Analyzing the impact of a discounted parameter on the reduction of collision time in Brownian particle trajectories

Using a set of nanosensors, we propose to study the existence of a finite tracking plan to detect a d-dimensional Brownian particle in the fluid. Each nanosensor moves iteratively along its path, alternating direction at random distances and velocities in both directions from its line's origin. These distances and velocities are expressed in terms of independent random variables that have probability density functions (PDFs) that are known. We further analyze the density of the random distances in our model by using the Fourier-Laplace transformation. Uncertainty is increased by the potential for a particle to collide with one of the nanosensors and the random sequence of turning points on each line. To account for this uncertainty, we might consider the search distance as a function of a discounted effort-reward parameter. The results of this analysis will give us the prerequisites needed to create a finite expectation value for the particle's first collision time with one of the nanosensors.