# Stochastics Seminars

Friday Jun 11, 2021. Strong stationary times for finite Heisenberg walks.

Speaker: Professor Laurent Miclo, Institut de Mathématiques de Toulouse, Toulouse School of Economics CNRS and University of Toulouse.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:

A strong stationary time associated to an ergodic Markov chain $(X_n)_n$ is a finite stopping time $\tau$ such that $\tau$ and $X_\tau$ are independent and $X_\tau$ is distributed according to the invariant probability.  Such a time enables to estimate the speed of convergence to equilibrium of $(X_n)_n$ in a probabilistic way. After presenting classical examples and general principles for the construction of strong stationary times, we will introduce a first strong stationary time for finite Heisenberg walks.

Friday May 14, 2021. Creeping of Lévy processes through curves

Speaker: Professor Loïc Chaumont, University d'Angers.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract: A real Lévy process X is said to creep upward if for some (and hence for all) x>0, with positive probability, the process X hits x at its first passage time above x. Necessary and sufficient conditions for a Lévy process to creep upward are well known as well as the probability to creep through each level x. In this work we study the creeping property of a Lévy process X issued from 0 through the graph of a continuous, non increasing function f such that f(0)>0. We give an expression of the probability that X creeps through f in terms of the drift coefficients of the upward ladder processes and the density of the entrance law of the excursions of X reflected at its infimum. This is a joint work with my PhD student Thomas Pellas.

Friday Apr 30, 2021. Convergence of ancestral processes from non-neutral Cannings models

Speaker: Dr. Jere Koskela, University of Warwick.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract: Cannings models are a wide class of stochastic models for the evolution of genetic traits in finite populations of fixed size. The population reproduces clonally and in discrete generations, with the next generation being formed from the current one by sampling a random vector specifying the family size for each individual, and with offspring inheriting the trait of their parent. When the model is neutral, i.e. the family size of an individual is independent of their trait, the ancestral tree embedded into a Cannings model by tracking parent-offspring relations backward in time is known to converge weakly to the Kingman coalescent under a natural condition on the third moments of family sizes, and a suitable rescaling of time. I will show that the same convergence takes place under analogous conditions in the non-neutral case, albeit with a less tractable time rescaling, and present some applications for sequential Monte Carlo algorithms, which can be thought of as examples of non-neutral Cannings models.

Friday Apr 23, 2021. The functional Breuer-Major theorem

Speaker: Professor Ivan Nourdin, University of Luxemburg.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract: Let $X$ be a zero-mean stationary Gaussian sequence of random variables with covariance function $p$ satisfying $p(0)=1$. Let $f$ be a function of Hermite rank $d$ bigger than 1. The celebrated Breuer–Major theorem asserts that, if $p^d$ is summable then the finite dimensional distributions of the normalized sum of $f(X_i)$ converge to those of a standard Brownian motion (up to some multiplicative constant). Surprisingly, and despite the fact this theorem has become over the years a prominent tool in a bunch of different areas, a necessary and sufficient condition implying the weak convergence in the space of càdlàg functions endowed with the Skorohod topology was still missing. We will explain in this talk how to fill this gap. This is joint work with David Nualart, based on a recent paper published in PTRF.

Friday Apr 16, 2021. Voronoi cells in random split trees

Speaker: Dr. Cécile Mailler, University of Bath.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:  Take k nodes uniformly at random in a n-node graph, and let k competing epidemics spread along edges at constant speed from these initial nodes, infecting nodes when they reach them. A node can only get infected by one of the epidemics, the first one that reaches it. We are interested in the sizes of the final territories of the k epidemics, which we call the Voronoi cells of the k initial nodes. In this joint work with Markus Heydenreich (Munich) and Alex Drewitz (Cologne), we prove limiting theorems for the sizes of the Voronoi cells of k nodes taken uniformly at random in an n-node random split tree.

Friday Mar 19, 2021. The Abelian sandpile model and its stochastic variants

Speaker: Dr. Thomas Selig, Xi'an Jiaotong-Liverpool University

13:00--14:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract: The sandpile model is a dynamic process on a graph G. At each unit of time, a grain of sand is added to a randomly selected vertex of G. When this causes the number of grains at that vertex to exceed a certain threshold (usually its degree), that vertex is said to be unstable, and topples, sending grains to (some of) its neighbours in G. This may cause other vertices to become unstable, and they topple in turn. A special vertex called the sink absorbs excess grains from the system, and so the process eventually stabilises. In the standard Abelian sandpile model (ASM), topplings are deterministic: one grain is sent to each neighbour of an unstable vertex. Stochastic variants of the ASM makes these topplings random: an unstable vertex sends grains to a random subset of its neighbours. There are a number of such variants in the literature, depending on how this random subset is chosen.

