Probability

Poisson-Voronoi percolation can be defined on any metric space $(M,d)$ with an infinite Radon measure $\mu$ as follows. For $\lambda>0$, consider a Poisson point process of intensity $\lambda\mu$ and associate to each point of the process its Voronoi cell, that is the set of all points in $M$ closer to this point than to any other point of the process. For $p\in(0,1)$, independently color each cell black with probability $p$ and consider the union of black cells.

For fixed $\lambda$, there is a phase transition reminiscent of classical Bernoulli percolation with a critical probability $p_c(\lambda)$, which is the infimal value of $p$ such that an unbounded cluster exists, and a possible different uniqueness threshold $p_u(\lambda)$, which is the infimal value of $p$ such that a unique unbounded cluster exists.

In this talk, I will review results about this model in Euclidean space $\mathbb R^d$ and also hyperbolic space $\mathbb H^d$ where interesting behavior of the functions $p_c(\lambda)$ and $p_u(\lambda)$ emerges. I will then discuss a new phenomenon, namely the fact that $p_u(\lambda)$ can become arbitrarily small, and its applications, in several situations of interest.

The talk is based on joint works with Jan Grebík and with Matteo D'Achille, Jan Grebík, Ali Khezeli and Amanda Wilkens.

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 (Hamburg) and Rik Versendaal (Delft).

We review the conformal mapping framework for planar particle aggregation and then visually motivate a Markovian simplification in the case where free particles are attracted to deposited particles. We explain the small-particle scaling limit for this simplification under smooth geometries with uniform boundary conditions and discuss ongoing work towards allowing for more general smooth boundary conditions.

The workshop will take place at the School of Mathematics (University of Leeds) and will bring together researchers in fractional, rough, and singular stochastic (partial) differential equations. Further information is available at https://conferences.leeds.ac.uk/frssd2026/

This talk addresses the efficient Monte Carlo estimation of rare-event probabilities associated with a broad class of McKean--Vlasov stochastic differential equations (MV-SDEs), which arise in the analysis of mean-field systems in statistical physics, mathematical finance, and collective behaviour models. Standard Monte Carlo methods become computationally infeasible in this setting due to the rapid growth of the estimator's relative variance (coefficient of variation) in the rare-event regime. Using stochastic optimal control, an optimal importance sampling measure change is constructed to minimise the variance of the resulting estimator. The resulting double-loop Monte Carlo (DLMC) estimator with importance sampling significantly mitigates this growth in the coefficient of variation. The framework is further extended to the multilevel Monte Carlo setting to reduce computational complexity, leveraging propagation-of-chaos and strong antithetic coupling to ensure that the level differences vanish in the mean-field limit. To address the discontinuity of the probability observable, a numerical smoothing technique is introduced to recover optimal variance convergence rates.  Numerical experiments on linear mean-field, Kuramoto, and Cucker--Smale models demonstrate computational savings of several orders of magnitude compared with standard Monte Carlo.

NOTES: online.
In a recent paper written jointly with Boaz Klartag, we prove that the variance of the Euclidean norm of any isotropic log-concave random vector is bounded above by a universal constant, not depending on the dimension. Thus, most of the mass of the random vector is concentrated in a thin spherical shell, whose width is order 1, while its radius is order root of the dimension. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of a given log-concave measure related to Eldan's stochastic localization and to the theory of non-linear filtering. Another ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture. In this talk, I'll first review the context and the history of the problem, before laying out the main steps of our proof.

NOTES: unusual time, online.
Abstract: Efficiently solving nonlinear stochastic optimal control problems remains a challenge with many applications. Existing optimality conditions typically rely on forward-backward stochastic differential equations (FBSDEs), which can be difficult to use in algorithms. In this talk, I will present new optimality conditions for stochastic optimal control, derived using rough path theory. This Pontryagin Maximum Principle uses the same Hamiltonian as in the deterministic setting, while avoiding FBSDEs. It unlocks the first indirect shooting method for stochastic optimal control, which only searches over the initial adjoint state and converges significantly faster than a direct method in numerical experiments. I will present applications to driving at the limits of handling and future directions towards robust and efficient uncertainty-aware control.

We consider coalescing random walks in 1+1 dimensional space-time, with a jump kernel that has finite moments up to order alpha. We view this system as a random set of coalescing paths, with one path starting from each point of space-time. When alpha>3, the diffusive scaling limit is known to be the Brownian web. We study the regime in which alpha is between 2 and 3. In this regime tightness fails (in the sense of continuous paths) due to erratic behaviour near the start times of some of the paths. We show that a surprising transition in behaviour occurs at alpha=9/4; when alpha>9/4 a diffusive scaling limit exists in which paths are essentially Brownian but some paths possess jumps at their initial times, whilst when alpha<9/4 tightness "truly" fails.