Tsiry Randrianasolo (Montan University Leoben, Austria)

Travelling wave solutions to the KPP equation with branching noise

The one-dimensional KPP-equation driven by space-time white noise, $$ \partial_t u = \partial_{xx} u + \theta u - u^2 + u^{\frac{1}{2}} dW, \qquad t>0, x \in \mathbb{R}, \theta>0, \qquad \qquad u(0,x) = u_0(x) \geq 0 $$ is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial non-negative finite-mass conditions. This SPDE arises for instance as the high density limit of particle systems which undergo branching random walks and allow for extra death due to overcrowding.

If $\theta$ is below a critical value $\theta_c$, solutions die out to $0$ in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$. For initial conditions that are ‘’uniformly distributed in space’’, a complete convergence result holds, that is, the corresponding solutions are all in the domain of attraction of a unique non-zero stationary distribution. What can be said for solutions with finite initial mass if we condition on their survival?

In this talk I start with an overview of the main probabilistic ideas and arguments of the above mentioned concepts. In a next step, I explain how to obtain a travelling wave solution to this SPDE. Here, the choice of the wave front marker plays an important role. Finally, I outline why an understanding of the behaviour of travelling wave solutions can help to answer questions on convergence of solutions with arbitrary initial conditions.

Metastability in diffusion processes and synchronization

Non-standard time & location: 17:15 @ WSC-S-U-4.02

The phase diagram of the complex branching Brownian motion energy model

We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and on the boundaries I/III and II/III, we prove a central limit theorem with a random variance. In Phase I and on the boundary I/II, we prove an a.s. and $L^1$ martingale convergence. All results are shown for any given correlation between the real and imaginary parts of the random energy. This is joint work with Lisa Hartung (Courant Institute, NYU)

Time-discretization of stochastic 2-D Navier-Stokes equations by a penalty-projection method

In this talk, I will present a time-discretization method of the stochastic incompressible Navier--Stokes problem using a penalty-projection method. Basically, the talk will consist of three parts. A brief introduction of the mathematical problem. Then an overview of the main computational issue that shares the stochastic and the deterministic form of Navier--Stokes. Different algorithm will be introduced: a main algorithm and in order to treat the nonlinear character of the equation two auxiliary algorithms. At the end of the day we will arrive at the convergence with rate in probability and a strong convergence of the main algorithm.

Convex Hulls of Random Walks: Expected Number of Faces

Consider a random walk $S_i= \xi_1+\dots+\xi_i$, $1\leq i\leq n$, starting at $S_0=0$, whose increments $\xi_1,\dots,\xi_n$ are random vectors in $\mathbb R^d$, $d\leq n$. We are interested in the properties of the convex hull $C_n:=\mathrm{Conv}(S_0,S_1,\dots,S_n)$. Assuming that the tuple $(\xi_1,\dots,\xi_n)$ is exchangeable and some general position condition holds, we derive an explicit formula for the expected number of $k$-dimensional faces of $C_n$ in terms of the Stirling numbers of the first and second kind. Generalizing the classical discrete arcsine law for the position of the maximum due to E. Sparre Andersen, we compute explicitly the probability that for given indices $0\leq i_1 < \dots < i_{k+1}\leq n$, the points $S_{i_1},\dots,S_{i_{k+1}}$ form a $k$-dimensional face of $\mathrm{Conv}(S_0,S_1,\dots,S_n)$. This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of a direct product of finitely many reflection groups of types $A_{n-1}$ and $B_n$. This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position. All formulae are distribution-free, that is do not depend on the distribution of the $\xi_k$'s. (Joint work with Vladislav Vysotsky and Dmitry Zaporozhets.)

Continuous spin models on annealed generalized random graphs

We study spin models where the spins take values in a general compact Polish space and interact via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution. We prove a variational formula for the corresponding annealed pressure.

We furthermore study classes of models with second order phase transitions which include models on an interval and rotation-invariant models on spheres, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model, but we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.

This is joint work with Christof Külske and Philipp Schriever.

Simple nested coalescents and two-level Fleming Viot processes

Simple nested coalescent has been introduced to model backwards in time the genealogy, of both, species trees and genes trees. In this setting, the Kingman case corresponds to binary coalescences in the species trees or the genes trees, but not simultaneously. On the other hand, two level Fleming Viot with two level selection arises in Dawson (2015) as the limit in distribution of multilevel multitype population undergoing mutation, selection, genetic drift and spatial migration. In this talk, I will establish a duality relation between Kingman nested coalescents and the two level Fleming Viot associated with a two-level multitype population with genetic drift. Using this relation we can read off the genealogy backwards in time of a Kingman nested.

