Schedule & Abstracts

7/8 15:00~15:40

Kijung Lee (Ajou University)

Title: Us and Brownian motions

Abstract: When we guess things of interest related to heat diffusion on a domain with (zero) Dirichlet condition, we owe Brownian motion(s) a lot. In this talk we appreciate its various helps by recalling and savoring some of the consequences. The talk is in informal manner.

7/8 15:50~16:30

Jae-Hwan Choi (KAIST)

Title: Sobolev regularity theory for stochastic reaction-diffusion-advection equations with spatially homogeneous colored noises and variable-order nonlocal operators

Abstract: In this presentation, we introduce Sobolev regularity theory for nonlinear stochastic reaction-diffusion-advection equations (SRDAEs). We focus on the existence, uniqueness, and regularity of solutions to nonlinear SRDAEs with spatially homogeneous colored noises and variable-order nonlocal operators in mixed norm Lq(Lp)-spaces. A new condition, the strongly reinforced Dalang's condition, is proposed for colored noise, facilitating a deeper understanding of the complex interactions between nonlinearities and stochastic forces. Furthermore, we establish the space-time Hölder-type regularity of solutions. This presentation is based on joint work with Beom-Seok Han and Daehan Park.

7/8 16:40~17:20

Sungbin Lim (Korea University)

Title: Score-based Generative Modeling through Stochastic Evolution Equations in Hilbert Spaces

Abstract: Diffusion models have recently gained significant attention in probabilistic machine learning due to their theoretical properties and impressive applications in generative AI, including Stable Diffusion and DALL-E. This talk will provide a brief introduction to the theory of score-based diffusion models in Euclidean space. It will also present recent findings on score-based generative modeling in infinite-dimensional spaces, based on the time reversal theory of diffusion processes in Hilbert space. This talk is based on the following paper presented at NeurIPS 2023 (https://openreview.net/forum?id=GrElRvXnEj).

7/8 17:20~18:00

Ildoo Kim (Korea University)

Title: An existence and uniqueness theory to stochastic partial differential equations with sign-changing time-measurable pseudo-differential operators driven by space-time white noises

Abstract: In this talk, we introduce a new weak formulation to guarantee existence and uniqueness of a solution to stochastic partial differential equations with sign-changing time-measurable pseudo-differential operators driven by space-time white noises.

7/9 09:40~10:40

Jae-Hyouk Lee (Ewha Womans University)

Title: Hitchhiker's guide to the Determinant

Abstract: Determinant is one of the fundamental mathematical notions appreciated in the researches of sciences and technologies. Here we consider three definitions of determinants and fundamental properties along the Group action and uniqueness of determinants. Moreover, we discuss well known examples of determinant and extend the idea of determinant via the notion of cofactors, and consider the notion of Pfaffian and the derivation of the determinant.  

7/9 11:30~12:10

Daehan Park (Kangwon National University)

Title: A Sobolev regularity theory of variable order for anisotropic space-time non-local equations

Abstract: In this presentation, we will consider Sobolev regularity of variable order for solutions to space-time non-local equations. In particular, our main object is spatial non-local operators related to vectors of independent subordinate Brownian motions. We impose scaling conditions for each component process to get an off and near-diagonal estimation of the related heat kernel. For the time non-local case, a more delicate estimation will be given. The presentation contains the existence, uniqueness, and estimation of solutions. This work is a collaboration with Jae-Hwan Choi and Jaehoon Kang. 

7/9 15:00~18:00

Free Discussion

7/10 09:20~10:00

Kyeong-Hun Kim (Korea University)

Title: Boundary value problems for SPDEs 

Abstract: So far, the zero boundary condition has been mainly imposed for the study of SPDEs. Obviously, this is not because other boundary conditions are less important. I believe this is mainly because the proper function spaces for the inhomogeneous Dirichlet problem and Neumann problem have not been well studied. 

In this talk, I will introduce some results related to inhomogeneous Dirichlet problem and Neumann problem. Main part of the talk is based on a calculus written about 20 years ago in Minnesota.

7/10 10:00~10:40

Hee-Sun Choi (Korea Atomic Energy Research Institute)

Title: Novel Series Representation for Function Approximation: Theoretical Foundations and Numerical Validation

Abstract: In this work, we introduce a novel series representation for approximating functions and rigorously prove its convergence. Our approach aims to enhance the current methodologies used in neural network modeling and function approximation. This new series representation is analyzed and compared with two fundamental theorems in the field: the Universal Approximation Theorem, which underpins Deep Neural Networks (DNNs), and the Kolmogorov-Arnold Approximation Theorem, which forms the basis for Kolmogorov-Arnold Networks (KANs). We delve into the theoretical aspects of our proposed representation, providing a sketch of proof on its convergence. This theoretical framework establishes the groundwork for understanding how our method can be leveraged in practical applications. By comparing our approach with the well-established DNN and KAN methodologies, we highlight the unique features and potential benefits of our representation. Despite being in the preliminary stages of research, our method exhibits structural similarities to the KAN approach, suggesting that it may inherit many of its advantages, such as reduced dimensionality and capability for mathematical symbolic representations. We substantiate our theoretical findings with numerical experiments that demonstrate the convergence and practical characteristics of our series representation. This research was supported by a grant from the Korea Atomic Energy Research Institute (KAERI) R&D Program and the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (RS-2023-00253853).

7/10 10:50~11:30

Jinsol Seo (KIAS)

Title: Lp estimates for Maximal operators related to Fourier multipliers

Abstract: In this talk, I discuss Lp estimates for maximal operators defined by Fourier multipliers, and introduce our results. Our method is based on fractional calculus, and function spaces similar to that of Krylov. This work is collaborated with Jin Bong Lee.

7/10 11:30~12:10

Junhee Ryu (Korea University)

Title: Sobolev estimates for degenerate linear equations on the upper half space

Abstract: In this talk, we present both divergence and nondivergence degenerate equations on the upper half space. The coefficient matrices of the equations are the product of $x_d^2$ and bounded uniformly elliptic matrices. Under a partially weighted mean oscillation assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces.