Topic
Infinitesimal generator
About: Infinitesimal generator is a research topic. Over the lifetime, 1317 publications have been published within this topic receiving 41100 citations.
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Book•
11 Feb 1992
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
Abstract: 1 Generation and Representation.- 1.1 Uniformly Continuous Semigroups of Bounded Linear Operators.- 1.2 Strongly Continuous Semigroups of Bounded Linear Operators.- 1.3 The Hille-Yosida Theorem.- 1.4 The Lumer Phillips Theorem.- 1.5 The Characterization of the Infinitesimal Generators of C0 Semigroups.- 1.6 Groups of Bounded Operators.- 1.7 The Inversion of the Laplace Transform.- 1.8 Two Exponential Formulas.- 1.9 Pseudo Resolvents.- 1.10 The Dual Semigroup.- 2 Spectral Properties and Regularity.- 2.1 Weak Equals Strong.- 2.2 Spectral Mapping Theorems.- 2.3 Semigroups of Compact Operators.- 2.4 Differentiability.- 2.5 Analytic Semigroups.- 2.6 Fractional Powers of Closed Operators.- 3 Perturbations and Approximations.- 3.1 Perturbations by Bounded Linear Operators.- 3.2 Perturbations of Infinitesimal Generators of Analytic Semigroups.- 3.3 Perturbations of Infinitesimal Generators of Contraction Semigroups.- 3.4 The Trotter Approximation Theorem.- 3.5 A General Representation Theorem.- 3.6 Approximation by Discrete Semigroups.- 4 The Abstract Cauchy Problem.- 4.1 The Homogeneous Initial Value Problem.- 4.2 The Inhomogeneous Initial Value Problem.- 4.3 Regularity of Mild Solutions for Analytic Semigroups.- 4.4 Asymptotic Behavior of Solutions.- 4.5 Invariant and Admissible Subspaces.- 5 Evolution Equations.- 5.1 Evolution Systems.- 5.2 Stable Families of Generators.- 5.3 An Evolution System in the Hyperbolic Case.- 5.4 Regular Solutions in the Hyperbolic Case.- 5.5 The Inhomogeneous Equation in the Hyperbolic Case.- 5.6 An Evolution System for the Parabolic Initial Value Problem.- 5.7 The Inhomogeneous Equation in the Parabolic Case.- 5.8 Asymptotic Behavior of Solutions in the Parabolic Case.- 6 Some Nonlinear Evolution Equations.- 6.1 Lipschitz Perturbations of Linear Evolution Equations.- 6.2 Semilinear Equations with Compact Semigroups.- 6.3 Semilinear Equations with Analytic Semigroups.- 6.4 A Quasilinear Equation of Evolution.- 7 Applications to Partial Differential Equations-Linear Equations.- 7.1 Introduction.- 7.2 Parabolic Equations-L2 Theory.- 7.3 Parabolic Equations-Lp Theory.- 7.4 The Wave Equation.- 7.5 A Schrodinger Equation.- 7.6 A Parabolic Evolution Equation.- 8 Applications to Partial Differential Equations-Nonlinear Equations.- 8.1 A Nonlinear Schroinger Equation.- 8.2 A Nonlinear Heat Equation in R1.- 8.3 A Semilinear Evolution Equation in R3.- 8.4 A General Class of Semilinear Initial Value Problems.- 8.5 The Korteweg-de Vries Equation.- Bibliographical Notes and Remarks.
11,637 citations
Book•
05 Apr 1977
TL;DR: In this paper, Liapunov functional for autonomous systems is used to define the saddle point property near equilibrium and periodic orbits for linear systems, which is a generalization of the notion of stable D operators.
