Applied Probability Seminar (2024-25)

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 Karen Habermann 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.

Venue: MB0.08 unless otherwise stated

Time: 11:00 unless otherwise stated

Academic Year 2024-25, Term 2

10 January 2025

Speaker: Miha Bresar

Title: Subexponential lower bounds for f-ergodic Markov processes

Abstract: In this talk I will describe a criterion for establishing lower bounds on the rate of convergence in f-variation of a continuous-time ergodic Markov process to its invariant measure. The criterion consists of novel super- and submartingale conditions for certain functionals of the Markov process. It provides a general approach for proving lower bounds on the tails of the invariant measure and the rate of convergence in f-variation of a Markov process, analogous to the widely used Lyapunov drift conditions for upper bounds. Our key technical innovation, which will be discussed in the talk, produces lower bounds on the tails of the heights and durations of the excursions from bounded sets of a continuous-time Markov process using path-wise arguments.

I will present applications of our theory to elliptic diffusions and Levy-driven stochastic differential equations with known polynomial/stretched exponential upper bounds on their rates of convergence. Our lower bounds match asymptotically the known upper bounds for these classes of models, thus establishing their rate of convergence to stationarity. The generality of our approach suggests that analogous to the Lyapunov drift conditions for upper bounds, our methods can be expected to find applications in many other settings. This is joint work with Aleks Mijatović.

17 January 2025

Speaker: Wilfrid Kendall

Title: Perfect Epidemics

Joint work with Stephen Connor (York)

Abstract: I will talk on some work I am engaged on with Stephen Connor, concerning a perfect simulation approach for making exact draws from an SIR epidemic when one observes only the removals.

24 January 2025

Speaker: Vassili Kolokoltsov

Title: Domains of quasi-attraction: Convergence rates for functional central limit theorems with stable laws and for CTRW (continuous time random walks)

Abstract: The talk will be devoted to the three new directions of research and three new groups of results: 1) Rates of convergence in the functional CLT with stable limits; 2) Domains of quasi-attraction as distributions, whose normalised sums of n i.i.d terms approach stable laws for large, but not too large n (full quantitative and qualitative description of this effect in a functional setting); 3) Rates of convergence of CTRWs, including Lévy walks and Lévy flights, to fractional evolutions. The ideas of the talk are taken from author's papers:

(1) The Rates of Convergence for Functional Limit Theorems with Stable Subordinators and for CTRW Approximations to Fractional Evolutions. Fractal Fract. (2023), 7, 335. https://doi.org/10.3390/fractalfract7040335Link opens in a new window

(2) Domains of Quasi Attraction: Why Stable Processes Are Observed in Reality? Fractal Fract. (2023), 7, 752. https://doi.org/10.3390/fractalfract7100752Link opens in a new window

(3) V. N. Kolokoltsov. Fractional Equations for the Scaling Limits of Lévy Walks With Position Depending Jump Distributions. MDPI Mathematics 2023, 11, 2566, https://www.mdpi.com/2227-7390/11/11/2566Link opens in a new window

31 January 2025

Speaker: Anne Schreuder (University of Cambridge)

Title: On Lévy-driven Loewner Evolutions

Abstract: This talk is about the behaviour of Loewner evolutions driven by a Lévy process. Schramm's celebrated version (Schramm-Loewner evolution), driven by standard Brownian motion, has been a great success for describing critical interfaces in statistical physics. Loewner evolutions with other random drivers have been proposed, for instance, as candidates for finding extremal multifractal spectra, and some tree-like growth processes in statistical physics. Questions on how the Loewner trace behaves, e.g., whether it is generated by a (discontinuous) curve, whether it is locally connected, tree-like, or forest-like, have been partially answered in the symmetric alpha-stable case. We consider the case of general Lévy drivers. Joint work with Eveliina Peltola (Bonn and Helsinki).

7 February 2025

No Seminar

14 February 2025

Speaker: Hirotatsu Nagoji (Kyoto University)

Title: Singularity of solutions to singular SPDEs

Abstract: In this talk, we discuss the condition for the marginal distribution of the solution to singular SPDEs on the d-dimensional torus to be singular with respect to the Gaussian law induced by the linearized equation. As applications of our result, we see the singularity of the Phi^4_3-measure with respect to the Gaussian free field measure and the border of parameters for the fractional Phi^4-measure to be singular with respect to the base Gaussian measure. This talk is based on a joint work with Martin Hairer and Seiichiro Kusuoka.

21 February 2025

Speaker: Isabella Gonçalves de Alvarenga

Title: A Version of Sharkovsky’s Theorem for Small Random Perturbations

Abstract: Sharkovsky’s Theorem establishes a non-usual ordering of natural numbers, in such a way that, if a dynamical system has a periodic point of a given period, it must also have periodic points of all larger periods in that ordering. But what happens to this result when the system undergoes small random perturbations? In this talk, we explore this question by introducing the framework of random dynamical systems and discussing suitable definitions of periodicity in this context. We will state a version of Sharkovsky’s Theorem for a specific class of random dynamical systems and outline the main mathematical tool used in its proof.

