Probability Workshop in Korea 2024
Title & Abstract
Soobin Cho (University of Illinois Urbana-Champaign)
Title: Heat kernel estimates for fractional Laplacian with supercritical killing potentials
Abstract: In this presentation, I will explore heat kernel estimates for $-(-\Delta)^{\alpha/2} + V$, where $\alpha \in (0,2)$ and the potential $V$ exhibits singularity at the origin, falling within the class of supercritical killing for the fractional Laplacian. Examples of such potentials include $|x|^{-\beta}$ with $\beta>\alpha$. The discussion includes a concise review of results for both (sub)-critical scenarios, emphasizing key distinctions. I will introduce a probabilistic proof, focusing on the approximate factorization formula for Dirichlet heat kernel estimates. Additionally, I aim to cover considerations for large-time estimates, contingent upon available time.
Jae-Hwan Choi (KAIST)
Title: On the trace theorem to generalized time-fractional equation in weighted Lebsegue spaces
Abstract: In this presentation, I will discuss the initial-time traces for solutions of evolution equations with generalized time-fractional derivatives in vector-valued and weighted Lebesgue spaces. Furthermore, I will also present results pertaining to Sobolev regularity theory for evolution equations, which have emerged as corollaries from our findings. This presentation is based on joint work with Jin Bong Lee, Jinsol Seo, and Kwan Woo.
Hong Chang Ji (IST)
Title: A Dynamical Approach to Local Laws in Random Matrices and its Application to non-Hermitian Edge Statistics
Abstract: Dyson Brownian motion, the eigenvalues of matrix-valued Brownian motion, has become the most standard and well-established approach to universalities for local eigenvalue statistics of Hermitian random matrices. The fundamental idea behind this approach is that the dynamic eigenvalues reach the 'microscopic' equilibrium much faster than the global equilibrium. In this talk, we discuss a manifestation of this principle in terms of local laws, which concern `mesoscopic' eigenvalue statistics. In particular, we present yet another simplification of the proof of local laws for non-Hermitian random matrices, making it less model-dependent. We will also discuss its application to edge universality for non-Hermitian random matrices.
Jaehoon Kang (SNU)
Title: Heat kernel estimates for symmetric jump processes with anisotropic jumping kernels
Abstract: We will discuss estimates of heat kernels for symmetric Markov processes on Euclidean spaces. We will first review heat kernel estimates for jump processes whose jumping kernels are comparable to isotropic functions. We will also check conditions that are equivalent to heat kernel estimates for jump processes. Then, we will consider jump processes with anisotropic jumping kernels and discuss their heat kernel estimates.
Minjoon Kim (POSTECH)
Title: Conditions of the Markov generator for non-exponential ergodicity of continuous-time Markov chains
Abstract: Continuous-time Markov chains is ergodic when its probability density converges over time to a stationary distribution. If this convergence is exponentially fast, the Markov chain is exponentially ergodic. Due to the complexity of deriving a closed form of the probability density, especially for the case of a countable state space, analytic criteria (e.g. Lyapunov functions) for exponential ergodicity have been investigated. However, its applicability is often limited to special cases such as birth and death models where negative drifts evidently exist. As an alternative, some `structural’ conditions for exponential ergodicity have been found by people in the literature of stochastic reaction networks where Markov chains with polynomial transition rates are often used to model the copy numbers of chemical species. Here the structural conditions roughly mean conditions solely depending on the transition vectors and the degree of the transition rates of the Markov chain. In this work, we introduce structural conditions of Markov chains that imply non-exponential ergodicity. These not only provide new avenues for verifying non-exponential ergodicity but also offer an easily applicable framework, particularly for continuous-time Markov chains in the form of chemical reaction networks.
Seonwoo Kim (KIAS)
Title: Spectral gap of the symmetric inclusion process: non-log-concave regime
Abstract: We consider the symmetric inclusion process (SIP) on a general finite graph. First, we establish universal upper and lower bounds for the spectral gap of SIP in terms of the spectral gap of the random walk on the same graph. In particular, in the log-concave regime, our bounds imply a version for SIP of Aldous' spectral gap conjecture. Next, in the non-log-concave regime, we establish various asymptotics of the spectral gap as the intensity tends to zero, and thereby obtain a counterexample to the Aldous conjecture in this regime. This talk is based on joint work with Federico Sau.
Jaehun Lee (KIAS)
Title: Heat kernel estimates of non-symmetric jump process with mixed polynomial growth
Abstract: In this talk, we consider the heat kernel estimates of non-symmetric and non-local operator with Holder continuous jumping kernel. In the first part, we construct the heat kernel of non-symmetric operator. Next, we establish the upper and lower bound for the heat kernel in the alpha-stable like case. Lastly, we introduce the heat kernel estimates for symmetric and non-symmetric jump processes with mixed polynomial growth.
Jaehun Lee (HKUST)
Title: Phase transition for the smallest eigenvalue of covariance matrices
Abstract: The smallest eigenvalue of a covariance matrix is well-understood under the finite fourth moment condition. Inspired by Aggarwal-Lopatto-Yau (2021), we investigate the case that each entry has an infinite fourth moment but still finite second moment, and find a phase transition from Tracy-Widom to Gaussian fluctuations for smallest eigenvalues. This is the joint work with Zhigang Bao and Xiaocong Xu.
