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We define an equation on a simple graph which is an extension of Tanaka's equation and the skew Brownian motion equation. We then apply the theory of transition kernels developed by Le Jan and Raimond and show that all the solutions can be classified by probability measures.
The Riemannian geometry of the space P m , of m × m symmetric positive definite matrices, has provided effective tools to the fields of medical imaging, computer vision and radar signal processing. Still, an open challenge remains, which consists of extending these tools to correctly handle the presence of outliers (or abnormal data), arising from excessive… (More)
We study a stochastic differential equation (SDE) driven by a finite family of independent white noises on a star graph, each of these white noises driving the SDE on a ray of the graph. This equation extends the perturbed Tanaka's equation recently studied by Prokaj  and Le Jan-Raimond  among others. We prove that there exists a coalescing… (More)
In a previous work, we have defined a Tanaka's SDE related to Walsh Brownian motion which depends on kernels. It was shown that there are only one Wiener solution and only one flow of mappings solving this equation. In the terminology of Le Jan and Raimond, these are respectively the stronger and the weaker among all solutions. In this paper, we obtain… (More)
Many signal and image processing applications, including texture analysis, radar detection or EEG signal classification, require the computation of a centroid from a set of covariance matrices. The most popular approach consists in considering the center of mass. While efficient, this estimator is not robust to outliers arising from the inherent variability… (More)
J o u r n a l o f P r o b a b i l i t y Electron. Abstract We study a simple stochastic differential equation (SDE) driven by one Brownian motion on a general oriented metric graph whose solutions are stochastic flows of kernels. Under some conditions, we describe the laws of all solutions. This work is a natural continuation of [17, 8, 10] where some… (More)
We review the construction of flows associated to Tanaka's SDE from . Using the skew Brownian motion, we 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 [1, 3, 9, 2].
Many signal and image processing applications, including SAR polarimetry and texture analysis, require the classification of complex covariance matrices. The present paper introduces a geometric learning approach on the space of complex covariance matrices based on a new distribution called Riemannian Gaussian distribution. The proposed distribution has two… (More)