Study on the existence of the transportation particle tracking model in the interactive medium

This paper assumes that the particle jumps randomly (Guassian jumps) from one point to another along one of the imaginary lines inside the interactive medium. Since this study was done in the space, we consider that the position of the particle at any time t has a multivariate distribution. The random waiting time of the particle for each Gaussian jump depends on its length. An identical set of programed nanosensors (with unit speed) were used to track this particle. Each line has a sensor that starts the tracking process from the origin. The existence of the necessary conditions which give the optimal search plan and the minimum expected value of the particle detection has been proven. This study is supported by a numerical example.