Stochastische Dynamik von Klimazuständen
Zusammenfassung der Projektergebnisse
This three year research project focused on the rigorous mathematical study of qualitative behaviour of solutions of non-linear stochastic differential equations arising in low-dimensional climate modelling. The research is mainly motivated by two paradigms of geophysical phenomena. The first one refers to the strong — almost deterministic — periodicity with which ice ages occurred during the last million years of Earth's history. Its physical and mathematical study leads to the concept of stochastic resonance, which captures an effect of strong amplification of weak external deterministic periodic forcing through transitions between different stable equilibria triggered by random noise. The weak external forcing of Earth's planetary trajectory is caused by the gravitation of heavy planets, the noise triggering ice ages by random variations of insolation or atmospheric processes. Another phenomenon is the physically still not entirely understood phenomenon of catastrophic climate instability during the last ice age, leading to the so-called Dansgaard-Oeschger events. The mathematical modelling approach suggested by physicists puts emphasis on the influence of non-Gaussian Levy noise with heavy tails (Levy flights). Surprisingly, despite the obvious demand of the physics community for a rigorous mathematical understanding of these effects, only few mathematical papers treated related questions prior to this project. In this project, a mathematical explanation of stochastic resonance for diffusions was given, in particular interpreting the mechanism of optimal periodic tuning, responsible for the periodicity of ice ages in low dimensional conceptual models of climate dynamics. Stochastic resonance for multi-stable diffusions triggered by time delayed feedback was interpreted, and a theory of finite horizon predictability was initiated for simple lowdimensional diffusion models. Thanks to its practical relevance, the theory of heavy-tailed jump diffusions developed a quite unexpected and promising momentum. It turned out that the effects of big jumps responsible for strongly non-local behaviour of random processes can be successfully used to construct diffusion models which fit quite well real paleo-climatic data. We developed first statistical testing methods based on modern power variations techniques to underpin this. We developed a mathematical theory of meta-stable behaviour of models obtained this way, namely small noise jump diffusions driven by Levy flights. This theory was generalised to annealed jump-diffusions and applied to problems of stochastic optimisation. We also showed that normal diffusion exit behaviour cannot be recovered from the one of jump diffusions by reducing the frequency with which big jumps occur.
Projektbezogene Publikationen (Auswahl)
- Limit theorems for p-variations of solutions of SDEs driven by additive non-Gaussian stable Levy noise
Hein, C., Imkeller, P. and Pavlyukevich, I.
- Two mathematical approaches to stochastic resonance. In: J.-D. Deuschel and A. Greven (eds.). Interacting Stochastic Systems, Springer, 2004
Herrmann, S., Imkeller, P. and Pavlyukevich, I.
- A two state model for noise-induced resonance in bistable systems with delays. Stochastics and Dynamics, 5, 247-270, 2005
Fischer, M. and Imkeller, P.
- First exit times of SDEs driven by stable Levy processes. Stochastic Processes and their Applications, 116(4), 611-642, 2006
Imkeller, P. and Pavlyukevich, I.
- Levy flights: transitions and metastability. Journal of Physics A: Mathematical and General, 39, L237-L246, 2006
Imkeller, P. and Pavlyukevich, I.
- Noise-induced resonance in bistable systems caused by delay feedback. Stochastic Analysis and Applications, 24(1), 135-194, 2006
Fischer, M. and Imkeller, P.
- Transition times and stochastic resonance for multidimensional diffusions with time periodic drift: A large deviations approach. Annals of Applied Probability, 16(4), 1851-1892, 2006
Herrmann, S., Imkeller, P. and Peithmann, D.
- Cooling down Levy flights. Journal of Physics A: Mathematical and Theoretical, 40, 12299-12313, 2007
Pavlyukevich, I.
- Levy flights, non-local search and simulated annealing. Journal of Computational Physics, 226(2), 1830-1844, 2007
Pavlyukevich, I.
- Large deviations and a Kramers' type law for self-stabilizing diffusions. Annals of Applied Probability, 18(4), 1379-1423, 2008
Herrmann, S., Imkeller, P. and Peithmann, D.
- Local Lyapunov exponents. Lecture Notes in Mathematics 1963, Springer 2008
Siegert, W.
- Metastable behaviour of small noise Levy-driven diffusions. ESAIM: Probability and Statistics, 12, 412-437, 2008
Imkeller, P. and Pavlyukevich, I.
- Ruin probabilities of small noise jump-diffusions with heavy tails. Applied Stochastic Models in Business and Industry, 24(1), 65-82, 2008
Pavlyukevich, I
- Simulated annealing for Levy-driven jump-diffusions. Stochastic Processes and their Applications, 118, 1071-1105, 2008
Pavlyukevich, I.
- First exit times for Levy-driven diffusions with exponentially light jumps. The Annals of Probability, 37(2), 530-564, 2009
Imkeller, P., Pavlyukevich, I. and Wetzel, T.
