# Lévy processes and random walks

## The University of Manchester

The Department of Mathematics at the University of Manchester will host the meeting on Levy processes and random walks, a three day-long workshop on Levy processes random and related areas, with 24 invited talks and opportunities to present posters. This workshop will celebrate 80th birthday of Ron Doney

## Invited speakers

Larbi Alili (University of Warwick)
A reformulation of Wiener-Hopf factorization for random walks and Levy processes.
Abstract.
I will talk about my work with Ron on a reformulation of the classical Wiener-Hopf factorization for random walks and Levy processes. This is applied to the study of the asymptotic behaviour of objects associated with the bivariate strict/weak ascending ladder processes.
Jean Bertoin (University of Zurich)
A model for an epidemic with contact tracing and cluster isolation, and a detection paradox
Abstract.
We determine the distributions of some random variables related to a simple model of an epidemic with contact tracing and cluster isolation and compute explicitly the asymptotic proportion of isolated clusters with a given size amongst all isolated clusters, conditionally on survival of the epidemic. Somewhat surprisingly, the latter differs from the distribution of the size of a typical cluster at the time of its detection; and we explain the reasons behind this seeming paradox.
Nick Bingham (Imperial)
General Regular Variation and Extremes
Abstract.
We survey the connections between extreme-value theory and regular variation, in one and higher dimensions, from the point of view of our recent work on general regular variation.
Francesco Caravenna (Universita degli Studi di Milano-Bicocca)
Renewal Theory, Disordered Systems, and Stochastic PDEs
Abstract.
Renewal theory for random walks with "heavy tails" (e.g. polynomially decaying) is both a very classical topic and an active domain of research. We present a key renewal theorem for triangular arrays of random walks with "ultra heavy tails", i.e. with asymptotic tail exponent alpha=0, which converge after rescaling to an explicit Levy process, the so-called Dickman subordinator. Besides its own interest, such a renewal theorem is the key to a fine understanding of a challenging model in statistical mechanics, the directed polymer in random environment, and to a singular stochastic PDE, the multiplicative Stochastic Heat Equation, in the critical space dimension d=2. We discuss the relations between these seemingly unrelated models, which led to important recent progress.
Based on joint work with Rongfeng Sun and Nikos Zygouras.
Loic Chaumont (University of Angers)
Creeping of Lévy Processes through Curves
Abstract.
A Lévy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph $\{(t, f (t)) : t \ge 0\}$ of any continuous, non increasing function $f$ such that $f (0)>0$, we give an expression of the probability that a bivariate subordinator $(Y, Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components $Y$ and $Z$. We apply this result to the creeping probability of any real Lévy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients.
This is a joint work with Thomas Pellas.
Ron Doney (University of Manchester)
The remainder in the Renewal Theorem
Abstract.
An old problem is that of adding additional terms in the asymptotic expansion of the renewal function in the situation where the mean is finite. Most work has been on the case of finite variance, but here we give some new results in the case that the tail of the distribution is regularly varying with index in $(-2,-1)$. We also discuss the corresponding local problem, i.e. the behaviour of the renewal mass function in the discrete case.
Nathalie Eisenbaum (Université Paris Cité)
A stochastic model for macrophages dynamics in atherosclerotic plaques
Abstract.
Several models of the macrophages dynamics in atherosclerotic plaques have been already proposed. Recently Meunier, Mouhot and Roux have had the idea to make use of the notion of local time to describe the visits of a macrophage cell to a lipid-enriched region. We will refine their model by introducing a factor taking into account the influence of the natural defenses or of a medical treatment.
Joint work with Nicolas Meunier.
Alison Etheridge (University of Oxford)
The motion of hybrid zones (and how to stop them)
Abstract.
Mathematical models play a fundamental role in theoretical population genetics and, in turn, population genetics provides a wealth of mathematical challenges. In this lecture we illustrate this by using a mathematical caricature of the evolution of genetic types in a spatially distributed population to demonstrate the role that the shape of the domain inhabited by a species can play in mediating the interplay between natural selection, spatial structure, and (if time permits) so-called random genetic drift (the randomness due to reproduction in a finite population).
Sonia Fourati (LPSM)
Bernstein functions and their complex analysis properties
Abstract.
Ron has always been a staunch supporter of the use of real variables for studying Lévy processes and stable, or non stable, one-dimensional laws. His virtuosity has amazed us with the wealth of the results he has obtained. In the past, I have sometimes used complex analysis to obtain a few results related to his. Today, I will go this way, but a little more frequently the other way round, in order to relate Lévy processes to statements involving complex analysis.
Christina Goldschmidt (University of Oxford)
The stable graph: the scaling limit of critical random graphs with i.i.d. random degrees having power-law tails
