We consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution ϕ(η) displaying asymmetric power law tails (i.e. ϕ(η) \sim c/η^{α+1} for large positive jumps and ϕ(η) \sim c/(γ|η|^{α+1}) for large negative jumps, with 0 < α< 2). In absence of boundaries and after a large number of steps n, the probability density function (PDF) of the walker position, x_n, converges to an asymmetric Lévy stable law of stability index αand skewness parameter β=(γ-1)/(γ+1). In particular the right tail of this PDF decays as c n/x_n^{1+α}. Much less is known when the walker is confined, or partially confined, in a region of the space. In this paper we first study the case of a walker constrained to move on the positive semi-axis and absorbed once it changes sign. In this case, the persistence exponent θ_+, which characterizes the algebraic large time decay of the survival probability, can be computed exactly and we show that the tail of the PDF of the walker position decays as c \, n/[(1-θ_+) \, x_n^{1+α}]. This last result can be generalized in higher dimensions such as a planar Lévy walker confined in a wedge with absorbing walls. Our results are corroborated by precise numerical simulations.
An important open problem in the theory of Lévy flights concerns the analytically tractable formulation of absorbing boundary conditions. Although numerical studies using the correctly defined nonlocal approach have yielded substantial insights regarding the statistics of first passage, the resultant modifications to the dynamical equations hinder the detailed analysis possible in the absence of these conditions. In this study it is demonstrated that using the first-hit distribution, related to the first passage leapover, as the absorbing sink preserves the tractability of the dynamical equati…
The conserved Manna model with a planar absorbing boundary is studied in various space dimensions. We present a heuristic argument that allows one to compute the surface critical exponent in one dimension analytically. Moreover, we discuss the mean field limit that is expected to be valid in d>4 space dimensions and demonstrate how the corresponding partial differential equations can be solved.
Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations <x(t)x(s)> ~ t^{2H} + s^{2H} - |t-s|^{2H}, where H, with 0<H<1 is called the Hurst exponent. For H = 1/2, x(t) is a Brownian motion, while for H unequal 1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P(x,t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P(x,t) ~ R(x/t^H)/t^H. Our objective is to compute the s…
We study the probability distribution function (pdf) of the position of a Lévy flight of index 0<α<2 in presence of an absorbing wall at the origin. The solution of the associated fractional Fokker-Planck equation can be constructed using a perturbation scheme around the Brownian solution (corresponding to α= 2), as an expansion in ε= 2 - α. We obtain an explicit analytical solution, exact at the first order in ε, which allows us to conjecture the precise asymptotic behavior of this pdf, including the first subleading corrections, for any α. Careful numerical simulations, as well as an exact c…
Scale-free surfaces, such as cones, remain unchanged under a simultaneous expansion of all coordinates by the same factor. Probability density of a particle diffusing near such absorbing surface at large time approaches a simple form that incorporates power-law dependencies on time and distance from a special point, such as apex of the cone, which are characterized by a single exponent $η$. The same exponent is used to describe the number of spatial conformations of long ideal polymer attached to the special point of a repulsive surface of the same geometry and can be used in calculation of en…
Lévy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal Lévy walk strategy for random searches
Federal University of Rio Grande do Norte（北里奥格兰德联邦大学）；Harvard-Smithsonian Center for Astrophysics（哈佛-史密森天体物理中心）；Universidade Federal de Pernambuco（伯南布哥联邦大学）；National Institute of Science and Technology of Complex Systems, Federal University of Rio Grande do Norte（复杂系统科学与技术国家研究所，北里奥格兰德联邦大学）
We study two different types of systems with many absorbing states (with and without a conservation law) and scrutinize the effect of walls/boundaries (either absorbing or reflecting) into them. In some cases, non-trivial structured absorbing configurations (characterized by a background field) develop around the wall. We study such structures using a mean-field approach as well as computer simulations. The main results are: i) for systems in the directed percolation class, a very fast (exponential) convergence of the background to its bulk value is observed; ii) for systems with a conservatio…
The master equation for a probability density function (pdf) driven by Lévy noise, if conditioned to conform with the principle of detailed balance, admits a transformation to a contractive strongly continuous semigroup dynamics. Given a priori a functional form of the semigroup potential, we address the ground-state reconstruction problem for generic Lévy-stable semigroups, for {\em all} values of the stability index $μ\in (0,2)$. That is known to resolve an invariant pdf for confined Lévy flights (e.g. the former jump-type process). Jeopardies of the procedure are discussed, with a focus on:…
