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 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana A1 - Kulik, Alexei Michajlovič T1 - Coupling distances between Lévy measures and applications to noise sensitivity of SDE N2 - We introduce the notion of coupling distances on the space of Lévy measures in order to quantify rates of convergence towards a limiting Lévy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Lévy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two Lévy 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)16 KW - Lévy diffusion approximation KW - coupling methods KW - Skorokhod' s invariance principle KW - statistical model selection Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-68886 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 - INPR A1 - Högele, Michael A1 - Ruffino, Paulo T1 - Averaging along Lévy diffusions in foliated spaces N2 - We consider an SDE driven by a Lévy noise on a foliated manifold, whose trajectories stay on compact leaves. We determine the effective behavior of the system subject to a small smooth transversal perturbation of positive order epsilon. More precisely, we show that the average of the transversal component of the SDE converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to the invariant measures on the leaves (of the unpertubed system) as epsilon goes to 0. In particular we give upper bounds for the rates of convergence. The main results which are proved for pure jump Lévy processes complement the result by Gargate and Ruffino for Stratonovich SDEs to Lévy driven SDEs of Marcus type. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)10 KW - Averaging principle KW - foliated diffusion KW - Lévy diffusions on manifolds KW - canonical Marcus integration Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64926 SN - 2193-6943 ER - TY - INPR A1 - Flandoli, Franco A1 - Högele, Michael T1 - A solution selection problem with small stable perturbations N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 8 KW - stochastic differential equations KW - singular drifts KW - zero-noise limit KW - Peano phenomena KW - non-uniqueness Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71205 SN - 2193-6943 VL - 3 IS - 8 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana A1 - Kulik, Alexei Michajlovič T1 - On the calibration of Lévy driven time series with coupling distances : an application in paleoclimate N2 - This article aims at the statistical assessment of time series with large fluctuations in short time, which are assumed to stem from a continuous process perturbed by a Lévy process exhibiting a heavy tail behavior. We propose an easily implementable procedure to estimate efficiently the statistical difference between the noisy behavior of the data and a given reference jump measure in terms of so-called coupling distances. After a short introduction to Lévy processes and coupling distances we recall basic statistical approximation results and derive rates of convergence. In the sequel the procedure is elaborated in detail in an abstract setting and eventually applied in a case study to simulated and paleoclimate data. It indicates the dominant presence of a non-stable heavy-tailed jump Lévy component for some tail index greater than 2. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 2 KW - time series with heavy tails KW - index of stability KW - goodness-of-fit KW - empirical Wasserstein distance Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69781 SN - 2193-6943 VL - 3 IS - 2 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Högele, Michael A1 - Pavlyukevich, Ilya T1 - Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed Lévy type noise N2 - We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors each of which supports a unique ergodic probability measure, which includes in particular the class of Morse–Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavytailed Lévy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a Itô, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions in case of inward pointing vector fields in the limit of ε-> 0 has been investigated by the authors. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique ε-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures. As examples we consider α-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 5 KW - hyperbolic dynamical system KW - Morse-Smale property KW - stable limit cycle KW - small noise asymptotic KW - multiplicative noise Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70639 SN - 2193-6943 VL - 3 IS - 5 PB - Universitätsverlag Potsdam CY - Potsdam 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 - INPR A1 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana T1 - Transportation distances and noise sensitivity of multiplicative Lévy SDE with applications N2 - This article assesses the distance between the laws of stochastic differential equations with multiplicative Lévy noise on path space in terms of their characteristics. The notion of transportation distance on the set of Lévy 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 2 KW - stochastic differential equations KW - multiplicative Lévy noise KW - Lévy type processes KW - heavy-tailed distributions KW - model selection KW - Wasserstein distance KW - time series Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-86693 SN - 2193-6943 VL - 5 IS - 2 PB - Universitätsverlag Potsdam CY - Potsdam 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 -