TY - JOUR A1 - Padash, Amin A1 - Chechkin, Aleksei V. A1 - Dybiec, Bartlomiej A1 - Pavlyukevich, Ilya A1 - Shokri, Babak A1 - Metzler, Ralf T1 - First-passage properties of asymmetric Levy flights JF - Journal of physics : A, Mathematical and theoretical N2 - Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the 'jump lengths'—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy 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 Lévy 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 Lévy 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 and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -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. KW - Levy flights KW - first-passage KW - search dynamics Y1 - 2019 U6 - https://doi.org/10.1088/1751-8121/ab493e SN - 1751-8113 SN - 1751-8121 VL - 52 IS - 45 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Högele, Michael A1 - Pavlyukevich, Ilya T1 - Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Levy type noise JF - Stochastics and dynamic N2 - We consider a finite dimensional deterministic dynamical system with finitely many local attractors K-iota, each of which supports a unique ergodic probability measure P-iota, perturbed by a multiplicative non-Gaussian heavy-tailed Levy noise of small intensity epsilon > 0. We show that the random system exhibits a metastable behavior: there exists a unique epsilon-dependent time scale on which the system reminds of a continuous time Markov chain on the set of the invariant measures P-iota. In particular our approach covers the case of dynamical systems of Morse-Smale type, whose attractors consist of points and limit cycles, perturbed by multiplicative alpha-stable Levy noise in the Ito, Stratonovich and Marcus sense. As examples we consider alpha-stable Levy perturbations of the Duffing equation and Pareto perturbations of a biochemical birhythmic system with two nested limit cycles. KW - Hyperbolic dynamical system KW - Morse-Smale property KW - physical SRB measures KW - stable limit cycle KW - small noise asymptotic KW - alpha-stable Levy process KW - multiplicative noise KW - Ito integral KW - Stratonovich integral KW - stochastic Marcus (canonical) differential equation KW - multiscale dynamics KW - metastability KW - embedded Markov chain KW - randomly forced Duffing equation KW - birhythmic behavior Y1 - 2015 U6 - https://doi.org/10.1142/S0219493715500197 SN - 0219-4937 SN - 1793-6799 VL - 15 IS - 3 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Pavlyukevich, Ilya A1 - Li, Yongge A1 - Xu, Yong A1 - Chechkin, Aleksei V. T1 - Directed transport induced by spatially modulated Levy flights JF - Journal of physics : A, Mathematical and theoretical N2 - In this paper we study the dynamics of a particle in a ratchet potential subject to multiplicative alpha-stable Levy noise, alpha is an element of(0, 2), in the limit of a noise amplitude epsilon -> 0. We compare the dynamics for Ito and Marcus multiplicative noises and obtain the explicit asymptotics of the escape time in the wells and transition probabilities between the wells. A detailed analysis of the noise-induced current is performed for the Seebeck ratchet with a weak multiplicative noise for alpha is an element of(0, 2]. KW - Levy flights KW - multiplicative noise KW - Seebeck ratchet KW - directed transport Y1 - 2015 U6 - https://doi.org/10.1088/1751-8113/48/49/495004 SN - 1751-8113 SN - 1751-8121 VL - 48 IS - 49 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Hoegele, Michael A1 - Pavlyukevich, Ilya T1 - The exit problem from a neighborhood of the global attractor for dynamical systems perturbed by heavy-tailed levy processes JF - Stochastic analysis and applications N2 - We consider a finite-dimensional deterministic dynamical system with the global attractor ? which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing ?. We perturb the dynamical system by a multiplicative heavy tailed Levy noise of small intensity E>0 and solve the asymptotic first exit time and location problem from D in the limit of E?0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of E, just as in the case when ? is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative -stable Levy noise. KW - alpha-stable Levy process KW - Canonical (Marcus) SDE KW - First exit location KW - First exit time KW - Global attractor KW - Ito SDE KW - Multiplicative noise KW - Regular variation KW - Stratonovich SDE KW - Van der Pol oscillator Y1 - 2014 U6 - https://doi.org/10.1080/07362994.2014.858554 SN - 0736-2994 SN - 1532-9356 VL - 32 IS - 1 SP - 163 EP - 190 PB - Taylor & Francis Group CY - Philadelphia ER -