Special Session 110: Stochastic Dynamics

Heat kernel estimates for anisotropic jump processes and Dirichlet problems

Jaehoon Kang Hankyong National University Korea Co-Author(s):

Abstract: In this talk, we consider symmetric jump processes with anisotropic jumping kernels given by unions of symmetric cones. We first review heat kernel estimates for such processes in the whole space and discuss their main features. We then briefly discuss the problem of Dirichlet heat kernel estimates in domains and some of the difficulties arising from anisotropy. We conclude with a discussion of an approach and some related computations.

AN EXISTENCE AND UNIQUENESS THEORY TO STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH PSEUDO-DIFFERENTIAL OPERATORS

Ildoo Kim Korea university Korea Co-Author(s):    Jae-Hwan Choi

Abstract: In this talk, we introduce a new weak formulation to guarantee existence and uniqueness of a solution to stochastic partial differential equations with pseudo-differential operators whose symbols are allowed to be signchanging.

The Dirichlet problem for stochastic partial differential equations (SPDEs) with nonlocal operators in $C^{1,\sigma}$ open sets

Kyeong Hun Kim Korea University Korea Co-Author(s):    Junhee Ryu

Abstract: In this talk I will introduce a Sobolev regularity theory for the Dirichlet problem of SPDEs having spatial non-local operators in $C^{1,\sigma}$ open sets. I will consider substantially large classes of nonlocal operators and generalized Gaussian noise. The existence and uniqueness of strong solutions in weighted Sobolev spaces will be introduced, along with maximal $L_p$-regularity estimates for the solutions.

Heat kernel estimates for Dirichlet forms degenerate at the boundary

Panki Kim Seoul National University Korea Co-Author(s):

Abstract: In this talk, we discuss estimates on the heat kernels of discontinuous symmetric Markov processes including ones with jump kernels degenerate at the boundary. There are new forms of the heat kernels estimates qualitatively different from all previously known heat kernel estimates. We also discuss the processes killed either by a critical potential or upon hitting the boundary. Their heat kernel estimates have the approximate factorization property with survival probabilities decaying as a power of the distance to the boundary, where the power depends on the critical potential. This talk is based on joint papers with Soobin Cho, Renming Song and Zoran Vondracek.

Transience time of the subcritical facilitated exclusion process

Seonwoo Kim Yonsei University Korea Co-Author(s):    Oriane Blondel, Clement Erignoux, Sanha Lee

Abstract: In this talk, we consider the facilitated exclusion process on the one-dimensional discrete N-torus. Because of the facilitating mechanism, the process freezes in finite time if the particle density of the initial configuration is subcritical, i.e., if it is smaller than (or equal to) 1/2. We prove that, starting from any subcritical Bernoulli product measure, the correct scale of the transience/freezing time is of order \log^3N. Based on a joint work with Oriane Blondel, Clement Erignoux and Sanha Lee.

Dynamical Methods in Random Matrix Theory and Applications to Correlated Models

JAEHUN LEE Institute of Science and Technology Austria (ISTA) Austria Co-Author(s):

Abstract: We review a recently developed dynamical method in random matrix theory that extends universality results to a broader class of random matrices. A key idea of the method is to derive generalized local laws by means of Green function comparison along two different stochastic flows. We will explain this approach with particular emphasis on correlated random matrices. As an application, we will also discuss linear eigenvalue statistics for correlated random matrix models.

Gamma Expansion of the Large Deviation Rate Functional for Diffusion Processes

Jungkyoung Lee Inha University Korea Co-Author(s):    Claudio Landim, Mauro Mariani

Abstract: The level two large deviation rate functional for a Markov process is closely related to its long-time behavior. When the process exhibits metastability, in particular a hierarchical structure across multiple time scales, this metastable behavior can be characterized through a Gamma expansion of the rate functional as established in Bertini, Gabrielli and Landim (Ann. Appl. Probab. 34(4): 3820-3869, 2024). In this talk, we discuss the Gamma expansion of the large deviation rate functional for reversible diffusion processes. This talk is based on joint work with Claudio Landim and Mauro Mariani.

