# Applied Probability Seminars

This is an informal seminar to give academic staff, visitors, graduate students, etc., based or hosted in Warwick (any department), the opportunity to know more about each other’s research on topics of probability and its uses in related areas such as mathematical statistics, statistical physics, computer sciences, analysis, et cetera.

PhD students and postdocs are particularly encouraged to contribute, as well as any external speaker from other departments or universities that you would like to invite. Please contact Daniel Valesin (daniel.valesin<at>warwick.ac.uk) if you wish to give a talk or if you have a name of speaker to suggest.

Below is a provisional schedule of the upcoming talks. Given the informal nature of the seminar, last minute changes may happen. Please join the Applied probability Seminar Teams channel to receive up to date information.

## Academic Year 2023-24, Term 3

**26 April 2024**

**Speaker:** **Daniel Ueltschi**

**Title: **Geometric correlation inequalities for spin systems

**Abstract: **In classical spin systems, the random variables are indexed by lattice sites. The behaviour and properties of these systems can be characterised by correlation functions (or covariance). In some cases it is possible to establish inequalities with respect to lattice sites. I will review some of them, such as Simon-Lieb-Rivasseau, Messager-Miracle-Sole, and Lees-Taggi. I will also discuss quantum spin systems (sometimes, these have a probabilistic setting) and point out that the Messager-Miracle-Sole inequalities hold for the spin-1/2 quantum XY model.

###### 3 May 2024

**Speaker:** **Samuel Johnston** (King's College)

**Title: **The largest and smallest fragment in the halving self-similar fragmentation process

**Abstract: **We study the simplest possible self-similar fragmentation process. The process starts at time zero with a single fragment of size 1, which has an exp(1) lifetime before splitting into two fragments of size 1/2. Thereafter, for a parameter q<1, a fragment of size 2^(-n) has an exp(q^n) lifetime before splitting into two fragments of sizes 2^(-n-1). We find that at large times the sizes of the largest and smallest fragment in the system can be characterised with high probability to specific integer powers of 1/2. Our approach draws on connections with branching random walks, point processes and q-combinatorics.

This is joint work with Piotr Dyszewski, Nina Gantert, Joscha Prochno and Dominik Schmid

###### 10 May 2024

**Speaker:** **Geronimo Uribe Bravo** (UNAM)

**Title: **A limit theorem for local times and an application to branching processes

**Abstract: **Consider a stochastic process which is regenerative at some state and which is approximated by a sequence of discrete time regenerative processes. We give conditions so that the naturally defined local time of the approximating processes (which counts the number of visits to the regenerative state) has a scaling limit. We discuss examples dealing with Lévy processes but mostly focus on a recent application to Bienaymé-Galton-Watson processes with immigration.

Joint work with Aleksandar Mijatovic and Ben Povar.

###### 17 May 2024

**Speaker:** **Konrad Anand** (Queen Mary University London)

**Title: **Lazy Depth-First Sampling of Spin Systems

**Abstract: **We examine the new sampling technique lazy depth-first sampling, particularly as applied to spin systems.

We are given a graph G = (V,E), q ‘spins’ or ‘colours,’ and vector b of weights for each spins, and a matrix A of weights of interactions between spins. Given a configuration \sigma, i.e., an assignment of a spin to each v \in V, the weight of the configuration is the product of the weights of each vertex and each edge. Normalizing gives us a probability distribution.

The goal is to sample a configuration on a subset of the graph. We do so in a lazy fashion, running a recursive algorithm which recurses as little as possible. This provides interesting perfect sampling results.

###### 24 May 2024

**Speaker:** **Karen Habermann**

**Title: **Score matching for simulating degenerate diffusion bridge processes

**Abstract: **Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics. Whilst simulating diffusion bridge processes is already difficult for a process with uniformly positive definite diffusivity matrix, further complications arise when considering a process with degenerate diffusivity matrix. For a wide class of such degenerate diffusion processes, which yet admit a positive smooth density everywhere, we handle these challenges and construct a method for bridge simulation which exploits recent progress in machine learning. A central object in the study of diffusion bridge processes is the score, that is, the logarithmic gradient of the density, which is needed to describe diffusion bridge processes, and we discuss several methods for estimating the score using neural networks. The results are illustrated by simulating diffusion bridge processes for the process where two independent one-dimensional Brownian motions are coupled with their Lévy area, also known as the canonical diffusion process on the Heisenberg group. Joint work with Erlend Grong (Bergen) and Stefan Sommer (Copenhagen).