Abstract.
Consider a graph with label set $\{1,2, \ldots,n\}$ chosen uniformly at random from those such that vertex i has degree $D_i$, where $D_1, D_2, \ldots, D_n$ are i.i.d. strictly positive random variables. The condition for criticality (i.e. the threshold for the emergence of a giant component) in this setting is $E[D^2] = 2 E[D]$, and we assume additionally that $P(D = k) \sim c k^{-(\alpha + 2)}$ as $k$ tends to infinity, for some $\alpha \in (1,2)$. In this situation, it turns out that the largest components have sizes on the order of $n^{\alpha/(\alpha+1)}$. Building on earlier work of Adrien Joseph, we show that the components have scaling limits which can be related to a forest of stable trees (à la Duquesne-Le Gall-Le Jan) via an absolute continuity relation. This gives a natural generalisation of the scaling limit for the Erdős-Renyi random graph (obtained in collaboration with Louigi Addario-Berry and Nicolas Broutin a few years ago, extending results of Aldous), which we call the stable graph.
This is joint work with Guillaume Conchon-Kerjan, which complements recent work on random graph scaling limits by various authors including Bhamidi, Broutin, Duquesne, van der Hofstad, van Leeuwaarden, Riordan, Sen, M. Wang and X. Wang.
Priscilla Greenwood (University of British Columbia)
Between sum and sup: a parametric family of operators on i.i.d. sequences
Abstract.
Arising from a model for solar energy storage, the "alpha-sun" recursive operation defines a smooth parametric family of operators on i.i.d. sequences. As alpha goes from 0 to 1 the operator goes from supremum to sum. After a brief diversion about discounted sums, we describe the domains of attraction of the three "alpha-sun" normed limit distributions. The limit family of densities of the Frechet-associated case satisfy a (daunting) recursive equation. An (ugly) integral representation and (nice) families of plotted functions have recently been obtained for this family of densities. This is joint work with Gerard Hooghiemstra in the 1980's and '90's, and with Nicholas Witte recently.
Claudia Klüppelberg (Technische Universitat Munchen)
Max-linear Bayesian networks
Abstract.
Graphical models can represent multivariate distributions in an intuitive way and, hence, facilitate statistical analysis of high-dimensional data. Such models are usually modular so that high-dimensional distributions can be described and handled by careful combination of lower dimensional factors. Furthermore, graphs are natural data structures for algorithmic treatment. Conditional independence and Markov properties are essential features for graphical models. Moreover, graphical models can allow for causal interpretation, often provided through a recursive system on a directed acyclic graph (DAG) and the max-linear model we introduced in 2018 is a specific example. In this talk I present some conditional independence properties of max-linear Bayesian networks and exemplify the difference to linear networks.
Andreas Kyprianou (University of Bath)
Attraction to and repulsion from patches on the hypersphere and hyperplane for isotropic $d$-dimensional $\alpha$-stable in dimension $d\geq 2$.
Abstract.
Consider a $d$-dimensional $\alpha$-stable processes in dimension $d\geq 2$. Suppose that D is a region of the unit sphere $S^{d-1} = {x \in R^d : |x| = 1}$. We construct the aforesaid stable Lévy process conditioned to approach $D$ continuously, either from inside $S^{d-1}$, from outside $S^{d-1}$ or in an oscillatory way. 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).
Ross Maller (Australian National University)
No-Tie Conditions for Large Values of Extremal Processes at Small Times
Abstract.
We give necessary and sufficient conditions for there to be no ties, asymptotically, among large values of a space-time Poisson point process evolving homogeneously in time. The convergence is at small times, in probability or almost sure.
This is joint work with Yuguang Ipsen, School of Finance, Actuarial Studies & Statistics, Australian National University.
Pierre Patie (Cornell University)
Is self-similarity unique?
Abstract.
Self-similarity is a fundamental and useful property in the study of stochastic processes. In this talk, we will revisit the classical notion of self-similarity resorting to group representation theory. We shall explain how this viewpoint enables to define self-similarity in a broader context. We will provide examples illustrating this idea and describe some interesting stochastic and analytical implications of this approach.
Phil Pollett (University of Queensland)
High-density limits for metapopulations with no occupancy ceiling
Abstract.
We consider a class of stochastic occupancy processes where there is no ceiling on the number of sites that can be occupied. We examine the limiting behaviour as the spatial density of sites increases. These limits are not only natural in the theory, but in practice as well, as they describe models with increasing environmental fidelity.
This is joint work with Liam Hodgkinson.
Victor Rivero(CIMAT)
Towards a Ray-Knight theorem for spectrally negative Lévy processes
Abstract.
It has been established by Kaspi and Eisenbaum that the local time process $(L^{x}_{\zeta}, x\in \mathbb{R})$, as a process in space, associated to an $\mathbb{R}$-valued Markov process $(X_t, 0\leq t\leq \zeta)$ bears itself the Markov property, if and only if, the process $X$ has continuous paths. In that case, the local time process can be described using branching processes. Besides, one can think of spectrally negative Lévy processes (SNLP) as those Lévy processes that bear properties closer to diffusions. Also, an analysis of the paths of SNLP allow to intuit that the local time process, as a process in space, associated to a SNLP killed at suitable stopping times, also bears a branching property. Even though the Markov property is not preserved. We will make this precise and describe the law of the local time process.
