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

Dunkl Hyperbolic Equations

We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.Comment: This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

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

A study of island network performance for streaming protocols

Nowadays video surveillance is a cornerstone of the security in the world. It provides real-time monitoring for alarm of the environment, for people as face recognition, for property as plate car numbers detection, and provides a recorded archive for investigation. With megapixel cameras becoming increasingly widespread, even the bandwidth exhaustion of corporate networks is becoming a real issue. In this research, study on the performance of the island network using streaming protocol of HTTP and RTSP to broadcast the IP camera when streaming executed on H264 and H.265 encoder was conducted. The research done on the real island network that build to use as test bed for the project, also used network emulator (NetEm) to inject the packet loss and delay to the island network to emulate real big network. Then the results were analysed by Wireshark packet analyser. Based on the results gained, it was found that HTTP over TCP has less packets when compared to RTSP. As a conclusion, Hypertext Transfer Protocol is a little superior and authoritative protocol to stream a video when compared to the RTSP protocol

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