Jinyeop Lee (LMU Munich)
Title: Deriving effective PDE from many-body Schrödinger equations
Abstract: In this talk, we derive effective dynamics governed by PDEs (e.g. Nonlinear Schrödinger equation, Vlasov-Poisson equation) from the N-body Schrödinger equation with interactions in the large N limit. For the derivation, we first visit quantum mechanics. Then, using the knowledge of many-body quantum mechanics given in the talk and with the suitable conditions for each situation, we derive PDEs describing the effective motion of the system.
Jung-Kyoung Lee (KIAS)
Title: Resolvent approach to Metastability of Langevin dynamics
Abstract: Langevin dynamics exhibits metastability in the presence of a potential with multiple equilibria. This complex behavior can be described by a Markov chain. Thanks to the recently developed resolvent approach, we can now fully understand the metastability of Langevin dynamics.
In this presentation, we introduce the resolvent approach as a key method for understanding the metastability and the results on metastability of Langevin dynamics. This is joint with Insuk Seo(Seoul National University) and Claudio Landim(Instituto de Matemática Pura e Aplicada).
Seungwoo Lee (SNU)
Title: Mixing of underdamped Langevin dynamics : from cut-off to Erying-Kramers formula
Abstract: The underdamped Langevin dynamics is a stochastic model describing evolution of thermostated molecular dynamics. In this talk, we discuss mixing behavior of the underdamped Langevin dynamics in the low temperature regime. First, if the force underlying the dynamics contains a unique stable point then we can provide sharp asymptotics of the mixing time of the underdamped Langevin Dynamics towards its stationary distribution, which behaves in log(1/temperature) when the temperature goes to zero(cut-off phenomenon). Second if the force admits multiple stable points then the dynamics moves from state to state in times behaving as e^{1/temperature} when the temperature goes to zero (meatastability). The main difficulty of the model is the degeneracy of the generator associated with the underdamped Langevin dynamics. This talk is based on two joint works with professor Insuk Seo and postdoc Ramil Mouad from Seoul National University.
Yong-Woo Lee (SNU)
Title: The rate of convergence of the real eigenvalue density for the elliptic Ginibre ensembles
Abstract: The elliptic Ginibre orthogonal ensemble (elliptic GinOE) belongs to a random matrix ensemble which interpolates between the Ginibre orthogonal ensemble (GinOE) and the Gaussian orthogonal ensemble (GOE). Although both the GinOE and the GOE possess eigenvalues on the real line, the real eigenvalue statistics of the GinOE is distinct from that of the GOE. Interpolating between them, it gives rise to two different regimes for the real eigenvalue statistics of the elliptic GinOE according to the non-Hermiticity parameter. In this talk, we will present the rates of convergence of the real eigenvalue densities for the elliptic GinOE in the two different regimes. Especially, the rates of convergence to the universal limit are shown to be dependent on the non-Hermiticity parameter. This is based on a joint work with Sung-Soo Byun.
Daehan Park (SNU)
Title: Diffusion equations with anisotropic non-local operators
Abstract: The parabolic equation is one of the main tools for mathematical elucidation of natural phenomena in diverse fields including mathematics, engineering, and biology. In this talk, we will consider parabolic equations with anisotropic non-local operators , where is an integro-differential operators with measurable coefficients. The simplest example of is anisotropic fractional Laplacian , and we also consider general operator related to subordinate Brownian motion. The existence, uniqueness, and estimation of solution will be considered.
Minjae Park (University of Chicago)
Title: Yang-Mills theory and random surfaces
Abstract: I will talk about some recent work on Yang-Mills theory and its connection to the theory of random surfaces. Rigorous construction of the continuum Yang-Mills measure is essential for the standard model in quantum physics and is one of the Millennium Prize Problems. I will explain how Wilson loop expectations (important observables in Yang-Mills) can be represented as surface sums in two settings:
- 2D continuum Euclidean Yang-Mills for classical Lie groups of any matrix size N (based on joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu)
- Any dimensional lattice Yang-Mills for classical Lie groups of any matrix size N and any inverse temperature β (based on joint work with Sky Cao and Scott Sheffield)
In addition, our probabilistic approaches provide alternative derivations and interpretations of several recent theorems, including Brownian motion limits (Dahlqvist), lattice string trajectories (Chatterjee and Jafarov), and surface sums (Magee and Puder).
Yongtak Sohn (MIT)
Title: Phase transitions of random constraint satisfaction problems
Abstract: The framework of random constraint satisfaction problems (CSPs) captures many fundamental problems in combinatorics, computer science, and statistics such as solving Boolean satisfiability problems or finding a classifier that fits the data perfectly. Based on deep, but non-rigorous methods from spin glass theory, statistical physicists have proposed a detailed picture of the solution space for random CSPs. In this talk, I will first describe the conjectured rich phase diagrams of random CSPs that exhibit long-range correlations. Then, I will highlight the recent progress in characterizing both the global and local geometry of solutions, particularly in the random regular NAE-SAT problem.
Jaeyun Yi (EPFL)
Title: Effect of nonlinearity on the support of solutions to stochastic heat equations.
Abstract: We discuss the support property of solutions to nonlinear stochastic heat equations. For deterministic PDEs, it is known that an integrability condition on the nonlinearity determines whether the support of a solution is compact or not. On the other hand, for SPDEs, the support properties have been studied only in the case of polynomial nonlinearity. We show that analogous integrability conditions on the nonlinearity of SPDE can ensure either the compact support property or the strict positivity of solutions. In the latter case, we also prove the uniqueness of the solution and the comparison principles. This is based on joint work with Kunwoo Kim and Beom-Seok Han.