# «Book of Abstracts Ruhr-Universität Bochum March 1 – 4, 2016 Plenary Talk 1 Martin Hairer1 01.03.2016 1 U. of Warwick, Coventry P1 | 10:00 On ...»

12th German Probability and Statistics Days

Stochastiktage Bochum [March 1 – 4, 2016]

Book of Abstracts

Ruhr-Universität Bochum

March 1 – 4, 2016

Plenary Talk 1

Martin Hairer1 01.03.2016

1 U. of Warwick, Coventry P1 | 10:00

On random rubber bands Audimaxx

A rubber band constrained to remain on a manifold evolves by trying to shorten its length,

eventually settling on some minimal closed geodesic, or collapsing entirely. It is natural to try to consider a noisy version of such a model where each segment of the band gets pulled in random directions. Trying to build such a model turns out to be surprisingly dicult and generates a number of nice geometric insights, as well as some beautiful algebraic and analytical objects.

Plenary Talk 2 Laszlo Erdös1 01.03.2016 1 Institute of Science and Technology Austria, Klosterneuburg P2 | 17:30 Local laws for eigenvalues of random matrices Audimaxx The empirical eigenvalue distribution of random matrix ensembles become deterministic as the size N of the matrix tends to innity. Individual eigenvalues uctuate but typically on a very small scale, comparable with the eigenvalue spacing. Hence the limiting density correctly describes the eigenvalues not only on large scales but also locally, on the smallest possible microscopic scale. A prominent random matrix ensemble is the sum of two hermitian matrices A and B conjugated with a Haar unitary U. We show that the eigenvalue density of A + U BU ∗ is given by the free convolution down to the optimal scale. Recent local laws for some other ensembles will also be discussed.

Planery Talk 3 Sandrine Dudoit1 02.03.2016 1 University of California, Berkeley, Biostatistics and Statistics, Berkeley P3 | 09:00 Identication of Novel Cell Types in the Brain Using Single-Cell Transcriptome Audimaxx Sequencing Single-cell transcriptome sequencing (scRNA-Seq), which combines high-throughput singlecell extraction and sequencing capabilities, enables the transcriptome of large numbers of individual cells to be assayed eciently. Proling of gene expression at the single-cell level for a large sample of cells is crucial for addressing many biologically relevant questions, such as, the investigation of rare cell types or primary cells (e.g., early development, where each of a small number of cells may have a distinct function) and the examination of subpopulations of cells from a larger heterogeneous population (e.g., classifying cells in brain tissues).

I will discuss some of the statistical analysis issues that have arisen in the context of a collaboration funded by the Brain Research through Advancing Innovative Neurotechnologies (BRAIN) Initiative, with the aim of classifying neuronal cells in the mouse somatosensory cortex.

**These issues, ranging from so-called low-level to high-level analyses, include:**

experimental design, exploratory data analysis (EDA) of scRNA-Seq reads, quality assessment/control (QA/QC), normalization to account for nuisance technical eects, cluster analysis to identify novel cell types, and dierential expression analysis to derive gene expression signatures for the cell types.

Planery Talk 4

We review several recent results for portfolio optimization under proportional transaction costs, such as a Tobin tax. Special emphasis will be put on nancial models based on fractional Brownian motion.

Planery Talk 5

In 1964 Alan James gave a remarkable classication of many of the eigenvalue distribution problems of multivariate statistics. We show how the classication readily adapts to contemporary "spiked models" high dimensional data with low rank structure. In particular we look at asymptotic approximations to the likelihood ratios when the number of variables grows proportionately with sample size or degrees of freedom. High dimensions bring phase transition phenomena, with quite dierent limit behavior for small and large spike strengths.

James' framework allows us to develop these in a unied way across problems such as signal detection, matrix denoising, regression and canonical correlations.

(Joint work with Alexei Onatski and Prathapa Dharmawansa).

Section 1: Stochastic Analysis

It has been long expected, and in various cases proved, that a stochastic heat equation with a multiplicative noise term can have solutions that develop very tall peaks, interspersed by vast space-time low-acitivity regions wherein the solution is quite small. We will describe some of the recent eorts to develop the mathematics behind such problems.

Time-permitting, recent connections to the macroscopic multifractal structure of the tall peaks of the solution will also be presented.

This talk is based on collaborative works with K. Kim, C. Mueller, S.-Y. Shiu, and Y. Xiao.

Using Malliavin calculus and Stein method techniques, we establish new lower bounds for the normal approximation in the Wasserstein distance of random variables that are functionals of a random Poisson measure. Our results generalize previous ndings by Nourdin and Peccati (2009, 2015), involving random variables living on a Gaussian space. We apply our theoretical result to obtain the optimal Berry-Esseen bounds for edge counting in random geometric graphs.

The talk is based on the following recent work.

Ehsan Azmoodeh, Giovanni Peccati (2015), Optimal Berry-Esseen bounds on the Poisson space. http://arxiv.org/abs/1505.02578.

Investigating the Lévy-Itô decomposition of Lévy processes with values in a locally convex Suslin space E, one encounters diculties like a non-metric structure and uncountable

**neighbourhood bases. In particular, the following questions are essential:**

Can one prove a.s. convergence of a series of independent elements dening a comQ1) pensated Poisson integral in E?

Are left limits and thus jumps measurable in the space of càdlàg functions (Q2) D([0, ∞); E) with values in these spaces?

In order to nd an almost surely converging compensated Poisson integral handling the small jumps of the process, one has to use results for series of random elements in a Banach space which is continuously embedded in the given state space. Also, measurability properties of paths do not translate immediately to measurability of the jump process without additional considerations.

