@unpublished{DebusscheHoegeleImkeller2013,
  author    = {Debussche, Arnaud and H{\"o}gele, Michael and Imkeller, Peter},
  title     = {The dynamics of nonlinear reaction-diffusion equations with small levy noise},
  series = {Lecture notes in mathematics : a collection of informal reports and seminars},
  volume    = {2085},
  journal   = {Lecture notes in mathematics : a collection of informal reports and seminars},
  publisher = {Springer},
  address   = {Berlin},
  isbn      = {978-3-319-00828-8; 978-3-319-00827-1},
  issn      = {0075-8434},
  doi       = {10.1007/978-3-319-00828-8_1},
  pages     = {1 -- 10},
  year      = {2013},
  abstract  = {Our primary interest in this book lies in the study of dynamical properties of reaction-diffusion equations perturbed by L{\´e}vy noise of intensity ? in the small noise limit ??0 .},
  language  = {en}
}
@article{DebusscheHoegeleImkeller2013,
  author    = {Debussche, Arnaud and H{\"o}gele, Michael and Imkeller, Peter},
  title     = {The Fine Dynamics of the Chafee-Infante Equation},
  series = {Lecture notes in mathematics : a collection of informal reports and seminars},
  volume    = {2085},
  journal   = {Lecture notes in mathematics : a collection of informal reports and seminars},
  publisher = {Springer},
  address   = {Berlin},
  isbn      = {978-3-319-00828-8; 978-3-319-00827-1},
  issn      = {0075-8434},
  doi       = {10.1007/978-3-319-00828-8_2},
  pages     = {11 -- 43},
  year      = {2013},
  abstract  = {In this chapter, we introduce the deterministic Chafee-Infante equation. This equation has been the subject of intense research and is very well understood now. We recall some properties of its longtime dynamics and in particular the structure of its attractor. We then define reduced domains of attraction that will be fundamental in our study and give a result describing precisely the time that a solution starting form a reduced domain of attraction needs to reach a stable equilibrium. This result is then proved using the detailed knowledge of the attractor and classical tools such as the stable and unstable manifolds in a neighborhood of an equilibrium.},
  language  = {en}
}
@article{DebusscheHoegeleImkeller2013,
  author    = {Debussche, Arnaud and H{\"o}gele, Michael and Imkeller, Peter},
  title     = {The stochastic chafee-infante equation},
  series = {Lecture notes in mathematics : a collection of informal reports and seminars},
  volume    = {2085},
  journal   = {Lecture notes in mathematics : a collection of informal reports and seminars},
  publisher = {Springer},
  address   = {Berlin},
  isbn      = {978-3-319-00828-8; 978-3-319-00827-1},
  issn      = {0075-8434},
  doi       = {10.1007/978-3-319-00828-8_3},
  pages     = {45 -- 68},
  year      = {2013},
  abstract  = {In this preparatory chapter, the tools of stochastic analysis needed for the investigation of the asymptotic behavior of the stochastic Chafee-Infante equation are provided. In the first place, this encompasses a recollection of basic facts about L{\´e}vy processes with values in Hilbert spaces. Playing the role of the additive noise processes perturbing the deterministic Chafee-Infante equation in the systems the stochastic dynamics of which will be our main interest, symmetric ?-stable L{\´e}vy processes are in the focus of our investigation (Sect. 3.1).},
  language  = {en}
}
@article{DebusscheHoegeleImkeller2013,
  author    = {Debussche, Arnaud and H{\"o}gele, Michael and Imkeller, Peter},
  title     = {Localization and metastability},
  series = {Lecture notes in mathematics : a collection of informal reports and seminars},
  volume    = {2085},
  journal   = {Lecture notes in mathematics : a collection of informal reports and seminars},
  publisher = {Springer},
  address   = {Berlin},
  isbn      = {978-3-319-00828-8; 978-3-319-00827-1},
  issn      = {0075-8434},
  doi       = {10.1007/978-3-319-00828-8_7},
  pages     = {131 -- 149},
  year      = {2013},
  abstract  = {In this chapter, equipped with our previously obtained knowledge of exit and transition times in the limit of small noise amplitude ??0 , we shall investigate the global asymptotic behavior of our jump diffusion process in the time scale in which transitions occur, i.e. in the scale given by ?0(?)=?(1?Bc?(0)),?,?>0 . It turns out that in this time scale, the switching of the diffusion between neighborhoods of the stable solutions ? ± can be well described by a Markov chain jumping back and forth between two states with a characteristic Q-matrix determined by the quantities ?((D±0)c)?(Bc?(0)) as jumping rates.},
  language  = {en}
}
@article{DebusscheHoegeleImkeller2011,
  author    = {Debussche, Arnaud and H{\"o}gele, Michael and Imkeller, Peter},
  title     = {Asymptotic first exit times of the chafee-infante equation with small heavy-tailed levy noise},
  series = {Electronic communications in probability},
  volume    = {16},
  journal   = {Electronic communications in probability},
  number    = {3-4},
  publisher = {Univ. of Washington, Mathematics Dep.},
  address   = {Seattle},
  issn      = {1083-589X},
  pages     = {213 -- 225},
  year      = {2011},
