The internal probability seminar is a regular meeting of the three teams of probability where one of its members presents one of its recent work / work in progress or something "everyone should know about". On the speaker it is very easy to prepare while it helps everybody to get a better understanding of what everyone is doing. For young researchers it is also the occasion to get feedback on your work and presentation skills in a friendly environment.

J. Thompson , , Seminar room MNO (5th floor)

Quantitative estimates for the derivative via Bismut formula

For a $C^2$ function $u$ and an elliptic operator $L$, we prove a quantitative estimate for the derivative $\mathrm{d}u$ in terms of local bounds on $u$ and $L u$. An integral version of this estimate is then used to derive a condition for the zero-mean value property of $\Delta u$. An extension to differential forms is also given. Our approach is basic, using only Itô’s formula and Bismut’s formula. This is joint work with Li-Juan Cheng and Anton Thalmaier. A preprint is available (arXiv: 1707.07121).

G. Dierickx , , 5A (MNO)

On a general Darling - Erdos theorem

We present a general version of the Darling-Erdos theorem for mean zero independent and identically distributed random vectors having a finite second moment. Furthermore, we will also discuss an integral test for upper/lower class functions, which are refinements of the multivariate Hartman-Wintner law of the iterated logarithm, for sums of such independent and identically distributed random vectors. Our main tool is a new strong approximation for independent random vectors.

This talk is based on a joint work with the speaker’s PhD advisor, prof. Dr. U. Einmahl (Vrije Universiteit Brussel). A preprint is available (ArXiv: 1608.04549).

We study the nodal intersections number of random Gaussian toral Laplace eigenfunctions (“arithmetic random waves”) against a fixed smooth reference curve. The expected intersection number is proportional to the square root of the eigenvalue times the length of curve, independent of its geometry. The asymptotic behaviour of the variance was addressed by Rudnick-Wigman; they found a precise asymptotic law for “generic” curves with nowhere vanishing curvature, depending on both its geometry and the angular distribution of lattice points lying on circles corresponding to the Laplace eigenvalue. They also discovered that there exist peculiar “static” curves, with variance of smaller order of magnitude, though did not prescribe what the true asymptotic behaviour is in this case. In this talk we study the finer aspects of the limit distribution of the nodal intersections number. For “generic” curves we prove the Central Limit Theorem (at least, for “most” of the energies). For the aforementioned static curves we establish a non-Gaussian limit theorem for the distribution of nodal intersections, and on the way find the true asymptotic behaviour of their fluctuations, under the well-separateness assumption on the corresponding lattice points, satisfied by most of the eigenvalues.

This talk is based on a joint work with Igor Wigman (King’s College London).

J. Thompson , , Seminar room MNO (6th floor)

Derivative and divergence formulae for diffusion semigroups

Suppose $V$ is a vector field on a smooth manifold and $P_{t}$ a semigroup generated by an elliptic operator. Then Bismut’s formula provides a probabilistic formula for the derivative $V(P_{t}f)$, not involving the derivatives of $f$. Bismut proved it using techniques from Malliavin calculus (MathSciNet: MR0755001), but a more elementary approach based on martingales was later developed by Elworthy and Li (MathSciNet: MR1297021) and Thalmaier (MathSciNet: MR1488139). In this talk, we prove analogous formulae for the derivative $P_{t}(V(f))$. For non-symmetric generators, such formulae correspond to the derivative of the heat kernel in the forward variable. As an application, our formulae can be used to derive various shift-Harnack inequalities.

In the paper Stein’s method on Wiener chaos Nourdin and Peccati (MathSciNet: MR2520122) combined the Malliavin calculus and Stein’s method of normal approximation to associate a rate of convergence to the celebrated fourth moment theorem of Nualart and Peccati. Their analysis, known as the Malliavin-Stein method nowadays, has found many applications towards stochastic geometry, statistical physics and zeros of random polynomials, to name a few. In this article, we further explore the relation between these two fields of mathematics. In particular, we construct a sequence of exchangeable pairs for multiple Wiener-Ito integrals and we compute its asymptotic linear and quadratic regressions. By combining our findings with E. Meckes’ infinitesimal version of exchangeable pairs, we can give another proof of the fourth moment theorem. Finally, we extend our result to the multidimensional case.

We prove an exact fourth moment bound for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result – that has been elusive for several years – shows that the so-called ‘‘fourth moment phenomenon’’, first discovered by Nualart and Peccati (MathSciNet: MR2118863) in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein’s method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a non-diffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux (MathSciNet: MR3050508) and Azmoodeh, Campese and Poly (MathSciNet: MR3150163).

