phenomenon. Then, we build a corrected pooled estimator that does not suffer from the super-efficiency phenomenon and allows for some heterogeneity in data. The new estimator essentially reverses the steps involved in constructing the above pooled estimator: we first smooth (by local averaging) on each subsample, and then isotonize the pooled smoothed data. Joint work with Moulinath Banerjee and Bodhisattva Sen.

Vlad Margarint (NYU Shanghai), Backward Loewner Differential Equation as a Singular Rough Differential Equation, the welding homeomorphism and new structural information about the SLE traces

Abstract: In this talk, I will give an overview of the Schramm-Loewner Evolutions (SLE) theory and present new results on this theory based on the analysis of a Singular Differential Equation that appears naturally in this context. This equation appears when extending the conformal maps to the boundary and can be thought of as a singular Rough Differential Equation (RDE), as in Rough Path Theory. In the study of RDEs, questions such as continuity of the solutions, the uniqueness/non-uniqueness of solutions depending on the behavior of parameters of the equation, appear naturally. We adapt these type of questions to the study of the backward Loewner differential equation in the upper half-plane, and the conformal welding homeomorphism. This view will allow us to obtain some new structural and geometric information about the SLE traces in the regime where they have double points.
This first part is a joint work with Dmitry Belyaev and Terry Lyons.
In the second part, I plan to cover the main ideas of an independent project that uses ideas from Quasi-Sure Stochastic Analysis through Aggregation in order to study SLE theory quasi-surely. This quasi-sure study will allow us to overcome some of the difficulties with the previous analysis that I will emphasize throughout the talk.

Benjamin Arras (Université de Lille), From generalized Mehler semigroups to stability results for Poincaré-type inequalities

Abstract: In this talk, I will present some recent results around Stein’s method for multivariate stable laws and generalized Mehler semigroup. This is based on joint works with Christian Houdré (GaTech).

Guenter Last (Karlsruhe Institute of Technology), Unbiased embedding of excursions into Brownian motion

Abstract: : In this talk, we discuss an embedding problem for a two-sided Brownian motion. We consider an excursion event with positive and finite Itô-measure and construct a stopping time such the two-sided Brownian motion centered around splits into three independent pieces: a time reflected Brownian motion on , an excursion distributed according to a conditional Itô law (given ) and a Brownian motion starting after this excursion. The proof relies on Palm theory for random measures and on excursion theory. Therefore we shall begin with a short review of some fundamental facts on invariant balancing transports of random measures. This talk is based on joint work with Wenpin Tang and Hermann Thorisson.

D. Yogeshwaran (ISI, Bangalore), Random minimal spanning acycles.

Abstract: It is well-known that extremal edge-weights on a minimal spanning tree, nearest-neighbour distances and connectivity threshold are inter-related for randomly weighted graphs. In this talk, we shall look at generalization of this result to randomly weighted simplicial complexes. The first part of the talk shall be about defining spanning acycles and establishing it to be a natural topological generalisation of spanning trees. We shall give the Kruskal’s algorithm to generate minimal spanning acycles. As a consequence of the Kruskal’s algorithm, we shall obtain a connection between minimal spanning acycles and persistent homology. We shall explore applications of these results in the context of random d-complexes and in particular, paying attention to extremal face-weights of the minimal spanning acycles on a complete d-complex with i.i.d. face weights. This is a joint work with Primoz Skraba and Gugan Thoppe. Time permitting, I will sketch some on-going work with Primoz Skraba on Euclidean minimal spanning acycles.