  abstract  = {This article studies the behavior of stochastic reaction-diffusion equations driven by additive regularly varying pure jump Levy noise in the limit of small noise intensity. It is shown that the law of the suitably renormalized first exit times from the domain of attraction of a stable state converges to an exponential law of parameter 1 in a strong sense of Laplace transforms, including exponential moments. As a consequence, the expected exit times increase polynomially in the inverse intensity, in contrast to Gaussian perturbations, where this growth is known to be of exponential rate.},
  language  = {en}
}
@unpublished{FlandoliHoegele2014,
  author    = {Flandoli, Franco and H{\"o}gele, Michael},
  title     = {A solution selection problem with small stable perturbations},
  volume    = {3},
  number    = {8},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71205},
  pages     = {43},
  year      = {2014},
  abstract  = {The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is a general alpha-stable process. It is proved that extremal solutions are selected and the probability of selection is computed. Detailed analysis of the characteristic function of an exit time form on the half-line is performed, with a suitable decomposition in small and large jumps adapted to the singular drift.},
  language  = {en}
}
@article{GairingHoegeleKosenkova2017,
  author    = {Gairing, Jan and H{\"o}gele, Michael and Kosenkova, Tetiana},
  title     = {Transportation distances and noise sensitivity of multiplicative Levy SDE with applications},
  series = {Stochastic processes and their application},
  volume    = {128},
  journal   = {Stochastic processes and their application},
  number    = {7},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0304-4149},
  doi       = {10.1016/j.spa.2017.09.003},
  pages     = {2153 -- 2178},
  year      = {2017},
  abstract  = {This article assesses the distance between the laws of stochastic differential equations with multiplicative Levy noise on path space in terms of their characteristics. The notion of transportation distance on the set of Levy kernels introduced by Kosenkova and Kulik yields a natural and statistically tractable upper bound on the noise sensitivity. This extends recent results for the additive case in terms of coupling distances to the multiplicative case. The strength of this notion is shown in a statistical implementation for simulations and the example of a benchmark time series in paleoclimate.},
  language  = {en}
}
@unpublished{GairingHoegeleKosenkova2016,
  author    = {Gairing, Jan and H{\"o}gele, Michael and Kosenkova, Tetiana},
  title     = {Transportation distances and noise sensitivity of multiplicative L{\´e}vy SDE with applications},
  volume    = {5},
  number    = {2},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-86693},
  pages     = {24},
  year      = {2016},
  abstract  = {This article assesses the distance between the laws of stochastic differential equations with multiplicative L{\´e}vy noise on path space in terms of their characteristics. The notion of transportation distance on the set of L{\´e}vy kernels introduced by Kosenkova and Kulik yields a natural and statistically tractable upper bound on the noise sensitivity. This extends recent results for the additive case in terms of coupling distances to the multiplicative case. The strength of this notion is shown in a statistical implementation for simulations and the example of a benchmark time series in paleoclimate.},
  language  = {en}
}
@unpublished{GairingHoegeleKosenkovaetal.2013,
  author    = {Gairing, Jan and H{\"o}gele, Michael and Kosenkova, Tetiana and Kulik, Alexei Michajlovič},
  title     = {Coupling distances between L{\´e}vy measures and applications to noise sensitivity of SDE},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68886},
  year      = {2013},
  abstract  = {We introduce the notion of coupling distances on the space of L{\´e}vy measures in order to quantify rates of convergence towards a limiting L{\´e}vy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the L{\´e}vy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two L{\´e}vy diffusions in terms of the couping distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data.},
  language  = {en}
}
@article{GairingHoegeleKosenkovaetal.2015,
  author    = {Gairing, Jan and H{\"o}gele, Michael and Kosenkova, Tetiana and Kulik, Alexei Michajlovič},
  title     = {Coupling distances between Levy measures and applications to noise sensitivity of SDE},
  series = {Stochastics and dynamic},
  volume    = {15},
  journal   = {Stochastics and dynamic},
  number    = {2},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0219-4937},
  doi       = {10.1142/S0219493715500094},
  pages     = {25},
  year      = {2015},
  abstract  = {We introduce the notion of coupling distances on the space of Levy measures in order to quantify rates of convergence towards a limiting Levy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Levy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two Levy diffusions in terms of the coupling distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data.},
  language  = {en}
}