Cutting-down random trees

Imagine that we destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. We then record the genealogy induced by this destruction process in a random rooted binary tree, the so-called cut-tree. The goal of this talk will be to show a general criterion for the convergence of the rescaled cut-tree in the Gromov-Prohorov topology to a real tree, when the underlying tree has a small height. In particular, we consider uniform random recursive trees, binary search trees, scale-free random trees. The approach relies in the introduction of a continuous version of the cutting-down procedure which we allow us to represent the destruction process up to a certain finite time as Bernoulli bond percolation.

Stochastic domination in space-time for the supercritical contact process

The contact process is a classical model for the spread of infections in a population. In this talk, we focus on the contact process in the supercritical regime for which infections may spread forever with positive probability. Our goal is to understand how this process behaves compared with a process having no spatial correlations. In particular, does the contact process stochastically dominate a non-trivial independent (in space) spin-flip process? Such questions were studied by Liggett and Steif (2006) who proved that, for the process on $\mathbb{Z}^d$, the upper invariant measure stochastically dominates a Bernoulli product measure. We present some space-time versions of their results for the contact process on general graphs. From our methods, we furthermore conclude strong (uniform) mixing properties for certain space-time projections of the contact process. Based on joint work with Rob van den Berg (CWI Amsterdam and VU Amsterdam).

Non-asymptotic analysis of non-linear forms in random elements

We review recent results on non-asymptotic analysis for the distributions of quadratic and almost quadratic forms in random elements with values in a Hilbert space. The study of almost quadratic forms is motivated by approximation problems in multidimensional mathematical statistics. A number of results are optimal - they can not be improved without additional assumptions. A unified approach will be proposed for constructing non-asymptotic approximations on the basis of the general result on approximation accuracy for symmetric functions of several variables.

Low-frequency estimation for moving average Lévy processes

This talk is devoted to statistical inference for the stochastic integrals of the type $$ Z_{t}=\int_{R} K(t-s) dL_{s},$$ where $K$ is a deterministic function, and $L$ is a Lévy process. In particular, I will present several new ideas yielding the construction of a consistent estimator for the Lévy density of $L$. Moreover, I intend to discuss the mixing properties, which draw particular interest from the theoretical point of view.

Biased random walk in random environments

We explain two models for biased random walks in random environment (biased random walk on percolation clusters, biased random walk among random conductances) which describe transport in an inhomogenous medium. We give an overview of typical questions and present several results about Einstein relation and monotonicity of the speed.

Approximation techniques for probabilistic inference --
a statistical physics perspective

Probabilistic, Bayesian methods provide important approaches to data modelling in the field of machine learning. Inference on unobserved variables or parameters typically requires the performance of high-dimensional integrals or sums. If the dimensionality of the problem is very large, one often has to apply approximate inference methods which approximate intractable multivariate probability distributions by tractable ones.

In this talk I will focus on the so-called “Expectation Propagation” message passing techniques for approximate inference in latent Gaussian variable models. These are related to (and were, in parts motivated by) the “Thouless-Anderson-Palmer” (TAP) mean field approach to spin-glass models in physics.

I will give an introduction to the basic idea behind this approach and then discuss a combination of ideas from random matrix theory and dynamical functional methods of statistical physics which could be used to improve the efficiency of such methods and to study the convergence of inference algorithms for large systems.

Convergence of Markov processes to the invariant measure with applications to SPDEs and stochastic delay equations

While convergence of finite-dimensional Markov processes (e.g., SDEs) to the invariant measure is quite well understood by now, less is known about the convergence of infinite-dimensional Markov processes (e.g., SPDEs). In the first part of the talk we explain how the classical methods (which are based on the construction of a Lyapunov function) can be extended to study convergence of infinite-dimensional Markov processes in the Wasserstein metric. This generalizes recent results of M. Hairer, J. Mattingly, M. Scheutzow (2011). In the second part of the talk we provide some specific applications to SPDEs and stochastic delay equations and discuss the arising challenges. (Joint work with Alexey Kulik and Michael Scheutzow)

[1] O. Butkovsky (2014). Subgeometric rates of convergence of Markov processes in the Wasserstein metric. Annals of Applied Probability, 24, 526-552.

[2] O. Butkovsky, M. Scheutzow (2017). Invariant measures for stochastic functional differential equations. arXiv:1703.05120.

Exchangeable ultrametrics and tree-valued Fleming-Viot processes

The Donnelly-Kurtz lookdown model contains an evolving genealogy. The genealogical tree of the population at each time can be described by an isomorphy class of a metric measure space to obtain a tree-valued Fleming-Viot process. The states of this process can be viewed as ergodic components of the genealogical distance matrices. In this context, we also discuss a general representation for exchangeable ultrametrics in terms of sampling from marked metric measure spaces.