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 optimising 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.

The proceedings of the summer 1999 Chorin workshop on stochastic climate models captures well the spirit of enthusiasm of the workshop participants engaged in research in this exciting field. It is amazing that nearly 25 years after the formal theory of natural climate variability generated by quasi-white-noise weather forcing was developed, and almost 35 years after J . M. Mitchell first suggested this mechanism as the origin of sea-surf ace-temperature fluctuations and climate variability, there have arisen so many fresh perspectives and new applications of the theory. The workshop has succeeded admirably in high­ lighting these new aspects while clarifying the position of stochastic climate modelling within the general framework of climate research and mathematical modelling. The organizers can be congratulated in bringing together leading researchers covering a wide range of scientific expertise, from mathematicians concerned with the derivation of stochastic models from first principles, to app­ lied climate modellers trying to understand the dynamics of the complex climate system. Following the first burst of stochastic modelling papers in the decade from the mid-seventies to the mid-eighties, in which the viability of the concept was demonstrated using relatively simple conceptual models, there was a lull of work in this field. One awaited the development of more sophisticated climate models with which one could carry out realistic quantitative analyses of the implications of stochastic forcing for the global climate system. Now that these models have become widely available, it is natural that one is witnessing a resurgence of stochastic modelling investigations.

The proceedings of the summer 1999 Chorin workshop on stochastic climate models captures well the spirit of enthusiasm of the workshop participants engaged in research in this exciting field. It is amazing that nearly 25 years after the formal theory of natural climate variability generated by quasi-white-noise weather forcing was developed, and almost 35 years after J . M. Mitchell first suggested this mechanism as the origin of sea-surf ace-temperature fluctuations and climate variability, there have arisen so many fresh perspectives and new applications of the theory. The workshop has succeeded admirably in high­ lighting these new aspects while clarifying the position of stochastic climate modelling within the general framework of climate research and mathematical modelling. The organizers can be congratulated in bringing together leading researchers covering a wide range of scientific expertise, from mathematicians concerned with the derivation of stochastic models from first principles, to app­ lied climate modellers trying to understand the dynamics of the complex climate system. Following the first burst of stochastic modelling papers in the decade from the mid-seventies to the mid-eighties, in which the viability of the concept was demonstrated using relatively simple conceptual models, there was a lull of work in this field. One awaited the development of more sophisticated climate models with which one could carry out realistic quantitative analyses of the implications of stochastic forcing for the global climate system. Now that these models have become widely available, it is natural that one is witnessing a resurgence of stochastic modelling investigations.

This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.

This book has two-fold aims. In a first part it gives an introductory, thorough and essentially self-contained treatment of the general theory of two-parameter processes that has developed since around 1975. Apart from two survey papers by Merzbach and Meyer it is the first text of this kind. The second part presents the results of recent research by the author on martingale theory and stochastic calculus for two-parameter processes. Both the results and the methods of these two chapters are almost entirely new, and are of particular interest. They provide the fundamentals of a general stochastic analysis of two-parameter processes including, in particular, so far inaccessible jump phenomena. The typical rader is assumed to have some basic knowledge of the general theory of one-parameter martingales. The book should be accessible to probabilistically interested mathematicians who a) wish to become acquainted with or have a complete treatment of the main features of the general theory of two-parameter processes and basics of their stochastic calculus, b) intend to learn about the most recent developments in this area.

This book has two-fold aims. In a first part it gives an introductory, thorough and essentially self-contained treatment of the general theory of two-parameter processes that has developed since around 1975. Apart from two survey papers by Merzbach and Meyer it is the first text of this kind. The second part presents the results of recent research by the author on martingale theory and stochastic calculus for two-parameter processes. Both the results and the methods of these two chapters are almost entirely new, and are of particular interest. They provide the fundamentals of a general stochastic analysis of two-parameter processes including, in particular, so far inaccessible jump phenomena. The typical rader is assumed to have some basic knowledge of the general theory of one-parameter martingales. The book should be accessible to probabilistically interested mathematicians who a) wish to become acquainted with or have a complete treatment of the main features of the general theory of two-parameter processes and basics of their stochastic calculus, b) intend to learn about the most recent developments in this area.

