Wed, 28 May 2025
11:00
L5

A central limit theorem and large deviations principle for the generalised Dean--Kawasaki equation with truncated noise on a bounded domain

Shyam Popat
(Mathematical Institute)
Abstract

We begin with motivation on how the study of SPDEs are relevant when interested in fluctuations of particle systems. 

We then present a law of large numbers, central limit theorem and large deviations principle for the generalised Dean--Kawasaki SPDE with truncated noise. 

Our main contribution is the ability to consider the equation on a general $C^2$-regular, bounded domain with Dirichlet boundary conditions. On the particle level the boundary condition corresponds to absorption and injection of particles at the boundary.

The work is based on discussions with Benjamin Fehrman and can be found at https://arxiv.org/pdf/2504.17094 

 

Wed, 30 Apr 2025
11:00
L5

Hydrodynamic limit of an active-passive lattice gas

Maria Bruna
(Mathematical Institute)
Abstract

In this talk, I will discuss a model mixture of active (self-propelled) and passive (diffusive) particles with non-reciprocal effective interactions (or forces that violate Newton’s third law). We derive the hydrodynamic PDE limit for the particle densities, which is not a Wasserstein gradient flow of any free energy, consistent with the microscopic model having non-equilibrium steady states. We study the emergence of collective behaviour, which includes phase separation and dynamical (travelling) steady states.

Thu, 13 Feb 2025

12:00 - 13:00
L3

Various

Various Speakers from OCIAM Year 2 Graduates
(Mathematical Institute)
Thu, 30 Jan 2025
16:00
L5

Market Making with fads, informed and uninformed traders.

Adrien Mathieu
(Mathematical Institute)
Abstract

We characterise the solutions to a continuous-time optimal liquidity provision problem in a market populated by informed and uninformed traders. In our model, the asset price exhibits fads -- these are short-term deviations from the fundamental value of the asset. Conditional on the value of the fad, we model how informed traders and uninformed traders arrive in the market. The market maker knows of the two groups of traders but only observes the anonymous order arrivals. We study both, the complete information and the partial information versions of the control problem faced by the market maker. In such frameworks, we characterise the value of information, and we find the price of liquidity as a function of the proportion of informed traders in the market. Lastly, for the partial information setup, we explore how to go beyond the Kalman-Bucy filter to extract information about the fad from the market arrivals.

Wed, 22 Jan 2025
11:00
L6

Adapted Wasserstein distance between continuous Gaussian processes

Yifan Jiang
(Mathematical Institute)
Abstract
Adapted Wasserstein distance is a generalization of the classical Wasserstein distance for stochastic processes. It captures not only the spatial information but also the temporal information induced by the processes. In this talk, I will focus on the adapted Wasserstein distance between continuous Gaussian processes. An explicit formula in terms of their canonical representations will be given. These results cover rough processes such as fractional Brownian motions and fractional Ornstein--Uhlenbeck processes. If time permits, I will also show that the optimal coupling between two 1D additive fractional SDE is driven by the synchronous coupling of the noise.
We introduce a 'causal factorization' as an infinite dimensional Cholesky decomposition on Hilbert spaces. This naturally bridges the probabilistic notion 'causal transport' and the algebraic object 'nest algebra'.  Such a factorization is closely related to the (non)canonical representation of Gaussian processes which is of independent interest. This talk is based on a work-in-progress with Fang Rui Lim.
Mon, 03 Mar 2025
16:30
L4

The Stein-log-Sobolev inequality and the exponential rate of convergence for the continuous Stein variational gradient descent method

Jakub Jacek Skrzeczkowski
(Mathematical Institute)
Abstract

The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information as a duality pairing between $H^{-1}$ and $H^{1}$, which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions. This is a joint work with J. A. Carrillo and J. Warnett. 

Mon, 20 Jan 2025
16:30
L4

Fluctuations around the mean-field limit for attractive Riesz interaction kernels in the moderate regime

Alexandra Holzinger
(Mathematical Institute)
Abstract

In this talk I will give a short introduction to moderately interacting particle systems and the general notion of fluctuations around the mean-field limit. We will see how a central limit theorem can be shown for moderately interacting particles on the whole space for certain types of interaction potentials. The interaction potential approximates singular attractive potentials of sub-Coulomb type and we can show that the fluctuations become asymptotically Gaussians. The methodology is inspired by the classical work of Oelschläger in the 1980s on fluctuations for the porous-medium equation. To allow for attractive potentials we use a new approach of quantitative mean-field convergence in probability in order to include aggregation effects. 

Tue, 05 Nov 2024
14:00
L5

María Reboredo Prado: Webs in the Wind: A Network Exploration of the Polar Vortex

María Reboredo Prado
(Mathematical Institute)
Abstract

All atmospheric phenomena, from daily weather patterns to the global climate system, are invariably influenced by atmospheric flow. Despite its importance, its complex behaviour makes extracting informative features from its dynamics challenging. In this talk, I will present a network-based approach to explore relationships between different flow structures. Using three phenomenon- and model-independent methods, we will investigate coherence patterns, vortical interactions, and Lagrangian coherent structures in an idealised model of the Northern Hemisphere stratospheric polar vortex. I will argue that networks built from fluid data retain essential information about the system's dynamics, allowing us to reveal the underlying interaction patterns straightforwardly and offering a fresh perspective on atmospheric behaviour.

Wed, 13 Nov 2024
11:00
L4

Flow equation approach for the stochastic Burgers equation

Andrea Pitrone
(Mathematical Institute)
Abstract

I will present the basic idea of the flow equation approach developed by Paweł Duch to study singular stochastic partial differential equations. In particular, I will show how it can be used to prove the existence of a solution of the stochastic Burgers equation on the one-dimensional torus.

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