Symmetric L\'evy flights in semi-infinite domain

Abstract

We study symmetric L\'evy flights in a semi-infinite domain $[0,\infty)$ with a reflecting and absorbing boundary at 0. To this end, we use the fractional differential equation that governs the L\'evy process. Incorporating the boundary conditions in L\'evy flights has been an open and tricky question, as the long jumps can lead to the L\'evy flights leaping over the boundary. We, for the first time, incorporate reflecting and absorbing boundary conditions for L\'evy flights and solve the fractional differential equation analytically to find the probability densities. Monte Carlo simulations are also performed for both boundary conditions to verify the results. Analytical and simulation results perfectly coincide for the reflecting boundary condition and, for the absorbing boundary condition, they coincide for the large abscissa values.