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This paper studies the sets of visit times to points x on the plane, where a standard two-dimensional Brownian motion makes a substantial number of visits. For this purpose, we introduce the concept of the "logarithmic scale" Minkowski dimension as a tool for measuring the sets of visit times. We prove that, almost surely, there is a point x such that the "logarithmic scale" Minkowski dimension of the set of visit times to x is 1. [ABSTRACT FROM AUTHOR]

This paper studies the opportunistic spectrum access (OSA) of the secondary users in a large-scale overlay cognitive radio (CR) network powered by renewable energy to improve both the energy and spectral efficiency of data transmission. Particularly, with energy harvesting module and energy storage module, the primary transmitters (PTs) and secondary transmitters (STs) are assumed to be able to collect ambient renewables and store them in batteries for future use. Upon harvesting sufficient energy, the corresponding PTs and STs (denoted by eligible PTs and STs) are then allowed to access the spectrum according to their respective medium access control (MAC) protocols. For the primary network, an Aloha type of MAC protocol is considered, under which the eligible PTs make independent decisions to access the spectrum with probability ρ p . For the secondary network, a threshold-based Opportunistic spectrum access (OSA) scheme, namely the primary transmitter assisted (PTA) protocol, is investigated, under which an eligible ST is allowed to access the spectrum only if the maximum signal power of the received pilots sent from the active primary transmitters (PTs) is lower than a certain threshold N ta . By applying tools from random walk theory and stochastic geometry, and assuming infinite battery capacity for energy storage module, we characterize the transmission probabilities of PTs and STs, respectively. With the obtained results of transmission probability, we then evaluate the coverage (transmission non-outage) performance of the overlay CR network powered by renewable energy. Simulations are provided to validate our analysis. [ABSTRACT FROM AUTHOR]

Abstract: Using the Poisson point process we model how the SiPM works and derive the non-linear response formula of the SiPM. Using this non-linear response formula we are able to capture the mean and variance saturation phenomena near the infinity and the linear behavior of the mean and variance near 0. We also introduce a different model of the SiPM. Under this different model we show the mean and variance saturation phenomena near the infinity and the linear behavior of the mean and variance near 0. We provide some open problems on this different model. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the exponential stability problem of second-order nonlinear stochastic evolution equations with Poisson jumps. By using the stochastic analysis theory, a set of novel sufficient conditions are derived for the exponential stability of mild solutions to the second-order nonlinear stochastic differential equations with infinite delay driven by Poisson jumps. An example is provided to demonstrate the effectiveness of the proposed result. [Copyright &y& Elsevier]

Abstract: In this note we provide a new proof of the log-Sobolev inequality on the path space of Poisson point processes. Our proof is elementary in the sense that it avoids the use of the martingale representation on Poisson spaces. Moreover, the weak Poincaré inequality for the weighted Dirichlet form is presented. [Copyright &y& Elsevier]

Consider a convex hull generated by a homogeneous Poisson point process in a cone in the plane. In the present paper the central limit theorem is proved for the joint probability distribution of the number of vertices and the area of a convex hull in a cone bounded by the disk of radius T (the center of the disk is at the cone vertex), for T → ∞. From the results of the present paper the previously known results of Groeneboom (1988) and Cabo and Groeneboom (1994) are followed, in which the central limit theorem was proved for the number of vertices and the area of the convex hull in a square by approximating the binomial point process by a homogeneous Poisson point process. [ABSTRACT FROM AUTHOR]

Consider a stationary Poisson point process in R d and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This construction gives rise to the well known random geometric graph. The number of edges of this graph is counted that have midpoint in the d -dimensional unit ball. A quantitative central limit theorem for this counting statistic is derived, as the space dimension d and the intensity of the Poisson point process tend to infinity simultaneously. [ABSTRACT FROM AUTHOR]