We propose a method exploiting the Suslin property of the state space to overcome these diculties. It is based on the fact that on comparable Suslin topologies, the respective Borel-σ-algebras coincide.

Concerning (Q1), we will give sucient conditions on the state space (satised e.g. by all separable Banach and Fréchet spaces and all common distribution spaces) to guarantee the embedding procedure of a separable Banach space. In order to obtain the desired measurability properties of the jumps (Q2), we introduce a continuous metric on the state space.

The proposed framework for E generalises and unies results for vector-valued Lévy processes in the case of separable Banach spaces [3] and Suslin duals of nuclear spaces [4].

In the end, we give an outlook to which extent these techniques can be used in order to prove implications between various convergences of random elements in the considered spaces.

**References:**

[1] F. Baumgartner. Stochastic Analysis for Lévy Processes. PhD thesis, Universität Innsbruck, 2015.

F. Baumgartner. Lévy processes with values in locally convex Suslin spaces.

[2] submitted, ArXiv 1510.00538, 2015.

E. Dettweiler. Banach space valued processes with independent increments and [3] stochastic integration. In Probability in Banach spaces IV (Oberwolfach 1982), volume 990 of Lectures Notes in Mathematics, pages 5483, 1983.

[4] A. S. Üstünel. Additive processes on nuclear spaces. Ann. Probab., 12(3):858868, 1984.

We study the two-dimensional snake-like pattern that arises in phase separation of alloys described by spinodal decomposition in the Cahn-Hilliard-Cook model. These are somewhat universal patterns due to an overlay of eigenfunctions of the Laplacian with a similar wavenumber. Similar structures appear in other models like reaction-diusion systems describing animal coats' patterns or vegetation patterns in desertication.

Our main result studies random functions given by cosine Fourier series with independent Gaussian coecients, that dominate the dynamics in the Cahn-Hilliard model. This is not a cosine process, as the sum is taken over domains in Fourier space that not only grow and scale with a parameter of order 1/ε, but also move to innity. Moreover, the model under consideration is neither stationary nor isotropic.

To study the pattern size of nodal domains we consider the density of zeros on any straight line through the spatial domain. Using a theorem by Edelman and Kostlan and weighted ergodic theorems that ensure the convergence of the moving sums, we show that the average distance of zeros is asymptotically of order ε with a precisely given constant.

where (Hpm ) shall denote the Haar functions on [0, 1]. Candidates for directions are on the one hand (Hölder-)continuous functions and on the other hand measures on the unit interval. We therefore work with dierent integral types.

Under suitable conditions we nd that the desired extensions are possible and we also identify certain limits beyond which one cannot hope to get reasonable directional derivatives. We also give a link to the Fournié-Cont calculus.

We study the approximation of stochastic partial equations (SPDEs) on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Here solutions are described by a slowly modulated periodic pattern.

Due to the unboundedness of the underlying domain a whole band of innitely many eigenfunctions changes stability. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, which is described by the modulation equation.

This is an SPDE that serves as the normal form for the bifurcation.

As a rst step towards a full theory of modulation equations for nonlinear SPDEs on unbounded domains, we focus, in the results presented here, mainly on the linear theory for one particular example, the Swift-Hohenberg equation. Here we need to estimate stochastic convolutions, where the integrands are explicitly given by Fourier multipliers.

These linear results are one of the key technical tools to carry over the deterministic approximation results to the stochastic case with additive forcing. One technical problem for establishing error estimates rises from the spatially translation invariant nature of space-time white noise on unbounded domains, which implies that at any time we can expect the error to be always very large somewhere in space.

It is well known from the results of A. Zvonkin, A. Veretennikov, N. Krylov, A. Davie, F. Flandoli, J. Mattingly, M. Scheutzow and other probabilists that ordinary dierential equations (ODEs) regularize in the presence of noise. Even if an ODE is "very bad" and has no solutions (or has many solutions), then the addition of a random noise leads almost surely to a "nice" ODE with a unique solution. We investigate the same phenomenon for a 1D heat equation with an irregular drift

A limit theorem for the moments in space of Brownian local time increments is presented.

As special cases for the second and third moments, the central limit theorems by Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen (Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab. 14, 2009; Electron.

Commun. Probab. 15, 2010) and Rosen (Séminaire de Probabilités XLIII, Springer, 2011) are contained. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally dierent approach by exclusively working in the space variable of the Brownian local time, which allows to give a unied argument for arbitrary orders. The main ingredients are Perkins' semimartingale decomposition, the Kailath-Segall identity and an asymptotic Ray-Knight Theorem by Pitman and Yor.

In this talk we analyze stochastic Volterra equations driven by spacetime Lévy noise, which includes certain semi-linear stochastic PDEs as a particular subclass. We discuss conditions on the Volterra kernel and the Lévy characteristics that ensure the existence and uniqueness of solutions on bounded or unbounded intervals. Based on this theory, we further investigate numerical schemes for the simulation of stochastic Volterra equations, and show, under reasonable assumptions, their Lp - and almost sure convergence to the true solution with explicit convergence rates. A simulation study visualizes the investigated processes.

This talk is partially based on joint work with Bohan Chen and Claudia Klüppelberg.

In numerical analysis for stochastic dierential equations, a general rule of thumb is that the optimal weak convergence rate of a numerical scheme is twice as large as the optimal strong convergence rate. However, for SPDEs the optimal weak convergence rate is dicult to establish theoretically. Recently, progress was been made by Jentzen, Kurniawan and Welti for semi-linear SPDEs using the so-called mild Itô formula. We consider this approach for the stochastic wave equation, and discuss the necessity of extending the theory to the Banach space setting in order to obtain optimal rates when the non-linear terms are given by Nemytskii operators.