Date/Time : 14 Apr 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Weijun XU, Peking University

Many stochastic PDEs arising from natural probabilistic models are classically “wrong” in the sense that they involve products between distributions. This type of ill-posedness has been existing even in ODEs. Ito’s theory of SDEs is probabilistic in nature. There is a lack of analytic stability, as can be seen from the existence of Ito-Stratonovich corrections.

We start from an old idea by Terry Lyons, now known as rough paths, that aimed to understand the nature of the analytic ill-posedness in SDEs. We will then see how these early attempts are gradually developed into a quite complete picture today for singular PDEs (regularity structures and para-controlled distributions). We shall focus on one particular example of singular SPDE, the 2D generalised parabolic Anderson model, to illustrate the coherence of this series of ideas.

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Date/Time : 24 Feb 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Shuhei Shibata, Kyushu University

We study the classical law of iterated logarithm for centered continuous Gaussian processes. Also, we introduce one of the asymptotic behavior of a non-Markovian stochastic process called elephant random walk as an application.

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Date/Time : 24 Feb 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Shota Osada, Kagoshima University

In this talk, we will consider the construction of $\mu$-reversible diffusion processes for a given point process $\mu$ on  $\mathbb{R}^d$. Specifically, we will discuss the Dirichlet form method to construct distorted Brownian motion on configuration spaces. For Poisson point processes, this corresponds to an infinite number of independent Brownian motions on $\mathbb{R}^d$. For correlated point processes, the interaction between particles is characterized by a function called the log-derivative, which describes the force exerted on one particle by the others. In previous results, the construction of diffusion processes using Dirichlet forms and their characterization using log-derivatives have been treated separately. In this talk, I will introduce that the existence of a logarithmic derivative implies the closability of the Dirichlet form. 

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Date/Time : 14 Feb 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Luca Makowiec, National University of Singapore

We will introduce Random Spanning Trees in Random Environment (RSTRE), a disordered system on spanning trees that interpolates between the Uniform Spanning Tree (UST) and Minimum Spanning Tree (MST) measures. Our primary goal is to study how local and global observables of the RSTRE depend on the inverse temperature beta. Of particular interest is the diameter of a (typical) spanning tree, as it is the first step towards establishing convergence to a non-trivial scaling limit. Furthermore, for the RSTRE on the complete graph with uniform disorder, we discuss a sharp transition of the local limit of the RSTRE to either the UST or MST local limit. This stands in contrast to our conjectured smooth transition of the diameter when beta is inside the so-called intermediate regime.

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Date/Time : 14 Feb 2025 11:00
Venue : S17 #05-11 (SR5)
Speaker : Benoît Corsini,

In this talk, I will present an upcoming result regarding the local evolution of Prim’s algorithm seen as a random process on graphs. To do so, I will start by introducing the concept of local limit for a sequence of graphs, before defining Prim’s algorithm and showing its relation to the minimum spanning tree. Finally, I will combine these two concepts to provide a representation of the limit, in particular its relation to invasion percolation and the percolation levels of a graph.

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Date/Time : 28 Nov 2024 13:30
Venue : S17 #04-05 (SR2)
Speaker : Peter Buhlmann, ETH Zurich

Nonparametric function estimation and prediction with moderate or large dimension of the covariates are particularly susceptible to extrapolation, because data points are typically far apart from each other in such moderate or higher dimension. Thus, there is a need to have machine learning methods that are extrapolation-aware, i.e. that automatically perform well (in a sense) when extrapolation occurs. Without such extrapolation-aware techniques, inference from standard machine learning and nonparametric procedures may be poor or invalid. We introduce a novel conceptual framework and introduce *Xtrapolation* which allows for extrapolation-aware inference with any ML algorithm.

This is joint work with Niklas Pfister (University of Copenhagen)

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Date/Time : 01 Nov 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Nathan Ross, University of Melbourne

Neural networks are random parametric functions, where the parameters are chosen according to stochastic gradient descent, based on the sample of training data. It has long been known that at initialisation, a neural network with “large” width and “fixed” depth can be well-approximated by a limiting Gaussian random field. But what quantities are well-approximated, and how wide does a neural network need to be for the approximation to be good? There has been a flurry of recent work trying to answer these and related questions, by deriving rates of convergence between the distribution of the neural network and the limiting Gaussian random field in various probability metrics, including some at the process level.

The aim of these lectures is to explain a general approach that can be applied to derive such rates at the process level, which uses Stein’s method, functional smoothing inequalities, and classical Gaussian process concepts such as Karhunen-Loeve expansion and Cameron-Martin/Reproducing Kernel Hilbert Space. The target audience is PhD students planning to do research in probability or statistics. In particular, the definitions and relevant properties of the concepts above will be covered in detail. Open questions and future directions will also be discussed.

The approach is based on joint work with (A.D. Barbour, K. Balasubramanian, L. Goldstein, A. Salim, and G. Zheng).

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Date/Time : 30 Oct 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Nathan Ross, University of Melbourne

Neural networks are random parametric functions, where the parameters are chosen according to stochastic gradient descent, based on the sample of training data. It has long been known that at initialisation, a neural network with “large” width and “fixed” depth can be well-approximated by a limiting Gaussian random field. But what quantities are well-approximated, and how wide does a neural network need to be for the approximation to be good? There has been a flurry of recent work trying to answer these and related questions, by deriving rates of convergence between the distribution of the neural network and the limiting Gaussian random field in various probability metrics, including some at the process level.

The aim of these lectures is to explain a general approach that can be applied to derive such rates at the process level, which uses Stein’s method, functional smoothing inequalities, and classical Gaussian process concepts such as Karhunen-Loeve expansion and Cameron-Martin/Reproducing Kernel Hilbert Space. The target audience is PhD students planning to do research in probability or statistics. In particular, the definitions and relevant properties of the concepts above will be covered in detail. Open questions and future directions will also be discussed.

The approach is based on joint work with (A.D. Barbour, K. Balasubramanian, L. Goldstein, A. Salim, and G. Zheng).

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Date/Time : 30 Oct 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Jimmy He, Ohio State University

The study of quantum groups and integrable systems have long been connected through the Yang-Baxter equation. Recently, these ideas have been introduced in probability to study integrable stochastic models. I will discuss a similar story for models with boundary, which are connected to quantum symmetric spaces through the reflection equation, as well as some applications to probability. The focus will be on illustrating the connections through illustrative examples and applications for a specific model called the six vertex model.

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