###### 31 May 2024

**Speaker:** **Kei Noba **(Institute of Statistical Mathematics, Japan)

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###### 7 June 2024

**Speaker:** **Christopher Dean**

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###### 14 June 2024

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###### 21 June 2024

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###### 28 June 2024

**Speaker:** **Matteo Mucciconi**

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## Past Seminars: Academic Year 2023-24, Term 1

**29 September 2023 (Welcome Week)**

**Venue is exceptionally MB2.24, time is exceptionally 14:00**

Speaker: **Abdelatif Bencherif-Madani **(Université Sétif 1 Ferhat Abbas)

**Title:** An entropic limit theorem for the boundary local time of a diffusion

**Abstract: **Let X be a symmetric diffusion reflecting in a C^3-bounded domain D in the Euclidean space of dimension n, n>=1. Assume that the diffusion matrix is of class C^2, bounded and non-degenerate. For t>0, and k, n positive integers, let N(n,t) be the number of dyadic intervals I_{n,k} of length 2^{-n}, k>=0, that contain a time s=<t such that X_s belongs to the boundary of D. For a suitable normalizing factor H(t), we prove, extending the one dimensional case, that a.s. for all t>0 the entropy functional N(n,t)/H(2^{-n}) converges to the boundary local time L_t as n\rightarrow \infty. Motivations of this study include boundary value problems in PDE theory, efficient Monte-Carlo simulation, and Finance.

**6 October 2023**

Speaker: **Emma Horton**

**Title:** Branching models for telomere evolution

**Abstract: **Telomeres are short sequences of DNA at the end of chromosomes, whose evolution is intrinsically related to biological ageing. As cells divide, telomeres become shorter, which can lead to a variety of diseases due degradation of DNA. We propose a branching process model to replicate the behaviour of telomeres during cell division and study their long-term distribution. We show that with our branching model, we recover the classical Hayflick limit, which is the number of doublings of the population.

This is based on joint work with Denis Villemonais and Coralie Fritsch (Université de Lorraine)

**13 October 2023**

Speaker: **Victor Rivero **(CIMAT)

**Title:** Recurrent Extensions and Stochastic Differential Equations

**Abstract: **In the 70's Itô settled the excursion theory of Markov processes, which is nowadays a fundamental tool for analyzing path properties of Markov processes. In his theory, Itô also introduced a method for building Markov processes using the excursion data, or by gluing excursions together, the resulting process is known as the recurrent extension of a given process. Since Itô's pioneering work the method of recurrent extensions has been added to the toolbox for building processes, which of course includes the martingale problem and stochastic differential equations. The latter are among the most popular tools for building and describing stochastic processes, in particular in applied models as they allow to physically describe the infinitesimal variations of the studied phenomena. In this work we answer the following natural question. Assume X is a real valued Markov process that dies at the first time it hits a distinguished point of the state space, say 0, which happens in a finite time a.s., also assume that X satisfies a stochastic differential equation, and finally that X admits a recurrent extension, say Y, that is a processes that behaves like X up to the first hitting time of 0 and for which 0 is a recurrent and regular state. If any, what is the SDE satisfied by Y? Our answer to this question allows us to describe the SDE satisfied by many Feller processes. We analyze various particular examples, as for instance the so-called Feller brownian motions and diffusions, which include their sticky and skewed versions, and also self-similar Markov processes, continuous state branching processes and real valued Levy processes.

**20 October 2023**

Speaker: **Andreas Kyprianou**

**Title:** General path integrals and stable SDEs

**Abstract: **The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. We study existence and uniqueness of weak solutions to driven by a (symmetric) -stable Lévy process, in the spirit of the classical Engelbert-Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence und uniqueness of weak solutions for . Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We prove integral tests for finiteness of path integrals under minimal assumptions.