Normal approximation via non-diffusive Markov generators

In a recent paper, M. Ledoux has used spectral theory to analyse the so-called fourth moment phenomenon by Nualart and Peccati as well as related asymptotic conditions for functionals of the stationary distribution of some Markov diffusion generator. His theory has then been extended and ramified in papers by Azmoodeh, Campese and Poly and by Campese, Nourdin, Peccati and Poly, respectively. In this talk, I will outline how one can complement the above mentioned theory to functionals of probability measures which are not necessarily the stationary distribution of a Markov diffusion. In particular, given a measure $\mu$ on a general space, I show how to construct a (non-diffusive) Markov generator with invariant measure $\mu$. Furthermore, I will present bounds on the normal approximation of such functionals. These bounds include, but are not restricted to, eigenfunctions of the involved Markov generator.

The classical Fourth Moment Theorem says that for a normalized sequence of multiple Wiener-Itô integrals (for a simple example take infinite linear combinations of products of independent Gaussians), convergence of just the fourth moment suffices to ensure convergence in distribution towards a standard Gaussian random variable. Since its discovery, several proofs and extensions of this result have been found, all of them heavily exploiting the rich structure of multiple integrals. In an exciting new development, it turned out that such Fourth Moment Theorems hold in much greater generality, namely for generic eigenfunctions of Markov diffusion generators with a certain chaotic property and target laws coming from almost the full Pearson class (examples being the Gaussian, Gamma and Beta distribution). We will present an overview of this new approach, which is accessible also to non-probabilists.

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit $d$-sphere ($d \geq 2$). We investigate the distribution of their defect i.e., the difference between the measure of positive and negative regions. Marinucci and Wigman studied the two-dimensional case giving the asymptotic variance (Marinucci and Wigman 2011) and a Central Limit Theorem (Marinucci and Wigman 2014), both in the high-energy limit. Our main results concern asymptotics for the defect variance and quantitative CLTs in Wasserstein distance, in any dimension. The proofs are based on Wiener-Itô chaos expansions for the defect, a careful use of asymptotic results for all order moments of Gegenbauer polynomials and Stein-Malliavin approximation techniques by Nourdin and Peccati. Our argument requires some novel technical results of independent interest that involve integrals of the product of three hyperspherical harmonics.

A preprint version is available (arXiv: 1605.03491).

A. Vidotto ,

Improved second order Poincaré inequality for functionals of Gaussian fields and applications

In this talk I present a new second order Poincaré inequality for functionals of a Gaussian field and, among many other possible applications, I show how to apply it to the trace of a power $p_n=o(n^{1/3})$ of a Wigner matrix, in order to give a quantitative CLT for the aforementioned, and to a non-linear functional of a Gaussian field.

This is a joint work in progress with my supervisor, Professor Giovanni Peccati.

Using a refinement of the concentration function, we show that the “eigenvalues” of the metric heat operator concentrate the space exponentially. The result has been known for a long time for the first eigenvalue (that is for operator satisfying a Poincaré inequality that is for CD(0, N)) but this is the first time we are able to take into account higher order eigenvalues. As a corollary, we obtain a bound on the growth of the eigenvalues which might possibly be used to bound the variance of the carré du champ without using the techniques of ACP (I have not thought about that in detail but it seems doable).

In the first place I will try to motivate why people are interested in random polynomials and its roots and I will do a brief review about the classical known results about the number of roots of random polynomials. Then, I will tell you about my recent efforts in the case of trigonometric polynomials of the form $\sum a_n \cos(n t)$ and on systems of (algebraic) polynomials.

C. Döbler ,

Normal approximation of generalized multilinear forms

This talk deals with a quantitative version of a Theorem by de Jong from 1990 which states that a homogeneous sum of a certain class of uncorrelated functionals of some independent data converges to a normal limit, if its fourth moment converges to 3 and if, additionally, some negligibility condition which extends the classical Feller condition holds true. This theorem generalizes known results about asymptotic normality of multilinear forms in independent random variables which have been the object of intense research in the past.

The proof relies on a combination of a suitable version of Stein’s method and the probabilistic structure behind the given random variables. This is joint work in progress with Giovanni Peccati.

