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We investigate the first-passage dynamics of symmetric and asymmetric Levy flights in a semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Levy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Levy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.
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Levy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments---the jump lengths---are drawn from an $alpha$-stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Levy Flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Levy Flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Levy Flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index $alpha$ and the skewness (asymmetry) parameter $beta$. The other approach is based on the stochastic Langevin equation with $alpha$-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.
We consider the problem of computing first-passage time distributions for reaction processes modelled by master equations. We show that this generally intractable class of problems is equivalent to a sequential Bayesian inference problem for an auxiliary observation process. The solution can be approximated efficiently by solving a closed set of coupled ordinary differential equations (for the low-order moments of the process) whose size scales with the number of species. We apply it to an epidemic model and a trimerisation process, and show good agreement with stochastic simulations.
Among Markovian processes, the hallmark of Levy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Levy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that exhibit very slow diffusion, logarithmic in time. These processes are path-dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the Central Limit Theorem is modified in this context, keeping the usual distinction between analytic and non-analytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.
Levy walks (LWs) are spatiotemporally coupled random-walk processes describing superdiffusive heat conduction in solids, propagation of light in disordered optical materials, motion of molecular motors in living cells, or motion of animals, humans, robots, and viruses. We here investigate a key feature of LWs, their response to an external harmonic potential. In this generic setting for confined motion we demonstrate that LWs equilibrate exponentially and may assume a bimodal stationary distribution. We also show that the stationary distribution has a horizontal slope next to a reflecting boundary placed at the origin, in contrast to correlated superdiffusive processes. Our results generalize LWs to confining forces and settle some long-standing puzzles around LWs.
We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Levy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulae when the stability index $alpha$ approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise.
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