**27 October 2023**

Speaker: **Conrado da Costa **(Durham University)

**Title:** Passage times for partially homogeneous reflected Random Walks on the quadrant

**Abstract:** In this talk we consider a partially homogeneous random walk on the quadrant with zero drift at the interior. The goal of the talk is to explain how to obtain qualitative and quantitative knowledge on the passage time of the origin and its moments. The focus of the talk is on the adaptation of the classical methods to our set up and the heuristics on how to obtain those classifications from both a probabilistic and a geometrical perspective. This is a joint work with Mikhail Menshikov and Andrew Wade.

**3 November 2023**

Speaker: **Sotiris Kotitsas**

**Title:** The two-dimensional heat equation with a random potential that is correlated in time

**Abstract:** We consider the PDE:

in the critical dimension d = 2 where V is a Gaussian random potential and β is the noise strength. We will focus in the case where the potential is not white in time and we will study the large scale fluctuations of the solution. This case was considered before in d ≥ 3 and for β small enough. It was proved that the fluctuations converge to the Edwards-Wilkinson limit with a nontrivial effective diffusivity and an effective variance. We prove that this result can be extended to d = 2. In particular we show that after tuning β accordingly and re-normalizing the large scale fluctuations of the solution, they converge in distribution to the Edwards-Wilkinson model with an explicit effective variance but with a trivial effective diffusivity. We able to prove this for all β below a critical value, after which the effective variance is infinite. Our main tools is the Feynman-Kac formula and a fine analysis of a specific Markov chain on the space of paths that was first introduced in d ≥ 3 for the same problem.

**10 November 2023**

Speaker: **Andreas Koller**

**Title:** Scaling limit of gradient models on $\Z^d$ with non-convex energy

**Abstract: **Random fields of gradients are a class of model systems arising in the study of random interfaces, random geometry, field theory and elasticity theory. The models we consider are characterised by an imposed boundary tilt and the free energy (called surface tension in the context of random interface models) as a function of tilt. Of interest are, in particular, whether the surface tension is strictly convex and whether the large-scale behaviour of the model remains that of the massless free field (Gaussian universality class). Where the Hamiltonian (energy) of the system is determined by a strictly convex potential, good progress has been made on these questions over the last two decades. Open problems include the conjecture that, in any regime of the parameters such that the scaling limit is Gaussian, its covariance (diffusion) matrix should be given by the Hessian of surface tension as a function of tilt. For models with non-convex energy fewer results are known. I will survey some recent advances in this direction using renormalisation group arguments and describe our result confirming the conjectured behaviour of the scaling limit for a class of non-convex potentials in the regime of low temperatures and small tilt. This is based on joint work with Stefan Adams.

**17 November 2023**

Speaker: **Brett Kolesnik **(Oxford)

**Title:** Rapidly sampling Coxeter tournaments

**Abstract: **A tournament is an orientation of the complete graph. An edge is a competitive game between its endpoints, directed away from the winner. The score sequence is the out-degree sequence. Kannan, Tetali and Vempala (1999) and McShine (2000) showed that tournaments with given score sequence can be rapidly sampled, via simple random walk on the interchange graphs of Brualdi and Li (1984). We prove an analogue for Coxeter tournaments, which involve collaborative and solitaire games, as well as the usual competitive games. Geometric connections with the Coxeter permutahedra recently introduced by Ardila, Castillo, Eur and Postnikov (2020) will be discussed. Joint work with three PhD students at Oxford: Matthew Buckland, Rivka Mitchell and Tomasz Przybyłowski.

**24 November 2023**

Speaker: **Marta Dai Pra **(Humboldt Universität)

**Title:** Xi-coalescents arising from populations undergoing bottlenecks

**Abstract: **Ancestry models can be used for predicting genetic diversity of a population, and comparing it to observed data. In population genetics most theory and statistical tests have been developed using the classical Wright-Fisher model and the Kingman coalescent. Nevertheless, organisms having a genealogy not well described by the Kingman coalescent are not rare. One example is the Atlantic cod which presents shallow genealogies and high-variance offspring number that might be rather described by a multiple merger coalescent. The aim of our work is to find a realistic individual-based model fitting these data. We present a spatially structured model undergoing localized, recurrent bottlenecks, and describe its ancestral lines. Depending on the severity of the bottleneck, we derive as scaling limits different structured Xi-coalescents featuring simultaneous multiple mergers and migrations.

This talk is based on ongoing joint work with A. Etheridge, J. Koskela, M. Wilke-Berenguer.

