[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Reduction methods in climate dynamics—A brief review

We prove the existence of a global random attractor for a certain class of stochastic partly dissipative systems. These systems consist of a partial and an ordinary differential equation, where both equations are coupled and perturbed by additive white noise. The deterministic counterpart of such systems and their long-time behaviour have already been considered but there is no theory that deals with the stochastic version of partly dissipative systems in their full generality. We also provide several examples for the application of the theory.

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Random attractors for stochastic partly dissipative systems

Propagation of uncertainty in dynamical systems is a significant challenge. Here we focus on random multiscale ordinary differential equation models. In particular, we study Hopf bifurcation in the fast subsystem for random initial conditions. We show that a random initial condition distribution can be transformed during the passage near a delayed/dynamic Hopf bifurcation: (I) to certain classes of symmetric copies, (II) to an almost deterministic output, (III) to a mixture distribution with differing moments, and (IV) to a very restricted class of general distributions. We prove under which conditions the cases (I)-(IV) occur in certain classes vector fields.

Uncertainty Transformation via Hopf Bifurcation in Fast-Slow Systems

In this work, we study the numerical approximation of local fluctuations of certain classes of parabolic stochastic partial differential equations (SPDEs). Our focus is on effects for small spatially correlated noise on a time scale before large deviation effects have occurred. In particular, we are interested in the local directions of the noise described by a covariance operator. We introduce a new strategy and prove a Combined ERror EStimate (CERES) for the five main errors: the spatial discretization error, the local linearization error, the noise truncation error, the local relaxation error to steady state, and the approximation error via an iterative low-rank matrix algorithm. In summary, we obtain one CERES describing, apart from modelling of the original equations and standard round-off, all sources of error for a local fluctuation analysis of an SPDE in one estimate. To prove our results, we rely on a combination of methods from optimal Galerkin approximation of SPDEs, covariance moment estimates, analytical techniques for Lyapunov equations, iterative numerical schemes for low-rank solution of Lyapunov equations, and working with related spectral norms for different classes of operators.

/pdf/combined-error-estimates-for-local-fluctuations-of-spdes-5evcocl62m.pdf

Combined Error Estimates for Local Fluctuations of SPDEs

In this document we review a geometric technique, called \emph{the blow-up method}, as it has been used to analyze and understand the dynamics of fast-slow systems around non-hyperbolic points. The blow-up method, having its origins in algebraic geometry, was introduced in 1996 to the study of fast-slow systems in the seminal work by Dumortier and Roussarie \cite{dumortier1996canard}, whose aim was to give a geometric approach and interpretation of canards in the van der Pol oscillator. Following \cite{dumortier1996canard}, many efforts have been performed to expand the capabilities of the method and to use it in a wide range of scenarios. Our goal is to present in a concise and compact form those results that, based on the blow-up method, are now the foundation of the geometric theory of fast-slow systems with non-hyperbolic singularities. We cover fold points due to their great importance in the theory of fast-slow systems as one of the main topics. Furthermore, we also present several other singularities such as Hopf, pitchfork, transcritical, cusp, and Bogdanov-Takens, in which the blow-up method has been proved to be extremely useful. Finally, we survey further directions as well as examples of specific applied models, where the blow-up method has been used successfully.

Christian Kuehn

Papers

Reduction methods in climate dynamics—A brief review

Random attractors for stochastic partly dissipative systems

Uncertainty Transformation via Hopf Bifurcation in Fast-Slow Systems

Combined Error Estimates for Local Fluctuations of SPDEs

A survey on the blow-up method for fast-slow systems