Lévy flights between absorbing boundaries: Revisiting the survival probability and the shift from the exponential to the Sparre-Andersen limit behavior

Solution of the space-fractional diffusion equation on bounded domains of superdiffusive phenomena

Space-fractional diffusion equations find widespread application in nature. They govern the anomalous dynamics of many stochastic processes, generalizing the standard diffusion equation to superdiffusive behavior. Strikingly, the solution of space-fractional diffusion equations on bounded domains is still an open problem. This is in part due to the difficulty of handling nonlocal boundary conditions ascribed to the space-fractional derivative, leading to the failure of standard methods. Here we revisit the space-fractional diffusion equation in one spatial dimension with bounded domains and present a solution in terms of weighted Jacobi polynomials. Calculated eigenvalues and eigenfunctions in the superdiffusive regime show remarkable agreement with results from numerical discretization of the space-fractional derivative operator and Monte Carlo simulations. To exemplify, we apply the proposed solution to obtain the exact mean residence time or mean first-passage time, first-passage-time distribution, and survival probability, in agreement with known results for the superdiffusive regime. The system of equations converges rather fast for the first eigensolutions, as is desirable for practical application purposes in superdiffusive phenomena.

Lévy flights versus Lévy walks in bounded domains

Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities is the discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. As a consequence, a well-developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flight and Lévy walk models on bounded domains, examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time, and stationary probability density functions. It is demonstrated that the similarity of the models is affected by the type of boundary conditions and the value of the stability index defining the asymptotics of the jump length distribution.