Abstract

Leonid Bogachev, University of Leeds

Title: Balanced pantograph equation revisited

Abstract: The \emph{balanced pantograph equation} is a functional-differential equation with rescaling, e.g.\ $y'(x)+y(x)=\frac12\myp y(2x)+\frac12\myp y(x/2)$ ($x\in\RR$). Of course, any constant $y(x)=\const$ is a solution of such an equation, but are there \emph{non-constant} bounded (and continuous) solutions? In this talk, we discuss this question for a more general equation $y(x)=\EE\bigl(y(\alpha (x-\beta))\bigr)$, with random $(\alpha,\beta)$. The critical parameter here is $K=\EE\bigl(\ln|\alpha|\bigr)$; e.g., it is easy to prove that if $K<0$ (subcritical case) then there are no non-constant bounded solutions. The critical ($K=0$) and supercritical ($K>0$) cases are more difficult, and we discuss some partial results in this direction. The approach is based on interpreting $y(x)$ as a harmonic function of the associated Markov chain $(X_n)$ $(X_0=x)$, and exploiting the iterated equation $y(x)=\EE_x\bigl(y(X_\tau)\bigr)$ (with a suitable stopping time~$\tau$) by virtue of Doob's optional stopping theorem applied to the martingale~$y(X_n)$.

The talk is based on joint work with Gregory Derfel (Beer Sheva) and Stanislav Molchanov (UNC-Charlotte).

Youness Boutaib, University of Liverpool

Title: Separation Capacity in Random Linear Reservoirs

Abstract: Recurrent neural networks (RNNs) constitute the simplest machine learning paradigm that is able to handle variable-length data sequences while tracking long-term dependencies and taking into account the temporal order of the received information. Reservoir computing (i.e. randomly choosing the connectivity matrix of the RNN) is a paradigm based on the idea that universal approximation properties can be achieved for several dynamical systems without the need to optimise all parameters. This technique simplifies the training of RNNs and has shown exceptional performance in a variety of tasks. Despite this, there is a fundamental lack in the mathematical understanding of the success of such approach. 

In this work, we explain this success by the separation capacity of such random reservoirs. In particular, we show that the expected separation capacity is characterised by the spectral analysis of a generalised matrix of moments - a classical object of interest in random matrix theory. As a byproduct of this result, we discuss how the parameters of the problem (dimension of the reservoir, geometry of the classes of time-series, the choice of the probability distribution, symmetries, etc.) impact the performance of the architecture. The analysis is complemented by probabilistic bounds and empirical insights, shedding light on the design of effective reservoir architectures for temporal learning tasks.

Giuliano Casale, Imperial College London

Title: Advances in matrix-analytic methods for fitting and analysing non-exponential queueing models

Abstract: This talk discusses our recent results in the area of matrix-analytic methods for queueing theory, focusing on two problems: (i) fitting matrix representations used to specify service processes in quasi-birth-death models and (ii) analysing queueing systems that can leverage these distributions and their extensions. On the first topic, we discuss some novel approaches that we have recently proposed based on Kronecker product compositions for fitting matrix-analytic distributions from measured datasets, improving accuracy and computational efficiency in obtaining high-order representations compared to state-of-the-art methods. On the second topic, we present  techniques that, again leveraging Kronecker product-composition, infer phase-type (PH) distributions from queue-length observations. Such method relies on a novel approximation for the underpinning quasi-birth death (QBD) process that significantly reduces the cost of the inference and that may also be used in general for approximate evaluation of PH/PH/1 queues. The talk is based on joint work with Julianna Bor, William J. Knottenbelt, Evgenia Smirni, and Andreas Stathopoulos.

Denis Denisov, University of Manchester

Title: Stable random walks in cones

Abstract: I will consider a multidimensional random walk killed on leaving a right circular cone with a distribution of increments belonging to the normal domain of attraction of an α-stable and rotationally-invariant law. I will discuss construction of a positive harmonic function for this random walk and related questions of   asymptotic tail behavior of exit time of the random walk from this cone and conditional functional limit theorems.