**1 December 2023**

Speaker: **Dario Spanò**

**Title: **Pólya urns, duality and eigenstructure for reversible Wright-Fisher-type processes

**Abstract: **Much of the tractability of neutral, parent-independent Wright-Fisher diffusions of population genetics come from the coexistence of a number of properties: they are polynomial diffusions; they have a moment dual (Kingman’s coalescent); they are time-reversible. I will describe the interplay of the above-mentioned properties under a perspective inspired by Bayesian statistics, and will show how this analysis can help in characterising new classes of reversible diffusions, whose eigenstructure is driven by a generalised moment dual process.

**8 December 2023**

Speaker: **Aria Ahari **

**Title: **Boundary crossing problems and functional transformations for Ornstein-Uhlenbeck processes

**Abstract: **We are interested in the law of the first passage time of driftless Ornstein-Uhlenbeck processes to time varying thresholds. We show that this problem is connected to the law of the first passage time of the process to some two-parameter family of functional transformations which, for specific values of the parameters, appears in a realisation of a standard Ornstein-Uhlenbeck bridge. We provide three different proofs of this connection. The first proof is based on a similar result to the case of the Brownian motion, the second uses a generalisation of the so-called Gauss-Markov processes and the third relies on the Lie group symmetry method applied to the Fokker-Planck equation of the Ornstein-Uhlenbeck process.

We investigate the properties of this transformation and study the algebraic and analytical properties of an involution operator which is used in constructing it. We also show that this transformation maps the space of solutions of Sturm-Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we discuss the interpretation of such transformations through the method of images and give new examples of curves with explicit first passage time densities.

Joint work with Larbi Alili and Massimilano Tamborrino.

## Past Seminars: Academic Year 2023-24, Term 2

Venue: MB0.08 (unless otherwise stated) or Online (Teams link will be provided in the Teams channel)

Time: 11:00 (unless otherwise stated) - followed by lunch in the Common Room.

**19 January 2024**

**Speaker: Marcelo Hilário (UFMG)**

**Title:** Percolation on randomly stretched lattices

**Abstract: **We study the existence/absence of percolation on a class of planar graphs. These graphs are randomly dilute versions of the square lattice, defined as follows: starting with all the sites in Z^2, we insert every horizontal edge connecting nearest neighbors. At this stage, the graph consists of infinitely many copies of the Z lattice parallel to one another, hence disconnected. We now insert only the vertical edges connecting nearest neighbors lying on {X_i} \times \Z, where X_1, X_2, X_3,... is an integer-valued renewal process. We perform Bernoulli percolation on the resulting graph and relate the question of whether it undergoes a non-trivial phase transition to the moments of the interarrivals of the renewal process.

This is a joint work with Marcos Sá, Remy Sanchis and Augusto Teixeira

**26 January 2024**

**Speaker: Tom Klose**

**Title: **Perturbation theory for the measure, revisited

**Abstract: **The measure is one of the easiest non-trivial examples of a Euclidean quantum field theory and has been constructed rigorously in the 1970’s. In this talk, I will revisit the perturbative expansion associated with the partition function of that measure and present a new proof that each of its coefficients converges as the ultraviolet cut-off is removed. Along the way, we will naturally encounter cumulant expansions, Feynman diagrams, and renormalisation Hopf algebras which I will introduce in a non-technical manner. This talk is based on joint work with Nils Berglund (Orléans), arXiv:2207.08555v2.

**2 February 2024 - Double feature!**

***** Exceptional location: S0.09, exceptional time: 10:00-12:00 *****

**Speaker 1: Thomas Hughes (University of Bath) **

**Title:** Stochastic PDE with the compact support property: old results and new

**Abstract: **A solution to the heat equation with non-negative, non-zero initial data is strictly positive. This property generalizes to most reasonable parabolic PDE, but not necessarily to stochastic PDE. The solution to a heat equation with multiplicative noise may be a compactly supported function, depending on the regularity of the noise coefficient. I will first discuss some classical theorems of this type when the equation has white Gaussian noise, and then discuss a recent result which proves the compact support property for solutions to a class of stochastic heat equations with white stable noise. Along the way we will develop some heuristics for why this property holds, sketch some proof techniques, and discuss connections with superprocesses.