Abstract: 1 Linear differential difference equations.- 1.1 Differential and difference equations.- 1.2 Retarded differential difference equations.- 1.3 Exponential estimates of x(?, f).- 1.4 The characteristic equation.- 1.5 The fundamental solution.- 1.6 The variation-of-constants formula.- 1.7 Neutral differential difference equations.- 1.8 Supplementary remarks.- 2 Retarded functional differential equations : basic theory.- 2.1 Definition.- 2.2 Existence, uniqueness, and continuous dependence.- 2.3 Continuation of solutions.- 2.4 Differentiability of solutions.- 2.5 Backward continuation.- 2.6 Caratheodory conditions.- 2.7 Supplementary remarks.- 3 Properties of the solution map.- 3.1 Finite- or infinite-dimensional problem?.- 3.2 Equivalence classes of solutions.- 3.3 Exponential decrease for linear systems.- 3.4 Unique backward extensions.- 3.5 Range in ?n.- 3.6 Compactness and representation.- 3.7 Supplementary remarks.- 4 Autonomous and periodic processes.- 4.1 Processes.- 4.2 Invariance.- 4.3 Discrete systems-maximal compact invariant sets.- 4.4 Fixed points of discrete dissipative processes.- 4.5 Stability and maximal invariant sets in processes.- 4.6 Periodic trajectories of ?-periodic processes.- 4.7 Convergent systems.- 4.8 Supplementary remarks.- 5 Stability theory.- 5.1 Definitions.- 5.2 The method of Liapunov functional.- 5.3 Liapunov functional for autonomous systems.- 5.4 Razumikhin-type theorems.- 5.5 Supplementary remarks.- 6 General linear systems.- 6.1 Global existence and exponential estimates.- 6.2 Variation-of-constants formula.- 6.3 The formal adjoint equation.- 6.4 The true adjoint.- 6.5 Boundary-value problems.- 6.6 Stability and boundedness.- 6.7 Supplementary remarks.- 7 Linear autonomous equations.- 7.1 The semigroup and infinitesimal generator.- 7.2 Spectrum of the generator-decomposition of C.- 7.3 Decomposing C with the formal adjoint equation.- 7.4 Estimates on the complementary subspace.- 7.5 An example.- 7.6 The decomposition in the variation-of-constants formula.- 7.7 Supplementary remarks.- 8 Linear periodic systems.- 8.1 General theory.- 8.2 Decomposition.- 8.3 Supplementary remarks.- 9 Perturbed linear systems.- 9.1 Forced linear systems.- 9.2 Bounded, almost-periodic, and periodic solutions stable and unstable manifolds.- 9.3 Periodic solutions-critical cases.- 9.4 Averaging.- 9.5 Asymptotic behavior.- 9.6 Boundary-value problems.- 9.7 Supplementary remarks.- 10 Behavior near equilibrium and periodic orbits for autonomous equations.- 10.1 The saddle-point property near equilibrium.- 10.2 Nondegenerate periodic orbits.- 10.3 Hyperbolic periodic orbits.- 10.4 Supplementary remarks.- 11 Periodic solutions of autonomous equations.- 11.1 Hopf bifurcation.- 11.2 A periodicity theorem.- 11.3 Range of the period.- 11.4 The equation $$\dot x(t) = - \alpha x(t - 1)[1 + x(t)]$$.- 11.5 The equation $$\dot x(t) = - \alpha x(t - 1)[1 - {x^2}(t)]$$.- 11.6 The equation $$\ddot x(t) + f(x(t))\dot x(t) + g(x(t - r)) = 0$$.- 11.7 Supplementary remarks.- 12 Equations of neutral type.- 12.1 Definition of a neutral equation.- 12.2 Fundamental properties.- 12.3 Linear autonomous D operators.- 12.4 Stable D operators.- 12.5 Strongly stable D operators.- 12.6 Properties of equations with stable D operators.- 12.7 Stability theory.- 12.8 General linear equations.- 12.9 Stability of autonomous perturbed linear systems.- 12.10 Linear autonomous and periodic equations.- 12.11 Nonhomogeneous linear equations.- 12.12 Supplementary remarks.- 13 Global theory.- 13.1 Generic properties of retarded equations.- 13.2 The set of global solutions.- 13.3 Equations on manifolds : definitions.- 13.4 Retraded equations on compact manifolds.- 13.5 Further properties of the attractor.- 13.6 Supplementary remarks.- Appendix Stability of characteristic equations.
5,799 citations
01 Jan 1998
TL;DR: In this paper, the authors explore questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties, using diffusion processes as a model of a Markov process with continuous sample paths.
Abstract: 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.
2,446 citations
TL;DR: In this article, the authors introduce a nonlinear expectation generated by a heat equation with infinitesimal generator G. The G-standard normal distribution is introduced and the canonical process is a G-Brownian motion.
Abstract: We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G--Brownian motion and derive the related Ito's formula. We have also give the existence and uniqueness of stochastic differential equation under our G-expectation. As compared with our previous framework of g-expectations, the theory of G-expectation is intrinsic in the sense that it is not based on a given (linear) probability space.
653 citations
TL;DR: It is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that the authors consider.
Abstract: A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in $\mbRn$. The nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficie...
566 citations