Ilya Pavlyukevich

Lévy flights, non-local search and simulated annealing

We solve a problem of non-convex stochastic optimisation with help of simulated annealing of Levy 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.

Lévy flights: transitions and meta-stability

We consider L?vy flights of stability index ? (0, 2) in a potential landscape in the limit of a small noise parameter. We give a purely probabilistic description of the random dynamics on the basis of a special decomposition of the driving L?vy 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 ??? of the noise intensity ?. This allows us to obtain meta-stability results for a jump diffusion in a double-well potential.

MODEL REDUCTION AND STOCHASTIC RESONANCE

We provide a mathematical underpinning of the physically widely known phenomenon of stochastic resonance, i.e. the optimal noise-induced increase of a dynamical system’s sensitivity and ability to amplify small periodic signals. The eect was rst discovered in energy-balance models designed for a qualitative understanding of global glacial cycles. More recently, stochastic resonance has been rediscovered in more subtle and realistic simulations interpreting paleoclimatic data: the Dansgaard-Oeschger and Heinrich events. The underlying mathematical model is a diusion in a periodically changing potential landscape with large forcing period. We study ’optimal tuning of the diusion trajectories with the deterministic input forcing by means of the spectral power amplication measure. Our results contain a surprise: due to small fluctuations in the potential valley bottoms the diusion | contrary to physical folklore | does not show tuning patterns corresponding to continuous time Markov chains which describe the reduced motion on the metastable states. This discrepancy can only be avoided for more robust notions of tuning, e.g. spectral amplication after elimination of the small fluctuations.

Stochastic resonance in two-state Markov chains

Abstract. In this paper we introduce a model which provides a new approach to the phenomenon of stochastic resonance. It is based on the study of the properties of the stationary distribution of the underlying stochastic process. We derive the formula for the spectral power aplification coefficient, study its asymptotic properties and dependence on parameters.

Marcus versus Stratonovich for systems with jump noise

The famous Ito–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, Levy flights or more general jump processes. In present communication we present an in-depth comparison of the Ito, 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.

FIRST EXIT TIMES OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY MULTIPLICATIVE LÉVY NOISE WITH HEAVY TAILS

In this paper, we study first exit times from a bounded domain of a gradient dynamical system Ẏt = -∇U(Yt) perturbed by a small multiplicative Levy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular, we determine the asymptotics of the first exit time of solutions of Ito, Stratonovich and Marcus canonical SDEs.

The Exit Problem from a Neighborhood of the Global Attractor for Dynamical Systems Perturbed by Heavy-Tailed Lévy Processes

We consider a finite-dimensional deterministic dynamical system with the global attractor 𝒜 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 𝒜. We perturb the dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ϵ > 0 and solve the asymptotic first exit time and location problem from D in the limit of ϵ↘0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of ϵ, just as in the case when 𝒜 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 α-stable Lévy noise.

Lévy ratchet in a weak noise limit: Theory and simulation

Abstract. We study the motion of a particle in a time-independent periodic potential with broken mirror symmetry under action of a Lévy-stable noise (Lévy ratchet). We develop an analytical approach to the problem based on the asymptotic probabilistic method of decomposition proposed by P. Imkeller and I. Pavlyukevich [J. Phys. A 39, L237 (2006); Stoch. Proc. Appl. 116, 611 (2006)]. We derive analytical expressions for the quantities characterizing the particle’s motion, namely for the splitting probabilities of the first escape from a single well, for the transition probabilities to other wells and for the probability current. We pay particular attention to the interplay between the asymmetry of the ratchet potential and the asymmetry (skewness) of the Lévy noise. Extensive numerical simulations demonstrate a good agreement with the analytical predictions for sufficiently small intensities of the Lévy noise driving the particle.

Stochastic resonance : a mathematical approach in the small noise limit

Heuristics of noise induced transitions Transitions for time homogeneous dynamical systems with small noise Semiclassical theory of stochastic resonance in dimension 1 Large deviations and transitions between meta-stable states of dynamical systems with small noise and weak inhomogeneity Supplementary tools Laplaces method Bibliography Index

Two Mathematical Approaches to Stochastic Resonance

We consider a random dynamical system describing the diffusion of a small-noise Brownian particle in a double-well potential with a periodic perturbation of very large period. According to the physics literature, the system is in stochastic resonance if its random trajectories are tuned in an optimal way to the deterministic periodic forcing. The quality of periodic tuning is measured mostly by the amplitudes of the spectral components of the random trajectories corresponding to the forcing frequency. Reduction of the diffusion dynamics in the small noise limit to a Markov chain jumping between its meta-stable states plays an important role.