The talk is based joint work with W.Cygan, Z. Palmowski and V. Wachtel.

ArXiv preprint:  https://arxiv.org/abs/2409.18200

Goncalo Dos Reis, University of Edinburgh

Title: Simulation of Interacting particle systems and mean-field SDEs: some recent results

Abstract: We review two results in the simulation for SDE of McKean-Vlasov type (MV-SDE). The first block of results addresses simulation of MV-SDEs having super-linear growth in the spatial and the interaction component in the drift, and non-constant Lipschitz diffusion coefficient. The 2nd block is far more curious. It addresses the study the weak convergence behaviour of the Leimkuhler–Matthews method, a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the stationary distribution of a one-dimensional McKean–Vlasov Stochastic Differential Equation (MV-SDE). The particular class under study is known as mean-field (overdamped) Langevin equations (MFL). We provide weak and strong error results for the scheme in both finite and infinite time. We work under a strong convexity assumption. Based on a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (of weak order convergence rate 3/2) than the standard Euler method (of weak order 1). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension of the IPS and this includes establishing uniform-in-time decay estimates for moments of the IPS, the Kolmogorov backward equation and their derivatives. The theoretical findings are supported by numerical tests.

This presentation is (loosely) based on the joint work [1], [2].

References:

[1] X. Chen, G. dos Reis, Wolfgang Stockinger, and Zac Wilde. “Improved weak convergence

for the long time simulation of Mean-field Langevin equations.” arXiv preprint arXiv:2405.01346 (2024).

[2] X. Chen, G. dos Reis. "Euler simulation of interacting particle systems and McKean-Vlasov SDEs with fully superlinear growth drifts in space and interaction" IMA Journal of Numerical Analysis, 2022, Vol. 42, No. 1.

Sergey Foss, Heriot-Watt University

Title: Heavy-tail asymptotics for the length of a busy period in a generalised Jackson network

Abstract: We consider a generalised Jackson network with finitely many servers, a renewal  input and i.i.d. service times at each queue. We assume the network to be stable and, in addition, the distribution of the inter-arrival times to have unbounded support. This implies that the length of a typical busy period B, which is the time between two successive idle periods, is finite a.s. and has a finite mean. We assume that the distributions of the service times with the heaviest tails belong to the class of so-called intermediate regularly varying distributions. We obtain the exact asymptotics for the probability P(B>x) as x grows to infinity. For that, we show that the Principle of a Single Big Jump holds: B takes a large value mainly due to a single unusually large service time.

The talk is based on a joint work with Linglong Yuan and Masakiyo Miyazawa.

Ayalvadi Ganesh, University of Bristol

Title: Social learning in multi-agent multi-armed bandits

Abstract: Consider a large number of agents, N, faced with the problem of choosing amongst a large number of options, K. The problem occurs repeatedly, and every time an agent chooses an option, it receives a random reward or payoff whose distribution depends on the option but not on the agent. The goal is to maximize the long-run payoff. The problem involves a trade-off between exploitation - choosing the option currently believed to be the best - and exploration - choosing possibly sub-optimal options in order to gain more information about their payoffs. The challenge is to optimize this trade-off.

If there were a single agent, then this is an instance of the multi-armed bandit problem with K arms., which has been studied extensively for decades. If no communication is allowed between agents, then it is N parallel instances of the multi-armed bandit problem.  If there are no communication constraints, then the agents act in aggregate as if they were a single agent. We are interested in the intermediate case where limited communication is allowed. We show that, even with limited communication, in the long run the system behaves in aggregate as if there were a single agent, i.e., as if there were no communication constraints.

Joint work with Abhishek Sankararaman, Ronshee Chawla, Sanjay Shakkottai, Conor Newton and Henry Reeve.

Robert Gaunt, University of Manchester

Title: Convex distance bounds for the stable central limit theorem via Stein’s method.