This is based in a work in process with Jos\'e Contreras, PhD student in CIMAT.
Mladen Savov (Sofia University ”St Kliment of Ohrid”)
Bivariate Bernstein-Gamma functions and asymptotic behaviour of exponential functionals on deterministic horizon
Abstract.
The properties of classical exponential functionals of L\'{e}vy processes have been investigated for almost thirty years now. The latest results have been obtained through the properties of the so-called Bernstein-Gamma functions. However, exponential functionals of L\'{e}vy processes on deterministic horizon have remained much less understood. There is a body of research in the study of their asymptotic properties when the horizon tends to infinity but yet there seems to be no unified approach in the treatment of this line of research. In this talk we present a new approach which seems posed to give a way to study this asymptotic behaviour in a larger generality. Our methodology is based on Mellin inversion, Tauberian theorems, including de Haan theory, and the study of the properties of the bivariate Bernstein-Gamma functions. We rely on various results in the literature but most crucially on a paper by Doney and Maller.
This is joint work with Martin Minchev.
Thomas Simon (Universite de Lille)
A characterisation of bell-shaped functions by their Fourier transform
Abstract.
A smooth curve $\mathbb R \to [0,+\infty]$ is said to be bell-shaped if it tends to zero at both infinities and if its $n$-th derivative vanishes exactly $n$ times on $\mathbb R$. We give a complete characterization of bell curves by the holomorphic extension of their Fourier transform. More precisely, we show that any bell curve is obtained by additive convolution of a Polya frequency function and an absolutely-then-completely-monotone function. We deduce that all bell-shaped probability densities are infinitely divisible and that the roots of their $n$-th derivatives have a linear growth when $n \to \infty$.
This is a joint work with M. Kwasnicki.
Stavros Vakeroudis (Athens University of Economics and Business)
Windings of planar Stochastic Processes
Abstract.
We start from the planar Brownian motion case where fine properties of their trajectories are discussed by using the skew-product representation and Bougerol's celebrated identity in law. Then, similar questions are addressed for complex-valued Ornstein-Uhlenbeck processes and for planar stable processes. For the last case, the methods applied in the planar Brownian motion setting are no longer valid and we shall use new methods invoking the continuity of the composition function and new techniques from the theory of self-similar Markov processes which involve the so-called Riesz--Bogdan--Zak transform. This approach allows to study similarly one-dimensional and possibly higher-dimensional windings.
Vitali Wachtel (University of Bielefeld)
Persistence of autoregressive sequences with logarithmic tails
Abstract.
We consider autoregressive sequences $X_n=aX_{n-1}+\xi_n$ and $M_n=\max\{aM_{n-1},\xi_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{\xi_k\}$. It is known that if $\mathbb P(\xi_1>x)\sim\frac{d}{\log x}$ with some $d\in(0,-\log a)$ then the chains $\{X_n\}$ and $\{M_n\}$ are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index $-1-d/\log a$. We also prove limit theorems for $\{X_n\}$ and $\{M_n\}$ conditioned to stay over a fixed level $x_0$. Furthermore, we study tail asymptotics for recurrence times of $\{X_n\}$ and $\{M_n\}$ in the case when these chains are positive recurrent and the tail of $\log\xi_1$ is subexponential.
Reflecting diffusions in generalized parabolic domains
Abstract.
For a multidimensional driftless diffusion in an unbounded, smooth, sub-linear generalized parabolic domain, with oblique reflection from the boundary, we give conditions under which either explosion occurs, if the domain narrows sufficiently fast at infinity, or else there is superdiffusive transience, which we quantify with a strong law of large numbers. For example, in the case of a planar domain, explosion occurs if and only if the area of the domain is finite.
This talk is based on joint work with Mikhail Menshikov and Aleksandar Mijatović.
Jon Warren (University of Warwick)
Some exceptional times for Brownian motion
Abstract.
The set of times at which a Brownian path has a local maxima is countable, dense, and, obviously, such times are determined by the local behaviour of the path. Boris Tsirelson once asked if there were any other sets of exceptional times with the same three properties ( other than local minima). In this talk I will give some examples, discuss their properties, and speculate as to what might be true generally.
Matthias Winkel (University of Oxford)
Increase of Lévy processes, and interval partition evolutions
Abstract.
It was Ron Doney who introduced me to the fluctuation theory of Lévy processes in general and to the notion of increase times, in particular. Increase times recently resurfaced in my ongoing research on branching interval partition evolutions and Fleming-Viot processes. The aim of my talk is to explain this connection starting from the Lévy processes side.
This is mainly based on joint work with Noah Forman, Soumik Pal, Douglas Rizzolo and Quan Shi.