28 February 2025

Speaker: Sam Olesker-Taylor

Title: A Randomised Approach to Sorting

Abstract: We introduce and analyse a new, extremely simple, randomised sorting algorithm:

choose a pair of indices {i, j} according to some distribution q;
sort the elements in positions i and j of the array in ascending order.

We prove that q_{i, j} ∝ 1/|j−i|, the harmonic sorter, yields an order -n(log n)² sorting time.

The sorter trivially parallelises in the asynchronous setting, yielding a linear speed-up. We also exhibit a low-communication, synchronous version with a linear speed-up.

7 March 2025

Speaker: Konrad Anand (University of Edinburgh)

Title: Sink-free orientations: a local sampler with applications

Abstract: On graphs of minimum degree at least 3, we show that there is an efficient deterministic approximate counting algorithm, a near-linear time sampling algorithm, and an efficient randomised approximate counting algorithm for sink-free orientations. All three algorithms are based on a local implementation of the sink popping method (Cohn, Pemantle, and Propp, 2002) under the partial rejection sampling framework (Guo, Jerrum, and Liu, 2019).

14 March 2025

No Seminar

Past Seminars: Academic Year 2024-25, Term 1

4 October 2024

Speaker: Oleg Zaboronski

Title: Work in progress: (i) Elliptic SPDE's and Dyson Brownian motions; (ii) from KPZ to TWD bypassing ASEP

Abstract: (i) It has been known since the work of Parisi-Sourlas in the 80's and its later rigorization by Landau et al and Gubinelli et all that elliptic SPDE's driven by white noise has certain exactly computable marginals. We find that particularly interesting examples of such marginals appear if the solution takes values in an interesting manifold. Unfortunately, a rigorous argument applicable to such situations is still lacking; (ii) There is a short derivation of the one-point height distribution for KPZ, which does not require either discretisation or the analysis of bound states for the attractive delta-Bose gas. Unfortunately, it relies on a non-rigorous analytical continuation from the KPZ with imaginary noise.

11 October 2024

Speaker: John Fernley

Title: Targeted immunization thresholds for the contact process on power-law trees

Abstract: Scale-free configuration models are intimately connected to power law Galton–Watson trees. It is known that contact process epidemics can propagate on these trees and therefore these networks with arbitrarily small infection rate, and this continues to be true after uniformly immunizing a small positive proportion of vertices. So, we instead immunize those with largest degree: above a threshold for the maximum permitted degree, we discover the epidemic with immunization has survival probability similar to without, by duality corresponding to comparable metastable density. With maximal degree below a threshold on the same order, this survival probability is severely reduced or zero. Based on joint work with Emmanuel Jacob (ENS de Lyon).

18 October 2024

Speaker: Hong Duong (University of Birmingham)

Title: Ergodicity and asymptotic limits for the generalized/relativistic Langevin dynamics

Abstract: We consider systems of interacting particles governed by the generalized/relativistic Langevin dynamics in the presence of singular repulsive interacting forces. For each system, we establish a rate of convergence toward the unique invariant probability measure, which relies on novel construction of Lyapunov functions. We also study asymptotic limits of these systems when passing to the limit the interested parameters (the small-mass limit and Newtonian limit, respectively).

This talk is based on joint works with H. D. Nguyen (University of Tennessee).

References

[1] M. H. Duong and H. D. Nguyen. Asymptotic analysis for the generalized Langevin equation with singular potentials. Journal of Nonlinear Science, Volume 34, article number 62, 2024.

[2] M. H. Duong and H. D. Nguyen. Trend to equilibrium and Newtonian limit for the relativistic Langevin equation with singular potentials. arXiv:2409.05645Link opens in a new window, 2024.

25 October 2024

Speaker: Lukas Gräfner

Title: Energy solutions and singular S(P)DEs: Beyond the subcritical regime

Abstract: Discovered by P. Gonçalves and M. Jara, energy solutions naturally arise as scaling limits of fluctuations of interacting particle systems and solve certain singular stochastic PDEs (SPDEs). In a general Hilbert space setting, we prove a weak well-posedness result for energy solutions to equations with quadratic non-linearity and Gaussian reference measure. A second result concerns well-posedness for SDEs with distributional drift.

Compared to popular pathwise approaches for SPDEs, we can bypass the contraint of scaling subcriticality, enabling us to tackle critical and supercritical cases. Additionally, because our framework is probabilistically weak, building on Markov generators and certain martingale problems, insight into the law of the solution is more immediate.

Applications include critical stochastic surface quasi-geostrophic equations and critical fractional stochastic Burgers equations, SDEs with supercritical distributional drift and certain infinite particle systems with non-summable interaction.