Abstract: The stable central limit theorem is a generalisation of the classical central limit theorem that relaxes the finite variance assumption. Quantifying the quality of the distributional approximation of the stable central limit theorem is a fundamental problem in probability, which has seen a number of contributions over years. In particular, there have been recent advances on this problem using Stein's method in which the stable central limit theorem is quantified with respect to Wasserstein-type distances. In this talk, we use Stein's method to derive optimal rates of convergence in the multivariate stable central limit theorem in terms of the convex distance, which is a natural multivariate generalisation of the Kolmogorov distance. Our bounds are derived by combining the recent advances on Stein's method for alpha-stable distributions together with a powerful smoothing technique.

This is a work in progress with Zixin Ye.

Wilfrid Kendall, University of Warwick

Title: Perfect Epidemics (joint work with Stephen Connor)

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. (Joint work with Stephen Connor.)

Dmitry Korshunov, University of Lancaster

Title: Tail asymptotics for a random walk stopped at random time

Abstract: We consider a random walk $\{S_n\}$ with a finite positive drift that is stopped at a random time $\tau$ having intermediate regularly varying distribution. We assume that the jump distribution is lighter than that of $\tau$. Under these conditions we show that the distributions of $S_{\tau}$ and of $M_{\tau} = \max_{k\le \tau} S_k$ are tail equivalent and their common tail asymptotics are determined by that of $\tau$, while $\{S_n\}$ only contributes via the law of large numbers. We consider the two cases, where $\tau$ is a random time independent of the future increments of the random walk or $\tau$ is arbitrary dependent of $S_n$.

In the course of the talk we will discuss why quite often the class of intermediate regularly varying distributions is more appropriate than the class of regularly varying distributions. We will also discuss in detail the importance of the notion of a random time independent of the future, which includes both stopping times and independent times.

Joint work with Sergey Foss (Edinburgh).

Jere Koskela, Newcastle University

Title: Jump diffusion models in proton beam therapy

Abstract: Proton beam radiotherapy is a key tool in cancer treatment. Unlike photons, protons interact weakly with surrounding tissue and are absorbed after a predictable path length, allowing more precise control over where their energy dose is delivered. Broadly speaking, state of the art treatment planning tools come in two varieties: highly detailed but computationally expensive Monte Carlo algorithms which simulate every physical interaction between a proton and the surrounding tissue, or simple curves fitted to measurement data, often assuming one-dimensional beam. I'll describe a jump diffusion model which is intermediate between these two regimes. The diffusion coefficient tractably absorbs many of the smaller nuclear interactions which greatly speeds up simulations, but the model retains three spatial dimensions and an explicit connection to the underlying physics, facilitating calibration and uncertainty quantification. This is joint and ongoing work with the MathRad team (especially Alastair Crossley, Karen Habermann, Emma Horton, Andreas Kyprianou, and Sarah Osman).

Andreas Kyprianou, University of Warwick

Title: The Brownian marble

Abstract: Fundamentally motivated by the two opposing phenomena of fragmentation and coalescence, we introduce a new stochastic object which is both a process and a geometry. The Brownian marble is built from coalescing Brownian motions on the real line, with further coalescing Brownian motions introduced through time in the gaps between yet to coalesce Brownian paths. The instantaneous rate at which we introduce more Brownian paths is given by $\lambda/g^2$ where $g$ is the gap between two adjacent existing Brownian paths. We show that the process “comes down from infinity” when $0<\lambda<6$ and the resulting space-time graph of the process is a strict subset of the Brownian Web on $\mathbb{R} x [0,\infty)$. When $\lambda\geq 6$, the resulting process “does not come down from infinity” and the resulting range of the process agrees with the Brownian Web.

                                                                                                                                                                     The Brownian marble

Khoa Lê, University of Leeds 

Title: Strong regularization of differential equations with integrable drifts by fractional noise

Abstract: We consider stochastic differential equations with integrable time-dependent  drift driven by additive fractional Brownian noise whose Hurst parameter is less than 1/2.  Under some subcriticality conditions, it is shown that such equation has a unique pathwise solution. Furthermore, stability with respect to all parameters is established.  Our strong uniqueness result can be considered as an extension of that from Krylov and R\"ockner (2005) for Brownian motion, it also improves upon previous results of Nualart and Ouknine (2003)  for dimension one. Our methods are built upon Lyons' rough path theory and the stochastic sewing lemma, complemented by the quantitative John--Nirenberg inequality for stochastic processes of vanishing mean oscillation. Joint work with Oleg Butskovsky and Toyomu Matsuda.

