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♦ Beom-Seok Han (POSTECH)

Title: Lp solvability for stochastic Burgers' equations perturbed by space-time white noise with time-fractional derivatives

Abstract: Burgers' equation is recognized as the fundamental nonlinear partial differential equation, and it is used in various fields. In this talk, we discuss Lp-solvability for stochastic Burgers' equations perturbed by space-time white noise with time-fractional derivatives. Additionally, we provide the Holder-regularity of the solution. 

♦ Hong Chang Ji (IST Austria)

Title: Eigenvalues, eigenvector overlaps, and singular values of non-Hermitian random matrices

Abstract: In this talk, we consider spectral properties of a large random non-Hermitian matrix X, whose entries consist of IID random variables. On macroscopic and mesoscopic scales, it is widely known that the eigenvalues of X are subject to the celebrated circular law, that is, they are asymptotically uniformly distributed on the unit disk in the complex plane. However, eigenvalues of X on microscopic scale remain largely untouched, mainly due to the instability of the spectrum of X. In this talk, we discuss recent developments on the three subjects in the title, all closely related to this instability. In particular, we show that the overlaps between the left and right eigenvectors X are bounded by the size of X in probability.

♦ Ildoo Kim (Korea University) 

♦ Jinsu Kim (POSTECH)

Title: Positive recurrence and mixing times for stochastically modeled biochemical reaction systems

Abstract: Reaction networks, which describe graphically biochemical interactions, have been used to study qualitative behaviors of biochemical systems based on their structural features. When a reaction network is modeled stochastically, we often use continuous-time Markov chains that represent the counts of species in the associated biochemical system. One of the main interests in study of those Markov chains is to find which structural conditions of the underlying reaction network imply positive recurrence and exponential ergodicity. In this talk, we discuss some tools for exploring those main problems: Lyapunov function approaches and spectral gap approaches.​

♦ Kunwoo Kim (POSTECH)

Title:  Long-time behavior of stochastic heat equations

Abstract: The long-time behavior of stochastic heat equations perturbed by space-time white noise depends on the spatial domain and the initial data. In this talk, we consider stochastic heat equations on a one-dimensional torus and the real line and investigate how the long-time behavior depends on the spatial domain and the initial data. This is based on joint work with Davar Khoshnevisan and Carl Mueller.

♦ Seonwoo Kim (SNU)

Title: Approximation method to metastability: an application to Ising/Potts models without external fields

Abstract: We introduce a new method to prove metastable behaviors, which is the H^1-approximation method of the equilibrium potential function. The strength of this method lies on the fact that one may avoid referring to complicated objects or principles in potential theory, such as the flow structure, the Dirichlet and Thomson principles, etc. As an application, we explain the metastable behavior of Ising/Potts models without external fields in the low-temperature regime. This talk is based on a joint work with Insuk Seo. 

♦ Panki Kim (SNU)

Title: Heat kernel upper bounds for symmetric Markov semigroups

Abstract: It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this talk, we discuss the equivalence among these and off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. We also discuss some applications too.

This talk is based on joint works with Zhen-Qing Chen, Takashi Kumagai and Jian Wang.

♦ Jaehun Lee (HKUST)

♦ Kyeongsik Nam (KAIST)

Title: Universality of Poisson-Dirichlet law for log-correlated fields

Abstract: It is widely conjectured that Poisson-Dirichlet behavior appears universally in the low-temperature disordered system. However, this principle has been verified only for the particular models which are exactly solvable. In this talk, I will  talk about the universal Poisson-Dirichlet behavior for the general log-correlated Gaussian fields. This is based on the ongoing work with Shirshendu Ganguly.

♦ Sung Chul Park (KIAS)

Title: Universality in the scaling limit of the planar Ising model

Abstract: The conjecture that the critical Ising model on the square lattice in two dimensions is conformally invariant has broadly been verified rigorously, thanks in large part to the discrete complex analytic formalism developed by Smirnov and others. In this talk I will give an overview of recent developments generalizing this setup in both the directions of thermal perturbation (near-criticality) and the type of the lattice (universality). I will end by describing ongoing research in the framework of Chelkak's s-embedding, which aims to replace assumptions based on specific lattice topology with ones largely based on geometric and analytic notions. Based on joint works with Chelkak, Wan, Mahfouf and others.

♦ Zoran Vondracek (University of Zagreb)

Title: Dirichlet forms with jump kernels degenerate at the boundary

Abstract: In this talk I will give an overview of recent results on Dirichlet forms and corresponding Markov processes with jump kernels degenerate at the boundary. I will discuss the general framework as well as motivating examples, and will describe some unexpected new features of potential theory and analysis of such Markov processes.

The talk is based on several joint papers with Soobin Cho, Panki Kim and Renming Song.

♦ Jaeyun Yi (KIAS)

Title: Fractal geometry of the parabolic Anderson model with spatial white noise

Abstract: In this talk, we discuss the fractal geometry of the parabolic Anderson model (PAM) with spatial white noise. The PAM is a default model that describes diffusion phenomena in random environments, which exhibits a localization property called intermittency or Anderson localization. We will show that tall peaks of PAM with spatial white noise in dimension two and three is macroscopically multifractal. More precisely, we compute the macroscopic Hausdorff dimension of the tall peaks of the solution to the PAM. As a byproduct, we obtain the spatial asymptotics of the PAM. The key idea of the proof is the estimation of the solution using the theory of paracontrolled distribution which is a modern tool for studying singular (S)PDEs.

♦ Hyun Jae Yoo (HKNU)

Title: Stochastic spin systems, symmetry, and entropy production 

Abstract: In this talk we discuss the interacting particle systems in the discrete spaces, e.g. lattice systems, with equilibrium measures the Gibbs measures for the system. We construct Markov jump processes resulting from spin flips or spin changes (namely, Glauber dynamics or Kawasaki dynamics). Then we discuss the symmetry and entropy production for the dynamics. The Gibbsian nature of the reference measures will greatly facilitate the analysis of the dynamical systems.