| Description |
1 online resource |
| Contents |
Ch. 0. Two-state random velocity model. 0.1. Two-state Markov chain -- 0.2. Random velocity model -- 0.3. Weak law and central limit theorem -- 0.4. Distribution functions of two-state model -- 0.5. Passage-time distributions -- 0.6. Asymptotic behavior with probability one -- ch. 1. Additive functionals of finite Markov chains. 1.1. Finite Markov chains -- 1.2. Asymptotic properties of the transition matrix -- 1.3. The weak law of large numbers and the central limit theorem -- 1.4. Recurrence properties -- 1.5. Limit theorems for discontinuous additive functionals -- 1.6. Proof of the Markov property -- ch. 2. General random evolutions. 2.1. Preliminaries on semigroups of operators -- 2.2. Construction of random evolution process -- 2.3. Discontinuous random evolutions -- 2.4. Limit theorems for random evolutions -- 2.5. Application to diffusion approximations -- 2.6. Martingale formulation of random evolution -- ch. 3. Applications to the Kinetic theory of gases. 3.1. Physical background -- 3.2. Stochastic solution of the linearized Boltzmann equation -- 3.3. Asymptotic analysis of the linearized Boltzmann equation -- ch. 4. Applications to isotropic transport on manifolds. 4.1. The Rayleigh problem of random flights -- 4.2. Isotropic transport process on a manifold -- 4.3. Applications to recurrence -- 4.4. Isotropic transport process of a frame field on a manifold -- ch. 5. Applications to stability of random oscillators. 5.1. Linear stochastic systems with multiplicative noise -- 5.2. Simple harmonic oscillator with small noise -- 5.3. Nilpotent linear systems with small noise |
| Summary |
Random evolution denotes a class of stochastic processes which evolve according to a rule which varies in time according to jumps. This is in contrast to diffusion processes, which assume that the rule changes continuously with time. Random evolutions provide a very flexible language, having the advantage that they permit direct numerical simulation-which is not possible for a diffusion process. Furthermore, they allow connections with hyperbolic partial differential equations and the kinetic theory of gases, which is impossible within the domain of diffusion proceses. They also posses great geometric invariance, allowing formulation on an arbitrary Riemannian manifold. In the field of stochastic stability, random evolutions furnish some easily computable models in which to study the Lyapunov exponent and rotation numbers of oscillators under the influence of noise. This monograph presents the various aspects of random evolution in an accessible and interesting format which will appeal to a large scientific audience |
| Bibliography |
Includes bibliographical references |
| Subject |
Stochastic processes.
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Semigroups.
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Stochastic Processes
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MATHEMATICS -- Probability & Statistics -- General.
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Semigroups
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Stochastic processes
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| Form |
Electronic book
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| ISBN |
9789812779359 |
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9812779353 |
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128363564X |
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9781283635646 |
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