Mateusz Majka, Heriot-Watt University

Title: Linear convergence of proximal descent schemes on the Wasserstein space

Abstract: In this talk, I will discuss how to use ideas at the intersection of probability, optimization and optimal transport, to tackle the problem of minimizing functions defined on the space of probability measures. The focus will be on proximal descent methods, inspired by the minimizing movement scheme introduced by Jordan, Kinderlehrer and Otto, for optimizing entropy-regularized functionals on the Wasserstein space. I will explain how to establish linear convergence under flat convexity assumptions, thereby relaxing the common reliance on geodesic convexity. The presented approach utilizes a uniform logarithmic Sobolev inequality (LSI) and the entropy "sandwich" lemma. The talk is based on a joint work with Razvan-Andrei Lascu, David Šiška and Łukasz Szpruch.

Aleksandar Mijatović, University of Warwick

Title: Non-asymptotic bounds on the forward process in denoising diffusions 

Abstract: Denoising diffusion probabilistic models (DDPMs) represent a recent advance in generative modelling that has delivered state-of-the-art results across many domains of applications. Despite their success, a rigorous theoretical understanding of the error within DDPMs, particularly the non-asymptotic bounds required for the comparison of their efficiency, remain scarce. Making minimal assumptions on the initial data distribution, allowing for example the manifold hypothesis, in this talk I will present explicit non-asymptotic bounds on the forward diffusion error in total variation (TV), expressed as a function of the terminal time T. 

The key idea is to parametrise multi-modal data distributions in terms of the distance $R$ to their furthest modes and consider forward diffusions with additive and multiplicative noise. Our analysis rigorously proves that, under mild assumptions, the canonical choice of the Ornstein-Uhlenbeck (OU) process cannot be significantly improved in terms of reducing the terminal time $T$ as a function of $R$ and error tolerance  $\epsilon>0$. Motivated by data distributions arising in generative modelling, we also establish a cut-off like phenomenon (as $R\to\infty$) for the convergence to its invariant measure in TV of an OU process, initialized at a multi-modal distribution with maximal mode distance $R$. This is joint work with Miha Brešar at Warwick. 

John Moriarty, Queen Mary University of London

Title: The one-shot problem: Solution to an open question of finite-fuel singular control with discretionary stopping

Abstract: I will present a novel 'one-shot' solution technique resolving a singular control problem of Karatzas et al. which has been open for a couple of decades. Unexpectedly, although the problem is convex, its waiting/inaction region is complex (that is, not necessarily connected). The state process may even visit both connected components, by a jump from one to the other. The analysis reveals more generally that when fuel is limited, perhaps contrary to intuition, the solution without fuel is not necessarily indicative of the solution for small amounts of fuel. To resolve this, we recommend solving the 'one-shot' problem, which is one of optimal stopping, prior to employing the usual 'guess and verify' solution approach. This is based on the joint work https://arxiv.org/abs/2411.04301  with Neofytos Rodosthenous. 

Arpan Mukhopadhyay, University of Warwick

Title: Load Balancing in Heterogeneous Systems 

Abstract: We consider the problem of assigning jobs with exponentially distributed job lengths arriving according to a Poisson process to a set of parallel first-come-first-serve queues. Each job must be assigned immediately upon arrival to a queue where it must stay until it is processed and leaves the system. The goal is to design an assignment policy that minimises the mean response time (i.e., the mean sojourn time) of jobs. When the queues have identical processing speeds, the solution is the well-known Join-the-Shortest-Queue policy wherein each job is assigned to the queue having the smallest queue length. However, the optimal policy remains unknown for the case where the queues have heterogeneous processing speeds. We will discuss some recent results where a speed-aware variant of the JSQ policy has been shown to be asymptotically optimal in specific scaling regimes such as the mean-field and the Halfin-Whitt regime. The talk, based on joint work with Sanidhay Bhambay (Durham) and Burak Buke (Edinburgh), will explore some of the key technical challenges in establishing these optimality results.

