Two-sided Green function estimates for killed subordinate Brownian motions

A subordinate Brownian motion is a L\'evy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is $-\phi(-\Delta)$, where $\phi$ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with $\phi$ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes, and much more. We give sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded $\kappa$-fat open set $D$. When $D$ is a bounded $C^{1,1}$ open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain a boundary Harnack principle in $C^{1,1}$ open sets with explicit rate of decay.Comment: 33 page

Minimal thinness for subordinate Brownian motion in half space

We study minimal thinness in the half-space H:=\{x=(\wt{x}, x_d):\, \wt{x}\in \R^{d-1}, x_d>0\} for a large class of rotationally invariant L\'evy processes, including symmetric stable processes and sums of Brownian motion and independent stable processes. We show that the same test for the minimal thinness of a subset of $H$ below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.Comment: 31 page

Boundary Harnack Principle for Subordinate Brownian Motions

We establish a boundary Harnack principle for a large class of subordinate Brownian motion, including mixtures of symmetric stable processes, in bounded $\kappa$-fat open set (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded $\kappa$-fat open sets with respect to these processes with their Euclidean boundary.Comment: 34 page

Ruin probabilities and decompositions for general perturbed risk processes

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs

Designing a Mobile App and Online Directory to Increase the Visibility of Environmental Organizations in a Community

Concern for environmental degradation, natural resource depletion, and public health has long been a necessary part of a strongly developed and sophisticated society. Humans are primarily responsible for the depletion of the natural environment as it supplies the commodities required to ensure the long-term survival of civilizations. As cultures continue to evolve, so does the practice of land stewardship and protection of resources. With the necessary expansion of industry and housing needed to support a rising human population, the management of natural resources, pollution control, and the environment has become increasingly important to people worldwide. Bernstein and Szuster (2019) suggest these concerns led to the environmental movement and environmentalism that largely emerged in the United States in the late 1960s. Environmentalism found its way into the public eye on the first Earth Day held on April 22, 1970, and was a catalyst in bringing about environmental legislation in the United States, such as the Clean Water Act and the Endangered Species Act. The first Earth Summit, attended by 113 nations, was held in Stockholm, Sweden, in 1972 and put environmentalism on an international stage (Bernstein and Szuster 2019). Environmentalism also surfaced in global political conversations through organizations like the Intergovernmental Panel on Climate Change (IPCC) and projects like the United Nations Sustainable Development Goals (SDGs)

China’s Stadium Diplomacy and its Determinants: A Typological Investigation of Soft Power

Since 1958, China has constructed over 140 sports facilities around the world. Previous research into stadium diplomacy lacks definitional clarity, has not systematically investigated the phenomenon, and crucially, has failed to explain why China employs stadium diplomacy where it does. This article defines the phenomenon and locates all known cases without temporal or geographic restrictions. We create a classification system and typology,permitting a comparison of theoretically-like types to develop and test a multi-determinant theory. We find empirical evidence that China employs stadium diplomacy to secure natural resources and to secure diplomatic recognition in line with the One-China policy. These findings have important implications for scholarship into the use of soft power within interstate rivalry, and the methodology demonstrates that a clear typology of soft power which is mutually exclusive and logically exhaustive can be created and is informative. Keywords: China, stadium diplomacy, soft power, interstate rivalr