Length of zero-set of random eigenfunctions on the torus

Arithmetic random waves are the Gaussian Laplace eigenfunctions on the two-dimensional torus. It was conjectured by Bogomolny and Schmit that the nodal domains of random (and chaotic) wavefunctions in the semi-classical limit are adequately described by the critical percolation theory. The connection with SLE (which describes the limit of some percolation problem and was supposed to approximate nodal lines of these random wavefunctions) gave additional support to this conjecture.

In this talk we show that the nodal length of random eigenfunctions on the torus converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener-Ito chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes - this is closely related to the so-called ‘‘obscure’’ Berry’s cancellation phenomenon). The rest of the argument relies on the precise analysis of the fourth order chaotic component.

This talk is based on the paper Non-Universality of Nodal Length Distribution for Arithmetic Random Waves (arXiv: 1508.00353), joint work with Domenico Marinucci (Università di Roma Tor Vergata), Giovanni Peccati (Université du Luxembourg) and Igor Wigman (King’s College London).

A. Vidotto ,

Markov generators in the no-diffusive case

Is it possible to adapt the Ledoux (MathSciNet: MR3050508) and Azmoodeh, Campese and Poly (MathSciNet: MR3150163) spectral methods to prove Fourth Moment Theorems in a framework that is not diffusive? Starting from Rademacher and Poisson settings, we try to give a definition of a quasi-diffusive generator. The main difficulty os to compute exact bounds on the error term in the ‘chain rule’ for the generator.

Classification of diffusive generators associated to orthogonal polynomial on the real line

A diagonalizable symmetric Markov generator gives rise to two orthonormal basis: one obtained from the Gram-Schmidt ortho-normalization procedure and the other given by the eigenfunctions. In the case the two coincide, the generator satisfies the property of chaos-eigenfunctions as defined in the previous seminar. On the real line an article of O. Mazet (Numdam: SPS_1997__31__40_0) shows that the only generators with orthogonal eigenfunctions are: Gaussian (Hermite polynomials); Exponential (Laguerre polynomials) and Beta (Jacobi). The case of the real plane is exhausted in a recent paper by D. Bakry, S. Orekov & M. Zani (arXiv: 1309.5632v2).

We will investigate two open questions:

Is there chaos-eigenfunctions outside of the orthogonal ones?

What if the dimension is greater than 2?

A. Vidotto ,

The Nualart-Peccati theorem for chaotic Markov diffusion generators

In 2009, Nourdin and Peccati used Stein-Malliavin approach to prove quantitative versions of so-called Fourth Moment Theorems (FMT) (MathSciNet: MR2520122). Their results state that, under certain circumstances, convergence of sequence of Gaussian functionnals towards a target distribution (for example Gaussian) is controlled by the convergence of the first four moments of the approximating sequence.

Recently, Ledoux (MathSciNet: MR3050508) and Azmoodeh, Campese and Poly (MathSciNet: MR3150163) provided a purely spectral point of view on the FMT. Indeed they show analogues of these results continue to hold in the more abstract setting of Markov diffusion generators. Moreover, in this more general setting, we can also cover other chaos structures and target distributions. As our own application we give a novel quantitative Fourth Moment Theorem for convergence towards a Beta distribution.

G. Zheng ,

Normal approximation and almost sure central limit theorem in the Rademarcher setting

In this talk, we study the central limit theorem and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we derive an approximate chain rule that refines the one discovered by I. Nourdin, G. Peccati and G. Reinert (MathSciNet: MR2735379) and then we deduce the bound on Wasserstein distance for normal approximation using the (discrete) Malliavin-Stein approach. We are also able to give the almost sure central limit theorem for a sequence inside a fixed Rademacher chaos using the Ibragimov-Lifshits criterion.

The Bakry-Emery theory of the Ricci curvature in the Riemannian setting

In classical Riemannian geometry, the Ricci curvature controls how the volume grows along parallel transport by geodesics. Informations on the Ricci curvature imply many infamous functionals inequalities (Levy-Gromov, Poincaré and so on). On a weighted Riemannian manifold, the hypothesis of Ricci bounded below is in fact equivalent to the well-known Böchner formula for fields deriving from a potential. Using this equality D. Bakry & M. Emery introduced the so-called curvature-dimension inequality characterizing bounds from below for the Ricci curvature (MathSciNet: MR889476).

This inequality is in fact purely algebraic and can be expressed for an abstract Markov generator. Roughly speaking, it measures the sub-commutation property of the generator with its so-called carré du champ. Generators satisfying the curvature dimension inequality, then all the inequalities proved in he Riemannian case stay valid in that abstract setting.