In this talk, we will expose some of the important results and recent developments of the ASM and introduce some of its more popular stochastic variants. We will then analyse in more detail one such variant, known as the Bernoulli stochastic sandpile model, in which unstable vertices flip a coin for each neighbour to decide whether to send a grain to it or not.

Friday Mar 12, 2021. k-CUT MODEL FOR THE BROWNIAN CONTINUUM RANDOM TREE

Speaker: Dr. Minmin Wang, University of Sussex.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract: To model the destruction of a resilient network, Cai, Holmgren, Devroye and Skerman introduced the k-cut model on a random tree, as an extension to the classic problem of cutting down random trees. Berzunza, Cai and Holmgren later proved that the total number of cuts in the k-cut model to isolate the root of a Galton–Watson tree with a finite-variance offspring law and conditioned to have n nodes, when divided by n^{1-1/2k}, converges in distribution to some random variable defined on the Brownian Continuum Random Tree. In this talk, we present a direct construction of the limit random variable, relying upon the Aldous–Pitman fragmentation process and a deterministic time change.

Slides from the speaker (seminar 2021.03.12)

Friday Feb 26, 2021. Interval-partition evolutions as limits of random walks on compositions, and their applications

Speaker: Dr. Quan Shi, Universität Mannheim.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:  A composition of a positive integer is a sequence of positive integers that sum to n. In this talk, we will introduce a family of interval-partition-valued diffusions as limits of random walks on integer compositions. These infinite-dimensional diffusions have Poisson--Dirichlet (pseudo-)stationary distributions and are closely related to the limits of random walks on integers partitions studied by Borodin--Olshanski and Petrov. I will also talk about some applications of our model in population genetics and continuum-tree-valued dynamics.

This talk is based on joint works with Noah Forman (McMaster), Douglas Rizzolo (Delaware), and Matthias Winkel (Oxford).

Slides from the speaker (seminar 2021.02.26)

Friday Feb 19, 2021. Quantitative two-scale stabilisation on the Poisson space

Speaker: Dr. Xiaochuan Yang, University of Bath.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract: The stabilisation theory was put forward in 2001 by Penrose and Yukich as a kind of high level abstraction of the famous CLT for minimal spanning trees of Kesten and Lee. Since its birth, this theory constitutes one of the most fundamental ideas for proving Gaussian approximation of stochastic geometric models, e.g. coverage processes, random tessellations, spatial networks etc. In a recent joint work with G. Peccati and R. Lachieze-Rey, we develop a quantitative stabilization theory which gives rates of multivariate Gaussian approximation for general stabilizing Poisson functionals, extending some estimates from a recent paper of Chatterjee and Sen on the rate of normal convergence of minimal spanning trees. Several examples are worked out to illustrate our results, including the online nearest neighbor graphs, edge statistics of Euclidean minimal spanning trees, and excursion of heavy tailed shot noise random fields. Based on the preprint https://arxiv.org/abs/2010.13362.

Slides from the speaker (seminar 2021.02.19)

Friday Dec 11, 2020. The scaling limit of a critical random directed graph.

Speaker: Dr. Robin Stephenson, University of Sheffield

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:  We consider the random directed graph D(n,p) with vertex set {1,2,...,n} in which each of the n(n − 1) possible directed edges is present independently with probability p. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at p = 1/n, with critical window p = 1/n + λn⁻⁴/³ for λ ∈ ℝ . We show that, within this critical window, the strongly connected components of D(n,p), ranked in decreasing order of size and rescaled by n⁻¹/³ , converge in distribution to a sequence (C₁,C₂, ...) of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs in the sense of an L^1 sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős–Rényi random graph G(n,p), whose scaling limit is well understood. We show that the limiting sequence (C₁,C₂, ...) contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability.

Joint work with Christina Goldschmidt.

Slides from the speaker (seminar 2020.12.11)

Friday Dec 04, 2020. Around the almost sure central limit theorem of Salem-Zygmund.

Speaker: Dr. Guillaume Poly, University of Rennes 1

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract: We will review some recent results around this limit theorem and provide a simple and self-contained proof. Then, we will discuss applications to the roots of random trigonometric polynomials.

Slides from the speaker (seminar 2020.12.04)

Friday Nov 20, 2020. Attraction to and repulsion from patches on the hypersphere and hyperplane for isotropic d-dimensional α-stable processes with index in α∈(0,1) and d≥2.

Speaker: Professor Andreas Kyprianou, University of Bath

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract: Consider a d-dimensional α-stable processes with index in α∈(0,1) and d≥2. Suppose that Ω is a region of the unit sphere S^{d−1} = {x ∈ R^d : |x| = 1}. We construct the aforesaid stable Lévy process conditioned to approach Ω continuously, either from inside S^{d−1}, from outside S^{d−1} or in an oscillatory way; all of which have zero probability. Our approach also extends to the setting of hitting bounded domains of (d-1)-dimensional hyperplanes. We appeal to a mixture of methods, appealing to the modern theory of self-similar Markov process as well as the classical potential analytic view.

