TY - JOUR A1 - Debussche, Arnaud A1 - Högele, Michael A1 - Imkeller, Peter T1 - Asymptotic first exit times of the chafee-infante equation with small heavy-tailed levy noise JF - Electronic communications in probability N2 - 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. KW - stochastic reaction diffusion equation with heavy-tailed Levy noise KW - first exit times KW - regularly varying Levy process KW - small noise asymptotics Y1 - 2011 SN - 1083-589X VL - 16 IS - 3-4 SP - 213 EP - 225 PB - Univ. of Washington, Mathematics Dep. CY - Seattle ER - TY - INPR A1 - Debussche, Arnaud A1 - Högele, Michael A1 - Imkeller, Peter T1 - The dynamics of nonlinear reaction-diffusion equations with small levy noise T2 - Lecture notes in mathematics : a collection of informal reports and seminars T2 - Lecture Notes in Mathematics N2 - Our primary interest in this book lies in the study of dynamical properties of reaction-diffusion equations perturbed by Lévy noise of intensity ? in the small noise limit ??0 . Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 U6 - https://doi.org/10.1007/978-3-319-00828-8_1 SN - 0075-8434 VL - 2085 SP - 1 EP - 10 PB - Springer CY - Berlin ER - TY - JOUR A1 - Debussche, Arnaud A1 - Högele, Michael A1 - Imkeller, Peter T1 - The Fine Dynamics of the Chafee-Infante Equation JF - Lecture notes in mathematics : a collection of informal reports and seminars JF - Lecture Notes in Mathematics N2 - 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. Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 U6 - https://doi.org/10.1007/978-3-319-00828-8_2 SN - 0075-8434 VL - 2085 SP - 11 EP - 43 PB - Springer CY - Berlin ER - TY - JOUR A1 - Debussche, Arnaud A1 - Högele, Michael A1 - Imkeller, Peter T1 - The stochastic chafee-infante equation JF - Lecture notes in mathematics : a collection of informal reports and seminars JF - Lecture Notes in Mathematics N2 - 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 Lévy 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 Lévy processes are in the focus of our investigation (Sect. 3.1). Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 U6 - https://doi.org/10.1007/978-3-319-00828-8_3 SN - 0075-8434 VL - 2085 SP - 45 EP - 68 PB - Springer CY - Berlin ER - TY - JOUR A1 - Debussche, Arnaud A1 - Högele, Michael A1 - Imkeller, Peter T1 - Localization and metastability JF - Lecture notes in mathematics : a collection of informal reports and seminars JF - Lecture Notes in Mathematics N2 - 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. Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 U6 - https://doi.org/10.1007/978-3-319-00828-8_7 SN - 0075-8434 VL - 2085 SP - 131 EP - 149 PB - Springer CY - Berlin ER - TY - JOUR A1 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana A1 - Kulik, Alexei Michajlovič T1 - Coupling distances between Levy measures and applications to noise sensitivity of SDE JF - Stochastics and dynamic N2 - 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. KW - Levy diffusion approximation KW - coupling methods KW - principle KW - statistical model selection Y1 - 2015 U6 - https://doi.org/10.1142/S0219493715500094 SN - 0219-4937 SN - 1793-6799 VL - 15 IS - 2 PB - World Scientific CY - Singapore 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 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana T1 - Transportation distances and noise sensitivity of multiplicative Levy SDE with applications JF - Stochastic processes and their application N2 - 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. KW - Stochastic differential equations KW - Multiplicative Levy noise KW - Levy type processes KW - Heavy-tailed distributions KW - Model selection KW - Wasserstein distance KW - Time series Y1 - 2017 U6 - https://doi.org/10.1016/j.spa.2017.09.003 SN - 0304-4149 SN - 1879-209X VL - 128 IS - 7 SP - 2153 EP - 2178 PB - Elsevier CY - Amsterdam ER -