45 research outputs found
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
solution of Tanaka's equation can be classified by pairs of probability
measures on , with mean 1/2. What happens at the first
vertex is governed by , and at the second by . For each vertex ,
we construct a sequence of stopping times along which the image of the whole
circle by is reduced to . 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 . To each edge of the
graph is associated an independent white noise, which drives on
this edge. This produces an interface at each vertex of the graph. We first do
our study on star graphs with rays. The case 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 has a
unique in law solution, which is a Walsh's Brownian motion. This solution is
strong if and only if .
Solution flows are also considered. There is a (unique in law) coalescing
stochastic flow of mappings \p solving . For , it is the
only solution flow. For , \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 . 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 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 and
if is the first time hits , then is a beta random variable
of the second kind. We also calculate \EE[L\_{\sigma\_0}], where is the
local time accumulated at the boundary, and is the first time
hits .Comment: Submitte
Application of Stochastic Flows to the Sticky Brownian Motion Equation
We show how the theory of stochastic flows allows to recover in an elementary
way a well known result of Warren on the sticky Brownian motion equation
Real Time Lidar and Radar High-Level Fusion for Obstacle Detection and Tracking with evaluation on a ground truth
20th International Conference on Automation, Robotics and Applications Lisbon sept 24-25, 2018— Both Lidars and Radars are sensors for obstacle detection. While Lidars are very accurate on obstacles positions and less accurate on their velocities, Radars are more precise on obstacles velocities and less precise on their positions. Sensor fusion between Lidar and Radar aims at improving obstacle detection using advantages of the two sensors. The present paper proposes a real-time Lidar/Radar data fusion algorithm for obstacle detection and tracking based on the global nearest neighbour standard filter (GNN). This algorithm is implemented and embedded in an automative vehicle as a component generated by a real-time multisensor software. The benefits of data fusion comparing with the use of a single sensor are illustrated through several tracking scenarios (on a highway and on a bend) and using real-time kinematic sensors mounted on the ego and tracked vehicles as a ground truth