**Speaker 2: Loïc Chaumont (University of Angers)**

**Title:** Lévy processes resurrected in the positive half-line

**Abstract: **A Lévy process resurrected in the positive half-line is a Markov process obtained by removing successively all jumps that makes it negative. A natural question, given this construction, is whether the resulting process is absorbed at 0 or not. We prove that the event corresponding to absorption at 0 satisfies a 0-1 law. Then we give conditions for absorption and conditions for non absorption bearing on the characteristics of the initial Lévy process.

This is a joint work with María Emilia Caballero and Víctor Rivero.

###### 9 February 2024

###### Speaker: Alex Watson (UCL)

**Title: **The Wiener-Hopf factorisation of Lévy processes

**Abstract: **The Wiener-Hopf factorisation of a Lévy process has two forms. The first describes how the process makes new maxima and minima, by decomposing it into two so-called 'ladder processes'. The second expresses its characteristic exponent as the product of two functions related to the ladder processes. Since the latter is analytic in nature, the question naturally arises: is such a decomposition unique? The answer has been known for killed Lévy processes since at least Rogozin's work in 1966, but appears to have remained open in general. We show that, indeed, uniqueness holds in all cases. This gives a solid foundation to the 'theory of friendship', which allows one to construct a Lévy process with known Wiener-Hopf factorisation. The results also hold for random walks. Joint work with Leif Döring (Mannheim), Mladen Savov (Sofia) and Lukas Trottner (Aarhus).

###### 16 February 2024

###### Speaker: Paul Chleboun

**Title: **Mixing times for Facilitated Exclusion Processes.

**Abstract: **We consider facilitated exclusion processes (FEP) in one dimension. These models belong to a class of kinetically constrained lattice gases. The process was introduced in the physics literature motivated by studying the active-absorbing phase transition. Under the dynamics, a particle can move to a neighbouring site provided that the target site is empty (the exclusion rule) and the other neighbour of the departure site is occupied (the constraint). These processes have recently attracted a lot of attention due to their interesting hydrodynamic limit behaviour. We examine the mixing time, the time to reach equilibrium, on an interval with closed boundaries and also with periodic boundary conditions. On the interval we observe that asymmetry significantly changes the mixing behaviour. The analysis naturally splits into examining the time to reach the ergodic configurations (irreducible component) followed by the time needed to mix on this set of configurations. This is joint work with James Ayre (Oxford).

###### 23 February 2024

###### Speaker: Leandro Chiarini (Durham University)

**Title: Branching random walk with overpopulation**

**Abstract: **We present a model of branching random walk where k particles are eliminated at unit rate if they occupy the same site, with k being a chosen parameter. We show that for k ≥ 3 on a regular tree graph, there exists a non-monotone phase transition: the process survives with positive probability for low or high branching rates, but the process dies out for intermediate branching rates. This is a joint work with Thomas Finn (Durham University) and Alexandre Stauffer (King's College London).

###### 1 March 2024

**Speaker: CANCELLED - Karen Habermann**

This talk will be moved to Term 3.

**8 March 2024**

**Speaker: Georgios Athanasopoulos**

**Title: **The exact solution of the 2D classical and 1D quantum Ising models via the Kac-Ward method

**Abstract: **Onsager proposed a closed-form expression for the free energy of the 2D classical Ising model in 1944. In 1952, Kac and Ward introduced an alternative elegant method of combinatorial nature. Kager, Lis and Meester in 2013 and Aizenman and Warzel in 2018 gave a rigorous proof of the latter. We extend their result to the triangular lattice, with coupling constants of arbitary sign. Furthermore, we derive rigorously the free energy of the 1D quantum Ising model. This is joint work with Daniel Ueltschi.

**15 March 2024**

**Speaker: Isabella Gonçalves de Alvarenga**

**Title: **Contact Process Against a Moving Barrier

**Abstract: **We consider a model representing a dynamic of a species reproducing on the right side of a randomly positioned barrier. We define it formally, and present connections of this model with the classical contact process and the multitype contact process. The main question is: given some instant of time, how close is the nearest living individual to the barrier? We prove that, under certain conditions, this distance is tight, and we sketch the proof. Under the same conditions, we also show results on the convergence in distribution of the process as seen by the barrier. We finish with some open questions about this model. Joint is an ongoing joint work with Daniel Valesin.