Search reliability and search efficiency of combined Lévy–Brownian motion: long relocations mingled with thorough local exploration

A combined dynamics consisting of Brownian motion and L\'evy flights is exhibited by a variety of biological systems performing search processes. Assessing the search reliability of ever locating the target and the search efficiency of doing so economically of such dynamics thus poses an important problem. Here we model this dynamics by a one-dimensional fractional Fokker-Planck equation combining unbiased Brownian motion and L\'evy flights. By solving this equation both analytically and numerically we show that the superposition of recurrent Brownian motion and L\'evy flights with stable exponent $\alpha<1$, by itself implying zero probability of hitting a point on a line, lead to transient motion with finite probability of hitting any point on the line. We present results for the exact dependence of the values of both the search reliability and the search efficiency on the distance between the starting and target positions as well as the choice of the scaling exponent $\alpha$ of the L\'evy flight component.