13:00-13:40 Jaehoon Lee (Seoul National University)
Title: Non-symmetric jump process with exponential decaying jumping kernel
Abstract: Consider a non-symmetric and non-local operator of the following form: $$ \mathcal{L}^\kappa f(x)= \lim_{\epsilon \rightarrow 0} \int_{|z|>\epsilon} (f(x+z)-f(x))\kappa(x,z)J(|z|)dz, \quad \kappa(x,z)=\kappa(x,-z). $$
Panki Kim, Renming Song and Zoran Vondracek obtained some heat kernel estimates of $\mathcal{L}^\kappa$ when $J$ be the L\'evy density of a symmetric L\'evy process in $\mathbb{R}^d$ with its L\'evy exponent satisfying a weak lower scaling condition at infinity. In this talk, we discuss the heat kernel estimates of $\mathcal{L}^\kappa$ when $J$ has additional exponentially decaying condition, i.e. $J(r) \le c_1 exp(-c_2 |r|^\beta)$, $r>1$ for some $c_1, c_2>0$ and $0 \le \beta \le 1$.
The talk is based on a joint work with Panki kim.

13:50-14:30 Yuichi Shiozawa (Okayama University)
Title: Bottom crossing probability for symmetric jump processes
Abstract: For symmetric Markov processes, lower rate functions are quantitative expressions of transience and non-point recurrence. The bottom crossing probability is some tail probability related to the lower rate function. In this talk, we determine the decay rate of the bottom crossing probability for symmetric jump processes under some conditions on heat kernel estimates.

14:40-15:20 Panki Kim (Seoul National University)
Title: Estimates of Dirichlet heat kernel for subordinate Brownian motions
Abstract: The transition density (if it exists) of Markov process is the heat kernel of the generator of the process. The transition density of a general Markov process rarely admits an explicit expression. Thus obtaining sharp estimates on transition density is a fundamental problem both in probability theory and in analysis. In this talk, we discuss the behavior of transition density (Dirichlet heat kernel) for subordinate Brownian motions in $C^{1,1}$-open subsets whose scaling orders are not necessarily strictly less than $2$. We consider subordinate Brownian motion both with and without Gaussian component Our estimate is sharp and explicitly expressed in terms of the distance to the boundary, Laplace exponent of subordinator and its derivative.
This talk is based on joint works with Ante Mimica and Joohak Bae.