@article{HoegelePavlyukevich2014,
  author    = {Hoegele, Michael and Pavlyukevich, Ilya},
  title     = {The exit problem from a neighborhood of the global attractor for dynamical systems perturbed by heavy-tailed levy processes},
  series = {Stochastic analysis and applications},
  volume    = {32},
  journal   = {Stochastic analysis and applications},
  number    = {1},
  publisher = {Taylor \& Francis Group},
  address   = {Philadelphia},
  issn      = {0736-2994},
  doi       = {10.1080/07362994.2014.858554},
  pages     = {163 -- 190},
  year      = {2014},
  abstract  = {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.},
  language  = {en}
}
@article{HoegelePavlyukevich2015,
  author    = {H{\"o}gele, Michael and Pavlyukevich, Ilya},
  title     = {Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Levy type noise},
  series = {Stochastics and dynamic},
  volume    = {15},
  journal   = {Stochastics and dynamic},
  number    = {3},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0219-4937},
  doi       = {10.1142/S0219493715500197},
  pages     = {26},
  year      = {2015},
  abstract  = {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.},
  language  = {en}
}