Joint work with Tsogzolmaa Saizmaa (National University of Mongolia), Sandra Palau (UNAM, Mexico) and Mateusz Kwasniki (Technical University of Wroclaw).

Thursday Nov 19, 2020. The van Dantzig problem and the Riemann hypothesis (organised jointly with Dynamics seminar)

Speaker: Professor Pierre Patie, University of Liverpool

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:  In this talk, we start by introducing the intriguing van Dantzig problem  which consists in characterizing the subset of Fourier transforms of probability measures on the real line that  remain invariant under the composition of the reciprocal map with a complex rotation.

We first focus  on the so-called Lukacs class of solutions that is the ones that belong to the set of Laguerre-Pόlya functions which are entire functions with only real zeros. In particular, we show that the Riemann hypothesis is equivalent to the membership to the Lukacs class of the Riemann ξ function.

We state several closure properties of this class including adaptation of known results of Pόlya, de Bruijn and Newman but also some new ones. We proceed by presenting a new class of entire functions, which is in bijection with a set of continuous negative definite functions, that are solutions to the van Dantzig problem and discuss the possibility of  the Riemann ξ function to belong to this class.

Friday Nov 13, 2020. Non-standard limits for a family of autoregressive stochastic sequences

Speaker: Professor Sergey Foss, Heriot-Watt University

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:

We consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We obtain a non-standard limit theorem where the limiting distribution is a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case where there is no the restart mechanism.

Joint work with Matthias Schulte.

Slides from the speaker (seminar 2020.11.13)

Friday 25th October 2019 - Fatou's Lemmas for Varying Probabilities and their Applications to Sequential Decision Making

Stochastics Seminar

Speaker: Distinguished Professor Eugene A. Feinberg, Stony Brook University (USA)

16:00 MATH-211

Abstract: The classic Fatou lemma states that the lower limit of expectations is greater or equal than the expectation of the lower limit for a sequence of nonnegative random variables. This talk describes several generalizations of this fact including generalizations to converging sequences of probability measures. The three types of convergence of probability measures are considered in this talk: weak convergence, setwise convergence, and convergence in total variation. The talk also describes the Uniform Fatou Lemma (UFL) for sequences of probabilities converging in total variation. The UFL states the necessary and sufficient conditions for the validity of the stronger inequality than the inequality in Fatou's lemma.

We shall also discuss applications of these results to sequential optimization problems with completely and partially observable state spaces. In particular, the UFL is useful for proving weak continuity of transition probabilities for posterior state distributions of stochastic sequences with incomplete state observations known under the name of Partially Observable Markov Decision Processes. These transition probabilities are implicitly defined by Bayes' formula, and general method for proving their continuity properties have not been available for long time.

This talk is based on joint papers with Pavlo Kasyanov, Yan Liang, Nina Zadoianchuk, and Michael Zgurovsky.

Wednesday 12th June 2019 - Comparison Theorem of Stochastic Differential Equations

Speaker: Chenggui Yuan (Swansea University)

16:00 MATH-117

Abstract: In this talk, the existence and uniqueness of strong solutions to distribution dependent neutral SFDEs are proved. We give the conditions such that  the order preservation of  these equations holds.  Moreover, we show these conditions are also necessary when the coefficients are continuous. Under sufficient conditions, the result extends the one in the distribution independent case, and the necessity of these conditions  is new even in distribution independent case.

Friday 1st March 2019 - Stochastic Porous Media Equations: An Entropy Approach

Speaker: Konstantinos Dareiotis (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany)

12:00 MATH-105

Abstract: Nonlinear diffusion equations describe macroscopic phenomena such as flow of gas or fluid in a porous medium, heat propagation with temperature-dependent conductivity, and the evolution of crowd-avoiding populations. Often, a source term is present that forces/removes mass into/from the system. This source term can be random. As a particular example, generalised stochastic porous media equations (SPME) appear as scaling limits of the empirical measure of interacting branching diffusion particle systems. The interaction leads to a non-linear, degenerate second order operator in the drift, while the randomness of the branching mechanism leads to a non-linear noisy source term. We will discuss the main difficulties towards well posedness of these equations and introduce the concept of entropy solutions. In the class of entropy solutions we obtain well-posedness, L 1 -contraction, and stability estimates. Our results cover the full range of powers of the porous medium operator and allow for nonlinearities in the noise that are 1/2-Hölder continuous. This is a joint work with Mát´e Gerencs´er and Benjamin Gess.