We solve a problem of non-convex stochastic optimisation with help of simulated annealing of Lévy flights of a variable stability index. The search of the ground state of an unknown potential is non-local due to big jumps of the Levy flights process. The convergence to the ground state is fast due to a polynomial decrease rate of the temperature.
We consider Lévy flights of stability index α ∈ (0, 2) in a potential landscape in the limit of small noise parameter. We give a purely probabilistic description of the random dynamics based on a special decomposition of the driving Lévy processes into independent small jumps and compound Poisson parts. We prove that escape times from a potential well are exponentially distributed and their mean values increase as a power ε −α of the noise intensity ε. This allows to obtain metastability results for a jump-diffusion in a double-well potential.
Lévy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments-the "jump lengths"-are drawn from an α-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 firstpassage 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 α and the skewness (asymmetry) parameter β. The other approach is based on the stochastic Langevin equation with α-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.∞ 0 t℘(t)dt can capture some aspects of this dynamics, ‡ the full information encoded in ℘(t) provides significant additional insight [114,115,116]. Here we study the first-passage properties for a general class of α-stable Lévy laws. We go beyond previous approaches [113,117,118,119,120,121,122,123,124,125,126,127,128] focusing on symmetric and one-sided α-stable relocation distributions and consider α-stable laws with arbitrary asymmetry in semi-infinite and bounded domains. Our approach is based on the convenient formulation of LFs in terms of the space-fractional diffusion equation. We derive these integro-differential equations for LFs based on general asymmetric α-stable distributions of relocation lengths in finite domains, and thus go beyond studies of the exit time and escape probability in bounded domain for symmetric LFs [129,130,131]. An important aspect of this study is that we complement our results with numerical analyses of the (stochastic) Langevin equation for LFs and show how both approaches complement each other.The paper is organised as follows. In section 2 we define Lévy stable laws and the associated fractional diffusion equation. In section 3 we set up our numerical model for the fractional diffusion equation and the associated Langevin equation. Moreover, a comparison between the numerical method and α-stable distributions for symmetric and asymmetric den...
The famous Itô-Stratonovich dilemma arises when one examines a dynamical system with a multiplicative white noise. In physics literature, this dilemma is often resolved in favour of the Stratonovich prescription because of its two characteristic properties valid for systems driven by Brownian motion: (i) it allows physicists to treat stochastic integrals in the same way as conventional integrals, and (ii) it appears naturally as a result of a small correlation time limit procedure. On the other hand, the Marcus prescription [IEEE Trans. Inform. Theory 24, 164 (1978); Stochastics 4, 223 (1981)] should be used to retain (i) and (ii) for systems driven by a Poisson process, Lévy flights or more general jump processes. In present communication we present an in-depth comparison of the Itô, Stratonovich, and Marcus equations for systems with multiplicative jump noise. By the examples of a real-valued linear system and a complex oscillator with noisy frequency (the Kubo-Anderson oscillator) we compare solutions obtained with the three prescriptions.
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