Probability

The randomised reproducing graph is a random graph model, with some
similarities to duplication graphs, which shows a number of interesting
phase transitions. This talk will describe work in progress constructing
a local limit for this model. (Joint with Simon Irons)

In this talk I will present a result on the construction of the unique invariant measure of the singular SDE with fractional Brownian noise (fBm), equipped with a linear damping. We build up on the theory of regularisation by noise, developed in recent years by Catellier, Gubinelli, Galeati and many others, and merge it with ergodic theory of fractional SDEs, studied by Hairer and his coauthors. We establish tightness in the usual regime of weak existence ($\alpha > 1/2-1/(2H)$, where $\alpha$ is Besov-Holder regularity of the drift and H is Hurst index of fBm), and uniqueness of invariant measure under usual condition on well-posedness for singular SDEs driven by fractional Brownian motion ($\alpha > 1 - 1/(2H)$). To this end we employ a modification of stochastic sewing, which also allows us to show Gaussian tails of the solution. Our approach does not require any assumption on the size of the drift with respect to the damping strength.

NOTES: unusual time.
We study a random tree model known as the Preferential Attachment tree with Vertex Death. Here, one can both add vertices to the tree as well as kill vertices. This model mimics the non-monotone growth of real-world networks, absent in classical preferential attachment models. One initialises the tree with a single root vertex labelled 1. At every step n, either a new vertex labelled $n+1$ is added to the tree and connected to an already present alive vertex, selected preferentially according to a function $b$, or an already present vertex is selected preferentially according to a function $d$ and killed. Killed vertices can make no new connections.

We are interested in the behaviour of the richest alive vertex $I_n$ (with the largest degree) and the oldest alive vertex $ O_n$ (with the smallest label) in the tree $T_n$ as n tends to infinity. When $ I_n$ converges almost surely, we say that a persistent hub exists. When $I_n$ does not converge but $I_n/O_n$is tight, we say that persistence occurs, and when $I_n/O_n$ diverges to infinity we say lack of persistence occurs. We uncover three distinct regimes in which behaviour is different: (1) The Infinite Lifetime regime, where we provide conditions under which a persistent hub exists almost surely. (2) The Rich Are Old regime, where we provide conditions under which either persistence or lack of persistence occurs. (3) The Rich Die Young regime, where lack of persistence always occurs. We shall discuss how the three regimes can be identified and what drives the behaviour observed in each regime. Partially joint work with Markus Heydenreich.

In this talk we consider the four-waves spatially homogeneous kinetic equation arising in weak wave turbulence theory from one-dimensional microscopic oscillator chains. This equation is sometimes referred to as the Phonon Boltzmann Equation. I will discuss the entropy maximisation problem, the collisional invariants, and properties of solutions of the kinetic equation near the Rayleigh-Jeans (RJ) thermodynamic equilibria, in the case where the microscopic model is the Fermi-Pasta-Ulam-Tsingou (FPUT) chain. This is based on joint works with Pierre Germain (Imperial College London), Joonhyun La (KIAS) and with Miguel Escobedo (Bilbao).

 In this talk, I will present recent results on the long-time stability of additive stochastic differential equations driven by fractional Brownian motion. The drift is decomposed into a singularity (that can be a geniune distribution) and a Lipschitz dissipativity. I will begin with motivations and examples where such equations naturally arise, before introducing the main analytical ideas used in the proofs, in particular regularisation by noise and comparison with the Ornstein–Uhlenbeck process. These tools allow us to establish a uniform-in-time bound on the moments of the solution together with a stability result with respect to the initial condition.
I will then briefly recall a numerical scheme for approximating solutions, which will serve as a basis to discuss ongoing projects and open questions, including coupling arguments under general dissipativity assumptions, approximation of Gibbs measures, and numerical approximations for the stochastic Allen–Cahn equation with singular drift.