Sub-critical Exponential random graphs

Kyeongsik Nam KAIST Korea Co-Author(s):    Shirshendu Ganguly; Kyeongsik Nam

Abstract: The exponential random graph model (ERGM) is a central object in the study of canonical ensembles in statistical physics. Due to the lack of exact solvability, many basic questions have remained unanswered. In this talk, I will introduce a series of concentration of measure results for the ERGM throughout the entire sub-critical regime, including a Poincare inequality and Gaussian concentration.

Boundedness of non-local operators with variable coefficients: probabilistic approach and applications

Daehan Park Kangwon National University Korea Co-Author(s):    Jae-Hwan Choi, Jaehoon Kang

Abstract: There has been growing interest in non-local operators in both analysis and probability. In particular, the structure of jumping kernels is closely connected to L\`evy processes, with the fractional Laplacian serving as the most fundamental example. In this presentation, we consider the boundedness of the non-local operators having variable coefficients in weighted Lebesgue spaces. We also provide some applications such as regularity theory of evolution equations and elliptic-type equations in weighted Lebesgue spaces and Krylov-type estimate for the processes generating our operators.

$L_p$-estimates for nonlocal equations with general L\`evy measures

Junhee Ryu KIAS Korea Co-Author(s):    Hongjie Dong

Abstract: In this talk, we consider time-dependent nonlocal operators associated with general L\`evy measures of order $\sigma \in(0,2)$. We allow the class of L\`evy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in $L_p$ spaces are established. We also demonstrate that, depending on the ranges of $\sigma$ and $d$, the operator can or cannot be treated in weighted mixed-norm spaces.

Inviscid Limit for the Two-Dimensional Navier-Stokes Equations: A Stochastic Lagrangian Approach

Jinsol Seo KIAS (Korea Institute for Advanced Study) Korea Co-Author(s):    Jae-Hwan Choi (KIAS), Chanwoo Kim (UW-Madison), and Dohyun Kwon (Yonsei University)

Abstract: I will discuss recent progress on the vanishing-viscosity limit from the two-dimensional Navier-Stokes equations to the Euler equations. Our approach is both Lagrangian and probabilistic. First, we develop a stochastic analogue of the DiPerna-Lions theory to construct and control stochastic Lagrangian flows associated with the viscous dynamics. Second, we establish a large-deviation principle that quantifies the convergence to the Euler dynamics.

Nonhomogeneous boundary condition for spectral nonlocal operators

Vanja Wagner University of Zagreb Croatia Co-Author(s):    Ivan Bio\v{c}i\`{c}

Abstract: The main interest of this talk will be the study of (semi-)linear nonlocal elliptic problems driven by spectral-type operators of the form $\psi(-L_{|D})$ in bounded $C^{1,1}$ domains $D\subset \mathbf R^d$. Here $\psi$ is a complete Bernstein function and $L_{|D}$ is the generator of a killed unimodal L\`evy process. We will present the general framework that covers and extends the theory of the spectral and interpolated fractional Laplacian. The focus will be on the analysis of the nonhomogeneous boundary condition via a weak $L^1$ trace-like boundary operator. This operator is formulated in terms of the Poisson potential with respect to the $d-1$ Hausdorff measure on $\partial D$. The methodology combines stochastic process techniques, potential theory, and spectral analysis.

Instantaneous shrinking of the support of solutions to stochastic PDEs.

Jaeyun Yi Korea Institute for Advanced Study Korea Co-Author(s):    Beom-Seok Han, Kunwoo Kim

Abstract: In this talk, we discuss the instantaneous shrinking of the support of solutions to one-dimensional parabolic stochastic PDEs driven by space-time white noise. It is well known that if the initial data has compact support, then the solution also has compact support for all later times. In deterministic cases, a much stronger result holds: the solution has compact support immediately after the initial time if the initial data merely vanishes uniformly at infinity. We show that a class of parabolic SPDEs also exhibits instantaneous shrinking. In particular, if the uniqueness in law holds, then all nonnegative solutions exhibit instantaneous shrinking.