Generalized diffusion and random search processes

We consider a one-dimensional Brownian search in the presence of trapping. The diffusion equation of the particle is represented by a memory kernel that enters the general waiting time probability density function. We find the general form of the first arrival time density, search reliability and efficiency and analyze several special cases of the memory kernel. We also analyze the Levy search in the presence of trapping in cases of single and multiple targets, as well as combined Levy-Brownian search strategies in case of a single target. The presented results are general and could be of interest for further investigation of different optimal search strategies, as well as in the animal foraging or spreading of contamination particles in the environment.