台大校區機率演講公告Probability Seminars on NTU Campus

(joint work with Shuenn-Jyi Sheu and Thi-Hien Nguyen)
In this talk, I want to present some results and open questions related to the analysis of the expansion of the spectral gap
$$ \\rho(c)=-\\sup\\{ {\\rm Re}(z);z\\in{\\rm Spec}(L_c)\\backslash\\{0\\}\\} $$
of the family of operators
$$ L_cf=\\Delta f+(-D+cC) x \\nabla f ; c\\in\\mathbb{R} $$
as $ c\\rightarrow\\infty $, where $ D $ is a symmetric positive definite $ m\\times m $-matrice and $ C $ is an $ m\\times m $-matrice satisfying $ \\langle Cx,Dx\\rangle=0 $. We will see that, due to some result from Matafune, Pallara and Priola, the question can be reduced to the study of the asymptotic behavior of the eigenvalues of the matrice $ E_c=-D+cC $ as $ c\\rightarrow\\infty $. The special relationship between $ D $ and $ C $ enables us to obtain some iterative method for determining the coefficients of the power series expansion of the eigen values and eigen vectors of the matrix family $ E_\\epsilon=-\\epsilon D+C $ as $ \\epsilon\\rightarrow0 $. This then yields information on the speed at which the gap $ \\rho_c $ converges as $ c\\rightarrow\\infty $. The method works well as long as the matrice $ C $ does not have eigen vectors with eigen space of dimension larger than one. Open questions remain in the general situation.

The two-dimensional stochastic heat equation (SHE) at criticality was introduced in the '90s. It arises from problems of statistical physics through several models of stochastic surface growth dynamics and the disordered system of a directed polymer in a random medium. Nevertheless, the equation has been known to pose difficulties to solution theories of stochastic partial differential equations despite the \"trivial\" form that the equation takes.
This talk will provide an overview of stochastic analytic descriptions of the two-dimensional SHE at criticality. Such descriptions for the discussion include the Feynman-Kac-type formulas for the moments and extend to the pathwise level by the martingale characterization of the solutions.

This talk offers a mathematical introduction to diffusion models in generative AI, based on their original formulations as presented in references [1] and [2], without introducing new insights. We will begin by discussing diffusion processes. Within the context of diffusion models, we will illustrate both the forward and reverse diffusion processes, highlighting their roles within the generative AI framework. The emphasis will be on the mathematical structure rather than intuitive explanations or theoretical justifications.

References
[1] Ho, J., Jain, A., and Abbeel, P. (2020). Denoising Diffusion Probabilistic Models.
[2] Sohl-Dickstein, J., Weiss, E. A., Maheswaranathan, N., and Surya Ganguli. (2015). Deep Unsupervised Learning using Nonequilibrium Thermodynamics. In International Conference on Machine Learning, 2256–2265

One of the main difficulties in solving stochastic partial differential equations (SPDEs) is the poor regularity of their solutions, which prevents the direct application of classical PDE techniques. However, rough path theory offers a way to solve this issue under suitable conditions. In this talk, I will introduce what rough path integration is and explain how it solves SPDEs in Martin Hairer's work. I will then introduce how this technique can be combined with the Littlewood-Paley theory to tackle our specific problem.

The primary goal of this lecture series is to introduce emerging topics in high- dimensional statistical inference. Specifically, we will focus on statistical physics models, such as the Ising model and its disordered variants. Parameterized by the temperature and external field, these models are spin systems, where the spin interactions are described by a structure matrix. While their Gibbs measures belong to the exponential family, understanding their corresponding parameter and structure estimation problems, based on a limited number of given samples, presents intriguing mathematical challenges. Throughout the series, we will highlight recent advancements, explain the approaches, and propose open problems that are particularly well-suited for graduate students and junior researchers to expand their research interests.

This talk offers a mathematical introduction to diffusion models in generative AI, based on their original formulations as presented in references [1] and [2], without introducing new insights. We will begin by discussing diffusion processes. Within the context of diffusion models, we will illustrate both the forward and reverse diffusion processes, highlighting their roles within the generative AI framework. The emphasis will be on the mathematical structure rather than intuitive explanations or theoretical justifications.

References
[1] Ho, J., Jain, A., and Abbeel, P. (2020). Denoising Diffusion Probabilistic Models.
[2] Sohl-Dickstein, J., Weiss, E. A., Maheswaranathan, N., and Surya Ganguli. (2015). Deep Unsupervised Learning using Nonequilibrium Thermodynamics. In International Conference on Machine Learning, 2256–2265

The Critical 2d Stochastic Heat Flow arises as a non-trivial solution of the Stochastic Heat Equation (SHE) at the critical dimension 2 and at a phase transition point. It is a log-correlated field which is neither Gaussian nor a Gaussian Multiplicative Chaos. We will review the phase transition of the 2d SHE, describe the main points of the construction of the Critical 2d SHF and outline some of its features and related questions. Based on joint works with Francesco Caravenna and Rongfeng Sun.

This is a mini course that aims at an audience of advanced undergraduate and graduate students as well as to researchers in probability, analysis and beyond. We will start with the robust method of the Lindeberg principle, which was originally used to prove the standard central limit theorem. We will then lift the Lindeberg principle to a setting of more complicated random objects such as multilinear polynomials and Wiener chaoses. The latter are intimately connected with recent developments in Stochastic PDEs and disordered systems and we will demonstrate how the Lindeberg principle can be used to obtain scaling limits of such models. We will then to move into nonlinear settings where chaos expansions are replaced by tree structures via the so called Butcher series. Relations to singular SPDEs will also be discussed.

Branching processes are stochastic processes that are often used to model the evolution of populations. They are also widely used in probability theory, biology (e.g., population genetics, epidemiology), and other fields. In this talk, we will introduce the most fundamental type of branching process, the Galton-Watson branching process, and review some classical limit theorems that describe the long-term behavior of populations. In addition, we will also discuss the coalescence problem, which provides us with a different perspective on investigating the history of the population.

Random walks on particle systems, such as the simple exclusion process (SEP, in which particles perform simple random walks that cannot occupy simultaneously the same position), have attracted notable attention in the last decade. In such models, say on the d-dimensional lattice, a walker moves at each unit of time to a random neighbouring vertex, with a drift that depends on whether its current location is occupied or not by a particle of the environment.
Compared to standard static environments, many classical techniques break down. A recurring question is then whether features of the static world are still present in the dynamic case, such as strong trapping effects that can force a transient walk to have zero speed. This is in particular relevant for the SEP, which is conservative and mixes slowly.
In dimension 1, we show on the contrary that the dynamic setting has a fundamentally different nature: the speed of the walker is a monotonic function of the density of the SEP, with at most one critical value where it could vanish.
The proof uses a comparison with a family of finite-range models to handle the long-range correlations, and an original coupling to circumvent the poor mixing properties of the exclusion process.
This is joint work with Daniel Kious and Pierre-François Rodriguez.