arXiv subjects

Joohak Bae

Publications and source records attributed to Joohak Bae.

3 recordsLinked to original sources

arXiv2023

Heat kernel estimates and their stabilities for symmetric jump processes with general mixed polynomial growths on metric measure spaces

In this paper, we consider a symmetric pure jump Markov process $X$ on a metric measure space with volume doubling conditions. Our focus is on estimating the transition density $p(t,x,y)$ of $X$ and studying its stability when the jumping kernel exhibits general mixed polynomial growth. Unlike previous work, in our setting, the rate function governing the jump growth may not be comparable to the scale function that determines whether $p(t,x,y)$ has near-diagonal or off-diagonal estimates. Under the assumption that lower scaling index of scale function is greater than $1$, we establish stabilities of heat kernel estimates. Additionally, if the metric measure space admits a conservative diffusion process with a transition density satisfying sub-Gaussian bounds, we generalize heat kernel estimates from [3, Theorems 1.2 and 1.4] using the rate function and the function $F$ related to walk dimension of underlying space. As an application, we prove the equivalence between a finite moment condition based on $F$ and a generalized Khintchine-type law of iterated logarithm at infinity for symmetric Markov processes.

math.PR↗

arXiv2018

Heat kernel estimates for symmetric jump processes with mixed polynomial growths

In this paper, we study the transition densities of pure-jump symmetric Markov processes in $ {\mathbb R}^d$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Lévy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment condition on such symmetric Markov process is equivalent to the Khintchine-type law of iterated logarithm at the infinity.

math.PR↗

arXiv2017

On estimates of transition density for subordinate Brownian motions with Gaussian components in $C^{1,1}$-open sets

We consider a subordinate Brownian motion $X$ with Gaussian components when the scaling order of purely discontinuous part is between $0$ and $2$ including $2$. In this paper we establish sharp two-sided bounds for transition density of $X$ in ${\mathbb R}^d$ and $C^{1,1}$-open sets. As a corollary, we obtain a sharp Green function estimates.

math.PR↗