Based on joint works with Nicolas Perkowski and Shyam Popat.

1 November 2024

Speaker: Erik Jansson (Chalmers University of Technology)

Title: An exponential-free structure preserving integrator for stochastic Lie-Poisson systems

Abstract: Lie-Poisson systems appear in many areas of physics. They exhibit a strong geometric structure, as they evolve on coadjoint orbits and have conserved quantities. By adding noise, it is possible to account for uncertainties and unresolved smaller scales. To ensure that the geometric properties of the equations, that for instance guarantees existence and uniqueness, are preserved, noise must be added in the right way. An important question is then how to numerically integrate these systems in a structure-preserving manner. In this talk, we describe an integrator for a class of stochastic Lie-Poisson systems driven by Stratonovich noise. We describe how its derivation follows from discrete Lie-Poisson reduction of the symplectic midpoint scheme for stochastic Hamiltonian systems. We discuss on how almost sure preservation of Casimir functions and coadjoint orbits under the numerical flow and strong and weak convergence rates of the proposed method may be proven.

8 November 2024

Speaker: Daniel Valesin

Title: The interchange-and-contact process

Abstract: We introduce a process called the interchange-and-contact process, which is defined on an arbitrary graph as follows. At any point in time, vertices are in one of three states: empty, occupied by a healthy individual, or occupied by an infected individual. Infected individuals recover with rate 1, and also infect healthy individuals in neighboring vertices with rate $\lambda$ . Additionally, each edge has a clock with rate $v$ , and when this clock rings, the states of the two vertices of the edge are exchanged. This means that particles perform an interchange process with rate $v$ , moving around and, when infected, carrying the infection with them. We study this process on $\mathbb{Z}^d$ , with an initial configuration where there is an infected particle at the origin, and every other vertex contains a healthy particle with probability $p$ and is empty with probability $1-p$ . We define $\lambda_c(v, p)$ as the infimum of the values of $\lambda$ for which the process survives with positive probability. We prove results about the asymptotic behavior of $\lambda_c$ when $p$ is fixed and $v$ is taken to zero and to infinity. Joint work with Daniel Ungaretti, Marcelo Hilário and Maria Eulalia Vares.

15 November 2024

Speaker: Elizabeth Baker (University of Copenhagen)

Title: Conditioning infinite-dimensional stochastic processes with applications to shapes

Abstract: Stochastic shape matching conditions stochastic processes to align with specific boundary shapes with applications in, for example, evolutionary biology. This talk explores methods to condition infinite-dimensional stochastic differential equations (SDEs) on a specific end-point at a given time, effectively creating an infinite-dimensional SDE bridge. To this end, we develop an infinite-dimensional analogue of Doob’s h-transform and combine it with score-matching techniques to sample from the bridged SDE. We demonstrate an application in evolutionary biology, where infinite-dimensional SDE bridges model how species' shapes evolve over time.

22 November 2024

Speaker: Isao Sauzedde

Title: Geometric formulas and Amperean area for planar Brownian loops

Abstract: The Stokes theorem asserts that some area delimited by a smooth curve can be computed as a line integral along that curve. The first part of this talk, which is partly inspired from geometric and rough path theoretic considerations, will be dedicated to explain what of the Stokes theorem remains when we consider Young or stochastic integration. In the second part of this talk, this will be used to present the renormalised Amperean area of a Brownian loop, an object I have recently constructed for its connection to the abelian Yang-Mills-Higgs model. The talk is based on two papers, "Lévy area without approximation", and "Renormalised Amperean of Brownian loops and Symanzik representation of the 2D Yang-Mills-Higgs fields", available on my webpage.

29 November 2024

Speaker: Osvaldo Angtuncio Hernández (CIMAT)

Title: The coalescent process of a multitype Bienaymé-Galton-Watson tree

Abstract: In this talk we discuss the genealogy of a sample of $k>=2$ individuals, from a continuos-time multitype Bienaymé-Galton-Watson tree that survives up to a large time. The technique used is via a change of measure, extending the work of Harris, Johnston and Roberts (2020). We will also discuss the law of the times when the particles in the sample coalesce, the types of the individuals when coalescing, and scaling limits of such laws. This is a joint work with Juan Carlos Pardo and Simon Harris.

6 December 2024

Speaker: Roger Tribe

Title: More unfinished results on Brownian Matrix Evolutions

Abstract: A Brownian Matrix process is a random matrix where the entries are defined using Brownian motions. Various cases are standard, depending on whether the matrix is real/complex and symmetric/non-symmetric. The aim is to describe the eigenvalues and eigenvectors evolving over time.

In the most famous case the eigenvalues follow Dyson Brownian motions and their law is well understood. Less is known in other cases, and I will discuss some bricks in the construction of a better understanding: multi-time distributions; overlaps of left/right eigenvectors.