In this talk, I will present a data-driven framework for incorporating Riemannian geometry into statistical modelling, with a particular focus on Gaussian process (GP) regression. High-dimensional data encountered in practice—such as point clouds, remote sensing measurements, or image collections—often concentrate near lower-dimensional manifolds with non-Euclidean geometry. Standard Euclidean GPs ignore this structure, leading to poor predictive performance and misleading uncertainty quantification. Our approach constructs GPs on complex or unknown manifolds by first learning a probabilistic atlas of the latent geometry, using tools such as autoencoders and latent variable models, and then defining stochastic processes that respect this geometry. This perspective connects ideas from stochastic differential equations on manifolds with statistical learning, allowing principled modelling of manifold-valued data. I will illustrate the method through simulations on the torus and applications to remote sensing of chlorophyll concentration in the Aral Sea, and Image point clouds. The talk will give an overview of how data-driven Riemannian geometry can inform statistical modelling more broadly, including directions towards diffusion based generative modelling and finding the shortest path on point cloud, while highlighting the role of stochastic processes in bridging geometry, statistics, and machine learning.

In this work, we seek an optimal short-term, continuous-time power procurement schedule to minimise operating expenditure and carbon footprint of cellular wireless networks equipped with energy storage capacity, and hybrid energy systems consisting of uncertain renewable energy sources. The network operator needs to ensure a certain QoS constraint with high probability. This probabilistic constraint prevents us from using dynamic programming to solve the continuous-time stochastic optimal control problem. We introduce a time-continuous Lagrangian relaxation approach tailored for real-time power procurement in cellular networks, overcoming tractability issues associated with probabilistic QoS constraints. The numerical solution procedure involves building an efficient upwind finite difference solver for the Hamilton--Jacobi--Bellman equation corresponding to the relaxed problem, and an effective stochastic sub-gradient method to efficiently navigate the stochastic problem structure. The proposed numerical approach is applied on a model cellular network base station based on the German power system and daily cellular traffic data. Our approach demonstrates computational efficiency, providing near-optimal solutions in practical timeframes.

Consider the complete bipartite graph with $n$ black vertices and $m=an$ white vertices. Edges in the graph can only exist between vertices of different colours. Equip the $mn/2$ edges of the graph with i.i.d Uniform (0, 1) weights. The minimum spanning tree of the graph with respect to these weights can be constructed using the so-called Prim’s Algorithm, which outputs a sequence of increasing subtrees $T_k$, where $T_k$ is a bipartite tree of $k$ vertices. Denote by $\rho_k$ the ratio of white vs black vertices in $T_k$. In a joint work with Félix Kahane, we give a complete characterisation of the asymptotic behaviours of $rho_k$ as both $k$ and $n$ tend to infinity (possibly with different speeds). In particular, our result implies that unless $m=n$ or $k=m+n$, the colour ratio we observe in $T_k$ converges to a quantity different from $m/n=a$. 

In this talk, a new approach for solving the problems of pricing and hedging derivatives is introduced in a general frictionless market setting. The method is applicable even in cases where an equivalent local martingale measure fails to exist. Our main results include a new superhedging duality for American options when wealth processes can be negative and trading strategies are subject to a cone constraint. This answers one of the questions raised by Fernholz, Karatzas and Kardaras. This is joint with Miklos Rasonyi

In this talk I will give an overview over recent applications of Energy solutions from the field of singular SPDEs to singular SDEs with distributional drift.

In the first part, I will present weak well-posedness results for Energy solutions in the case of, up to perturbations, divergence-free drifts which lie in certain scaling-supercritical function spaces and can have even more singular but local blow-ups.

In the second part and depending on time, I will present a recent construction of the 1-D self-repelling Brownian polymer (SRBP), which is formally the solution to an SDE with a certain path-dependent distributional drift. Our approach exploits the fact that the SDE is formally in one-to-one correspondence with a singular SPDE that can be solved uniquely in the sense of Energy solutions. We then give a unique dynamic characterization of the law of the SRBP and show that the process is superdiffusive and not self-avoiding.

Based on joint works with Nicolas Perkowski and Harry Giles.