Stochastic flows related to Walsh Brownian motion

We define an equation on a simple graph which is an extension of Tanaka equation and the skew Brownian motion equation. We then apply the theory of transition kernels developped by Le Jan and Raimond and show that all the solutions can be classified by probability measures.Comment: Electronic journal of probability, 16, 1563-1599 (2011

On flows associated to Tanaka's SDE and related works

We review the construction of flows associated to Tanaka's SDE from [9] and give an easy proof of the classification of these flows by means of probability measures on [0, 1]. Our arguments also simplify some proofs in the subsequent papers [2, 3, 7, 4]

On the Cs\'aki-Vincze transformation

Cs aki and Vincze have de fined in 1961 a discrete transformation T which applies to simple random walks and is measure preserving. In this paper, we are interested in ergodic and assymptotic properties of T . We prove that T is exact : \cap_{k\geq 1} \sigma(T^k(S)) is trivial for each simple random walk S and give a precise description of the lost information at each step k. We then show that, in a suitable scaling limit, all iterations of T "converge" to the corresponding iterations of the continous L evy transform of Brownian motion. Some consequences are also derived from these two results.Comment: Title changed and various other modification

Tanaka's equation on the circle and stochastic flows

We define a Tanaka's equation on an oriented graph with two edges and two vertices. This graph will be embedded in the unit circle. Extending this equation to flows of kernels, we show that the laws of the flows of kernels $K$ solution of Tanaka's equation can be classified by pairs of probability measures $(m^+,m^-)$ on $[0,1]$, with mean 1/2. What happens at the first vertex is governed by $m^+$, and at the second by $m^-$. For each vertex $P$, we construct a sequence of stopping times along which the image of the whole circle by $K$ is reduced to $P$. We also prove that the supports of these flows contains a finite number of points, and that except for some particular cases this number of points can be arbitrarily large.Comment: To appear in ALEA Lat. Am. J. Probab. Math. Sta

Stochastic flows and an interface SDE on metric graphs

This paper consists in the study of a stochastic differential equation on a metric graph, called an interface SDE $(\hbox{ISDE})$. To each edge of the graph is associated an independent white noise, which drives $(\hbox{ISDE})$ on this edge. This produces an interface at each vertex of the graph. We first do our study on star graphs with $N\ge 2$ rays. The case $N=2$ corresponds to the perturbed Tanaka's equation recently studied by Prokaj \cite{MR18} and Le Jan-Raimond \cite{MR000} among others. It is proved that $(\hbox{ISDE})$ has a unique in law solution, which is a Walsh's Brownian motion. This solution is strong if and only if $N=2$. Solution flows are also considered. There is a (unique in law) coalescing stochastic flow of mappings \p solving $(\hbox{ISDE})$. For $N=2$, it is the only solution flow. For $N\ge 3$, \p is not a strong solution and by filtering \p with respect to the family of white noises, we obtain a (Wiener) stochastic flow of kernels solution of $(\hbox{ISDE})$. There are no other Wiener solutions. Our previous results \cite{MR501011} in hand, these results are extended to more general metric graphs. The proofs involve the study of $(X,Y)$ a Brownian motion in a two dimensional quadrant obliquely reflected at the boundary, with time dependent angle of reflection. We prove in particular that, when $(X\_0,Y\_0)=(1,0)$ and if $S$ is the first time $X$ hits $0$, then $Y\_S^2$ is a beta random variable of the second kind. We also calculate \EE[L\_{\sigma\_0}], where $L$ is the local time accumulated at the boundary, and $\sigma\_0$ is the first time $(X,Y)$ hits $(0,0)$.Comment: Submitte

Energy Efficiency in Cache Enabled Small Cell Networks With Adaptive User Clustering

Using a network of cache enabled small cells, traffic during peak hours can be reduced considerably through proactively fetching the content that is most probable to be requested. In this paper, we aim at exploring the impact of proactive caching on an important metric for future generation networks, namely, energy efficiency (EE). We argue that, exploiting the correlation in user content popularity profiles in addition to the spatial repartitions of users with comparable request patterns, can result in considerably improving the achievable energy efficiency of the network. In this paper, the problem of optimizing EE is decoupled into two related subproblems. The first one addresses the issue of content popularity modeling. While most existing works assume similar popularity profiles for all users in the network, we consider an alternative caching framework in which, users are clustered according to their content popularity profiles. In order to showcase the utility of the proposed clustering scheme, we use a statistical model selection criterion, namely Akaike information criterion (AIC). Using stochastic geometry, we derive a closed-form expression of the achievable EE and we find the optimal active small cell density vector that maximizes it. The second subproblem investigates the impact of exploiting the spatial repartitions of users with comparable request patterns. After considering a snapshot of the network, we formulate a combinatorial optimization problem that enables to optimize content placement such that the used transmission power is minimized. Numerical results show that the clustering scheme enable to considerably improve the cache hit probability and consequently the EE compared with an unclustered approach. Simulations also show that the small base station allocation algorithm results in improving the energy efficiency and hit probability.Comment: 30 pages, 5 figures, submitted to Transactions on Wireless Communications (15-Dec-2016

iPhone forensics methodology and tools

iPhone mobile devices are rapidly overtaking the new generation of mobile phones market, especially among the young generation. It is also gaining a lot of popularity among security specialists and fancy gadgets for collectors. The device is considered as a “special” mobile phone due to its ability to perform multi-operations if not multitasking. It can therefore be used as a entertainment media device, a camera, a GPS, Internet surfing via Wi-Fi technology, Internet Mobile Edge Services, personal organizer, and finally performing as a cell phone with all the usual services including sms, and so forth. However, the difference between the iPhone and the other conventional phones vendors is its ability to store and process huge volume of data which is supported by decent computing capabilities of the iPhone processor. As part of every technology, such a device can be used for legal and illegal activities. Therefore the potential risks from such “special” technology are not limited to the possibility of containing illegal materials, such as audios and visuals, including explicit materials, images, documents and the possibility of propagating malicious activities rapidly. Such modification can breach or tamper with the telecommunications network authorities and regulations. The goal of this paper is to focus on both the logical and the physical extraction of the iPhone generation one through the extraction of the iPhone flash drive NAND memory chip and also the logical extraction of data onto the second generation of iPhone using various techniques and methods at our disposal

An Exclusion zone for Massive MIMO With Underlay D2D Communication

Fifth generation networks will incorporate a variety of new features in wireless networks such as data offloading, D2D communication, and Massive MIMO. Massive MIMO is specially appealing since it achieves huge gains while enabling simple processing like MRC receivers. It suffers, though, from a major shortcoming refereed to as pilot contamination. In this paper we propose a frame-work in which, a D2D underlaid Massive MIMO system is implemented and we will prove that this scheme can reduce the pilot contamination problem while enabling an optimization of the system spectral efficiency. The D2D communication will help maintain the network coverage while allowing a better channel estimation to be performed

Caching Improvement Using Adaptive User Clustering

In this article we explore one of the most promising technologies for 5G wireless networks using an underlay small cell network, namely proactive caching. Using the increase in storage technologies and through studying the users behavior, peak traffic can be reduced through proactive caching of the content that is most probable to be requested. We propose a new method, in which, instead of caching the most popular content, the users within the network are clustered according to their content popularity and the caching is done accordingly. We present also a method for estimating the number of clusters within the network based on the Akaike information criterion. We analytically derive a closed form expression of the hit probability and we propose an optimization problem in which the small base stations association with clusters is optimized