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 optimising 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.

The proceedings of the summer 1999 Chorin workshop on stochastic climate models captures well the spirit of enthusiasm of the workshop participants engaged in research in this exciting field. It is amazing that nearly 25 years after the formal theory of natural climate variability generated by quasi-white-noise weather forcing was developed, and almost 35 years after J . M. Mitchell first suggested this mechanism as the origin of sea-surf ace-temperature fluctuations and climate variability, there have arisen so many fresh perspectives and new applications of the theory. The workshop has succeeded admirably in high­ lighting these new aspects while clarifying the position of stochastic climate modelling within the general framework of climate research and mathematical modelling. The organizers can be congratulated in bringing together leading researchers covering a wide range of scientific expertise, from mathematicians concerned with the derivation of stochastic models from first principles, to app­ lied climate modellers trying to understand the dynamics of the complex climate system. Following the first burst of stochastic modelling papers in the decade from the mid-seventies to the mid-eighties, in which the viability of the concept was demonstrated using relatively simple conceptual models, there was a lull of work in this field. One awaited the development of more sophisticated climate models with which one could carry out realistic quantitative analyses of the implications of stochastic forcing for the global climate system. Now that these models have become widely available, it is natural that one is witnessing a resurgence of stochastic modelling investigations.

The proceedings of the summer 1999 Chorin workshop on stochastic climate models captures well the spirit of enthusiasm of the workshop participants engaged in research in this exciting field. It is amazing that nearly 25 years after the formal theory of natural climate variability generated by quasi-white-noise weather forcing was developed, and almost 35 years after J . M. Mitchell first suggested this mechanism as the origin of sea-surf ace-temperature fluctuations and climate variability, there have arisen so many fresh perspectives and new applications of the theory. The workshop has succeeded admirably in high­ lighting these new aspects while clarifying the position of stochastic climate modelling within the general framework of climate research and mathematical modelling. The organizers can be congratulated in bringing together leading researchers covering a wide range of scientific expertise, from mathematicians concerned with the derivation of stochastic models from first principles, to app­ lied climate modellers trying to understand the dynamics of the complex climate system. Following the first burst of stochastic modelling papers in the decade from the mid-seventies to the mid-eighties, in which the viability of the concept was demonstrated using relatively simple conceptual models, there was a lull of work in this field. One awaited the development of more sophisticated climate models with which one could carry out realistic quantitative analyses of the implications of stochastic forcing for the global climate system. Now that these models have become widely available, it is natural that one is witnessing a resurgence of stochastic modelling investigations.

This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.

This book has two-fold aims. In a first part it gives an introductory, thorough and essentially self-contained treatment of the general theory of two-parameter processes that has developed since around 1975. Apart from two survey papers by Merzbach and Meyer it is the first text of this kind. The second part presents the results of recent research by the author on martingale theory and stochastic calculus for two-parameter processes. Both the results and the methods of these two chapters are almost entirely new, and are of particular interest. They provide the fundamentals of a general stochastic analysis of two-parameter processes including, in particular, so far inaccessible jump phenomena. The typical rader is assumed to have some basic knowledge of the general theory of one-parameter martingales. The book should be accessible to probabilistically interested mathematicians who a) wish to become acquainted with or have a complete treatment of the main features of the general theory of two-parameter processes and basics of their stochastic calculus, b) intend to learn about the most recent developments in this area.

This book has two-fold aims. In a first part it gives an introductory, thorough and essentially self-contained treatment of the general theory of two-parameter processes that has developed since around 1975. Apart from two survey papers by Merzbach and Meyer it is the first text of this kind. The second part presents the results of recent research by the author on martingale theory and stochastic calculus for two-parameter processes. Both the results and the methods of these two chapters are almost entirely new, and are of particular interest. They provide the fundamentals of a general stochastic analysis of two-parameter processes including, in particular, so far inaccessible jump phenomena. The typical rader is assumed to have some basic knowledge of the general theory of one-parameter martingales. The book should be accessible to probabilistically interested mathematicians who a) wish to become acquainted with or have a complete treatment of the main features of the general theory of two-parameter processes and basics of their stochastic calculus, b) intend to learn about the most recent developments in this area.