In this paper, we study the Schr\"odinger operator $\Delta-V$, where $V$ is a supercritical non-negative potential belonging to large class of functions containing form $b|x|^{-(2+2\beta)}$, $b, \beta>0$. We obtain two-sided estimates on heat kernel $p(t, x, y)$ along with for corresponding Green function. Unlike case fractional $-(-\Delta)^{\alpha/2}-V$, $\alpha\in (0, 2)$, killing dealt in [11], present case, decays 0 exponentially as $x$ or $y$ tends origin.

In this paper we discuss non-local operators with killing potentials, which may not be in the standard Kato class. We first factorization of their Dirichlet heat kernels metric measure spaces. Then establish explicit estimates under critical killings $C^{1,1}$ open subsets $\mathbb{R}^d$ or $\mathbb{R}^d\setminus\{0\}$. The decay rates our come from values multiplicative constants potentials. Our method also provides an alternative and unified proof main results \cite{CKS10a, CKS10b, CKS-AOP}.

Many companies are focusing on remote working since the Covid-19 pandemic. One difference between environment and face-to-face setting is absence of natural awareness, which promotes social interaction helps coordination work flow. To maintain benefits awareness in a environment, it important to first understand information workers disclose need for awareness. In this paper, we aim investigate disclosed needed, provide insights mitigate two. We conducted case study with an actual workgroup...

In this paper, we study transition density functions for pure jump unimodal Lévy processes killed upon leaving an open set D. Under some mild assumptions on the density, establish two-sided Dirichlet heat kernel estimates when D is C 1 , . Our result covers case that densities of are regularly varying whose indices equal to Euclidean dimension. This first results such lower scaling index not necessarily strictly bigger than

In this paper, we discuss estimates on transition densities for subordinators, which are global in time. We establish the sharp two-sided subordinators whose L\'evy measures absolutely continuous and decaying mixed polynomial orders. Under a weaker assumption measures, also obtain precise asymptotic behaviors of at infinity. Our results cover geometric stable Gamma much more.

We study weak Harnack inequality and a priori H\"older regularity of harmonic functions for symmetric nonlocal Dirichlet forms on metric measure spaces. Our analysis relies three main assumptions: the existence strongly local form with sub-Gaussian heat kernel estimates, tail estimate jump outside balls energy comparability condition. establish robustness our results, ensuring that constants in estimates remain bounded, provided order scale function appearing condition, maintains certain...

Continuing from Cho, Kim, and Lee [<italic>General Law of iterated logarithm for Markov processes: Limsup law</italic>, arXiv:2102,01917v3], in this paper, we discuss general criteria forms liminf laws (LIL) continuous-time processes. Under some minimal assumptions, which are weaker than those Cho et al., establish LIL at zero (at infinity, respectively) metric measure spaces. In particular, our assumptions law the form truly local so that can cover highly space-inhomogenous cases. Our...

The goal of this work is to develop a general theory for non-local singular operators the type $$ L^{\mathcal{B}}_{\alpha}f(x)=\lim_{\epsilon\to 0} \int_{D,\, |y-x|>\epsilon}\big(f(y)-f(x)\big) \mathcal{B}(x,y)|x-y|^{-d-\alpha}\,dy, and L f(x)=L^{\mathcal{B}}_{\alpha}f(x) - \kappa(x) f(x), in case $D$ $C^{1,1}$ open set $\mathbb{R}^d$, $d\ge 2$. function $\mathcal{B}(x,y)$ above may vanish at boundary $D$, killing potential $\kappa$ be subcritical or critical. From probabilistic point view...

Privacy is essential to fully enjoying the benefits of social media. While fear around privacy risks can sometimes motivate management, negative impact such fear, particularly when it perceived as unaddressable (i.e., "dysfunctional" fear), significantly harm teen well-being. In a co-design study with 136 participants aged 13-18, we explored how teens protect their without experiencing heightened fear. We identified seven different sources dysfunctional `fear hostile environment' and...

Our purpose is to investigate the potential use of chatbots for information sharing and social connection within a co-living space. To this end, we designed chatbot residents space based on following principles: (1) The range shared limited three areas derived from similarities residents, it takes 'give-and-take QnA' structure, where one should answer question another resident after they ask question. (2) Conversation resemble human-like dialogue reveal presence other residents. 19 used week...

In this paper, we study estimates on tail probabilities $\mathbb{P}(S_r \ge t)$ of several classes subordinators under mild assumptions the its L\'evy measure. As an application that result, obtain two-sided for fundamental solutions general homogeneous time fraction equations including those with Dirichlet boundary conditions.

In this paper, we discuss general criteria of limsup law iterated logarithm (LIL) for continuous-time Markov processes. We consider minimal assumptions LILs to hold at zero(at infinity, respectively) in metric measure spaces. establish under local near zero (near on uniform bounds the expectations first exit times from balls terms a function $\phi$ and tails jumping kernel $\psi$. The main result is that simple ratio test functions $\psi$ completely determines whether there exists positive...

Continuing from arXiv:2102.01917v2, in this paper, we discuss general criteria and forms of liminf laws iterated logarithm (LIL) for continuous-time Markov processes. Under some minimal assumptions, which are weaker than those establish LIL at zero (at infinity, respectively) metric measure spaces. In particular, our assumptions law the form truly local so that can cover highly space-inhomogenous cases. Our results all examples arXiv:2102.01917v2 including random conductance models with long...

It is difficult for a human operator to find roll, pitch, yaw (RPY) that indicates the desired direction of unmanned aerial vehicle (UAV) in three-dimensional space. Herein, controller UAV was developed allowing controlling without finding RPY information. The algorithm implemented automatically calculated information from normal vector end effector. designed using parallel mechanism. joint angles were measured potentiometers estimate Five subjects participated an experiment control space...

In this paper, we discuss the laws of iterated logarithm (LIL) for occupation times Markov processes $Y$ in general metric measure space both near zero and infinity under some minimal assumptions. We first establish LILs (truncated) on balls $B(x,r)$ radii $r$ up to an function $\Phi (r)$, which is mean exit time $Y$, by showing that $\Phi$ optimal. Our result covers regardless transience recurrence process. assumptions are truly local particular at our truncated $r \mapsto\int_0^{ \Phi...

The goal of this paper is to prove the existence heat kernels for two types purely discontinuous symmetric Markov processes in upper half-space $\mathbb R^d$ with jump degenerate at boundary, and establish sharp two-sided estimates on kernels. are form $J(x,y)=\mathcal B(x,y)|x-y|^{-\alpha-d}$, $\alpha\in (0,2)$, where function $\mathcal B$ depends four parameters may vanish boundary. Our results first non-local operators type we consider conservative closed kernel $J(x,y)$. Depending...

In this paper, we establish sharp two-sided estimates for transition densities of a large class subordinate Markov processes. As applications, show that the parabolic Harnack inequality and H\"older regularity hold functions such processes, derive Green function estimates.