## Location

The meeting will take place in the Department of Mathematics of the University of Manchester.

Alan Turing Building is building 46 on the Campus Map. For more details on reaching the venue, please see Maps and Travel.

The room has stepless access.

## Programme

 26th of July 27th of July 28th of July 08:30 Registration & coffee 08:55 Opening 09:00 Andreas Kyprianou 09:00 Jean Bertoin 09:00 Loïc Chaumont 09:45 Phil Pollett 09:45 Sonia Fourati 09:45 Ross Maller 10:30 Coffee break 10:30 Coffee break 10:30 Coffee break 11:00 Francesco Caravenna 11:00 Ron Doney 11:00 Matthias Winkel 11:45 Jon Warren 11:45 Mladen Savov 11:45 Victor Rivero 12:30 Lunch 12:30 Lunch 12:30 Lunch 14:00 Christina Goldschmidt 14:00 Vitali Wachtel 14:00 Larbi Alili 14:45 Nathalie Eisenbaum 14:45 Andrew Wade 14:45 Claudia Klüppelberg 15:30 Coffee break 15:30 Coffee break 15:30 Coffee break 16:00 Alison Etheridge 16:00 Pierre Patie 16:00 Stavros Vakeroudis 16:45 Nick Bingham 16:45 Priscilla Greenwood 16:45 Thomas Simon 17:30 Wine reception and poster session 18:30 Conference dinner 17:30 Discussion and close

Coffee, lunches and a reception for all participants will be provided.

## Abstracts

Abstracts(in progress)

## Registration

The conference will be run in hybrid mode with talks given either offline or online. The participants can also participate either in person or online. There is an opportunity for early-career researchers and research students to present a poster. We have limited support for some participants presenting posters with preference given to EPSRC funded students. The deadline for abstract submission is Tuesday 31st of May and decisions about the financial support will be communicated by Friday 3rd of June.

## Contact

The local organising committee is:

• Denis Denisov(University of Manchester)
• Andreas Kyprianou(University of Bath)
• Ronnie Loeffen(University of Manchester)
• Jon Warren(University of Warwick)
• Tusheng Zhang(University of Manchester)

The scientific committee is:

• Loic Chaumont(University of Angers)
• Andreas Kyprianou(University of Bath)
• Pierre Patie(Cornell University)
• Victor Rivero(CIMAT)
• Jon Warren(University of Warwick)