7 research outputs found
Large deviations and a Kramers' type law for self-stabilizing diffusions
We investigate exit times from domains of attraction for the motion of a
self-stabilized particle traveling in a geometric (potential type) landscape
and perturbed by Brownian noise of small amplitude. Self-stabilization is the
effect of including an ensemble-average attraction in addition to the usual
state-dependent drift, where the particle is supposed to be suspended in a
large population of identical ones. A Kramers' type law for the particle's exit
from the potential's domains of attraction and a large deviations principle for
the self-stabilizing diffusion are proved. It turns out that the exit law for
the self-stabilizing diffusion coincides with the exit law of a potential
diffusion without self-stabilization and a drift component perturbed by average
attraction. We show that self-stabilization may substantially delay the exit
from domains of attraction, and that the exit location may be completely
different.Comment: Published in at http://dx.doi.org/10.1214/07-AAP489 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Transition times and stochastic resonance for multidimensional diffusions with time periodic drift: A large deviations approach
We consider potential type dynamical systems in finite dimensions with two
meta-stable states. They are subject to two sources of perturbation: a slow
external periodic perturbation of period and a small Gaussian random
perturbation of intensity , and, therefore, are mathematically
described as weakly time inhomogeneous diffusion processes. A system is in
stochastic resonance, provided the small noisy perturbation is tuned in such a
way that its random trajectories follow the exterior periodic motion in an
optimal fashion, that is, for some optimal intensity . The
physicists' favorite, measures of quality of periodic tuning--and thus
stochastic resonance--such as spectral power amplification or signal-to-noise
ratio, have proven to be defective. They are not robust w.r.t. effective model
reduction, that is, for the passage to a simplified finite state Markov chain
model reducing the dynamics to a pure jumping between the meta-stable states of
the original system. An entirely probabilistic notion of stochastic resonance
based on the transition dynamics between the domains of attraction of the
meta-stable states--and thus failing to suffer from this robustness defect--was
proposed before in the context of one-dimensional diffusions. It is
investigated for higher-dimensional systems here, by using extensions and
refinements of the Freidlin--Wentzell theory of large deviations for time
homogeneous diffusions. Large deviations principles developed for weakly time
inhomogeneous diffusions prove to be key tools for a treatment of the problem
of diffusion exit from a domain and thus for the approach of stochastic
resonance via transition probabilities between meta-stable sets.Comment: Published at http://dx.doi.org/10.1214/105051606000000385 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Large deviations and exit time asymptotics for diffusions and stochastic resonance
Diese Arbeit behandelt die Asymptotik von Austritts- und Übergangszeiten für gewisse schwach zeitinhomogene Diffusionsprozesse. Darauf basierend wird ein probabilistischer Begriff der stochastischen Resonanz (SR) studiert. Techniken der großen Abweichungen spielen eine zentrale Rolle. Im ersten Teil der Arbeit (Kapitel 1-3) werden Resultate aus der Theorie der großen Abweichungen für zeithomogene Diffusionen rekapituliert. Es werden die klassischen Resultate von Freidlin und Wentzell und Erweiterungen dieser Theorie präsentiert, und es wird an das Kramers''sche Austrittszeitengesetz erinnert. Teil II befasst sich mit dem Phänomen der SR, d.h. mit Periodizitätseigenschaften von Diffusionen. In Kapitel 4 werden physikalische Maße zur Messung der Periodizität diskutiert. Deren Nachteile legen es nahe, einem alternativen, probabilistischen Ansatz zu folgen, der hier behandelt wird. Das 5. Kapitel dient der Herleitung eines gleichmäßigen Prinzips der großen Abweichungen für Diffusionen mit schwach zeitabhängigem, periodischem Drift. Die Gleichmäßigkeit des Prinzips ermöglicht die exakte Bestimmung exponentieller Übergangsraten in Kapitel 6, das die zentralen Ergebnisse des 2. Teils beinhaltet. Hierdurch wird die Maximierung gewisser Übergangswahrscheinlichkeiten ermöglicht, was zum in Kapitel 7 studierten Resonanzbegriff führt. Teil III der Arbeit setzt sich mit der Asymptotik von Austrittszeiten sogenannter selbststabilisierender Diffusionen auseinander. In Kapitel 8 wird der Zusammenhang zwischen interagierenden Teilchensystemen und selbststabilisierenden Diffusionen erläutert und die Existenz- und Eindeutigkeitsfrage behandelt. Das 9. Kapitel dient dem Studium der großen Abweichungen dieser Klasse von Diffusionen. In Kapitel 10 wird das Kramers''sche Austrittszeitengesetz auf selbststabilisierende Diffusionen übertragen, und in Kapitel 11 wird der Einfluß der selbststabilisierenden Komponente auf das Austrittszeitengesetz illustriert.In this thesis, we study the asymptotic behavior of exit and transition times of certain weakly time inhomogeneous diffusion processes. Based on these asymptotics, a probabilistic notion of stochastic resonance (SR) is investigated. Large deviations techniques play the key role throughout this work. In the first part (Chapters 1-3) we recall the large deviations theory for time homogeneous diffusions. We present the classical results due to Freidlin and Wentzell and extensions thereof, and we remind of Kramers'' exit time law. Part II deals with the phenomenon of stochastic resonance. That is, we study periodicity properties of diffusion processes. In Chapter 4 we explain the paradigm of stochastic resonance and discuss physical notions of measuring periodicity of diffusions. Their drawbacks suggest to follow an alternative probabilistic approach, which is treated in this work. In Chapter 5 we derive a large deviations principle for diffusions subject to a weakly time dependent periodic drift term. The uniformity of the obtained large deviations bounds w.r.t. the system''s parameters plays a key role for the treatment of transition time asymptotics in Chapter 6, which contains the main result of the second part. The exact exponential transition rates obtained here allow for maximizing transition probabilities, which finally leads to the announced probabilistic notion of resonance studied in Chapter 7. In the third part we investigate the exit time asymptotics of a certain class of so-called self-stabilizing diffusions. In Chapter 8 we explain the connection between interacting particle systems and self-stabilizing diffusions, and we address the question of existence. The following Chapter 9 is devoted to the study of the large deviations behavior of these diffusions. In Chapter 10 Kramers'' exit law is carried over to our class of self-stabilizing diffusions. Finally, the influence of self-stabilization is illustrated in Chapter 11
Stochastic resonance: a mathematical approach in the small noise limit
Stochastic resonance is a phenomenon arising in a wide spectrum of areas in the sciences ranging from physics through neuroscience to chemistry and biology. This book presents a mathematical approach to stochastic resonance which is based on a large deviations principle (LDP) for randomly perturbed dynamical systems with a weak inhomogeneity given by an exogenous periodicity of small frequency. Resonance, the optimal tuning between period length and noise amplitude, is explained by optimizing the LDP's rate function. The authors show that not all physical measures of tuning quality are robust with respect to dimension reduction. They propose measures of tuning quality based on exponential transition rates explained by large deviations techniques and show that these measures are robust. The book sheds some light on the shortcomings and strengths of different concepts used in the theory and applications of stochastic resonance without attempting to give a comprehensive overview of the many facets of stochastic resonance in the various areas of sciences. It is intended for researchers and graduate students in mathematics and the sciences interested in stochastic dynamics who wish to understand the conceptual background of stochastic resonance