Zhongmin Qian, University of Oxford

Title: Functional integral representation for flows past a solid wall

Abstract:  In this talk I present a new stochastic integral representation for the solutions of incompressible viscous fluid flows, established by using diffusion bridge duality. I then explain how to apply this integral representation to Large Eddy Simulation for boundary turbulent flows.

Gesine Reinert, University of Oxford

Title: Exponential random graph models analysed using Stein's method

Abstract: Exponential random graph models are popular models for the analysis of social networks, due to their flexibility and to their ability of capturing some complex dependence in networks. Mathematically, their analysis is hindered by their probability distribution given only up to a normalising constant. Moreover usually only one observation from the network is available. Using ideas from Stein's method we can make progress though; we can characterize the model using a Stein operator and devise a kernelized Stein goodness-of-fit test based on this characterisation. We generate synthetic samples from the model underlying  the observed data by mimicking the Stein operator dynamics. Finally, in a simplified model we  use Stein equations to estimate the model parameters via a so-called Stein estimator. This is based on joint works with Nathan Ross, Wenkai Xu, and Adrian Fischer

Perla Sousi, University of Cambridge

Title: The cutoff phenomenon for random walks

Abstract: I will talk about the mixing time which is the time it takes for a random walk to reach equilibrium. My focus will be on the cutoff phenomenon observed when the transition to equilibrium happens abruptly in time. I will survey the developments in the last 30 years and present a recent universality result for graphs with a random matching that was obtained in collaboration with J. Hermon and A. Sly.

Thirupathaiah Vasantam, Durham University

Title: Service-the-Longest-Queue Among $d$ Choices Policy for Quantum Entanglement Switching

Abstract: An Entanglement Generation Switch (EGS) is a quantum network hub that provides entangled states to a set of connected nodes by enabling them to share a limited number of hub resources. As entanglement requests arrive, they join dedicated queues corresponding to the nodes from which they originate. We propose a load-balancing policy wherein the EGS queries nodes for entanglement requests by randomly sampling d of all available request queues and choosing the longest of these to service. This policy is an instance of the well-known power-of- d-choices paradigm previously introduced for classical systems such as data-centers. In contrast to previous models, however, we place queues at nodes instead of directly at the EGS, which offers some practical advantages. Additionally, we incorporate a tunable back-off mechanism into our load-balancing scheme to reduce the classical communication load in the network. To study the policy, we consider a homogeneous star network topology that has the EGS at its center, and model it as a queueing system with requests that arrive according to a Poisson process and whose service times are exponentially distributed. We provide an asymptotic analysis of the system by deriving a set of differential equations that describe the dynamics of the mean-field limit and provide expressions for the corresponding unique equilibrium state. Consistent with analogous results from randomized load- balancing for classical systems, we observe a significant decrease in the average request processing time when the number of choices d increases from one to two during the sampling process, with diminishing returns for a higher number of choices. We also observe that our mean-field model provides a good approximation to study even moderately-sized systems. This is a joint work with Guo Xian Yau (TU Delft), Gayane Vardoyan (UMass, Amherst). 

Andrew Wade, Durham University

Title: Perimeter of the convex hull of hyperbolic Brownian motion

Abstract: In this talk, I will relate the expected hyperbolic length of the perimeter of the convex hull of the trajectory of Brownian motion in the hyperbolic plane to an expectation of a certain exponential functional of a one-dimensional real-valued Brownian motion and hence derive small- and large-time asymptotics for the expected hyperbolic perimeter. In contrast to the case of Euclidean Brownian motion with non-zero drift, the large-time asymptotics are a factor of two greater than the lower bound implied by the fact that the convex hull includes the hyperbolic line segment from the origin to the endpoint of the hyperbolic Brownian motion. This is joint work with Chinmoy Bhattacharjee and Rik Versendaal.