Diffusions and Waves - Henryk Gzyl

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Résumé :
In this book several connections between probability theory and wave propagation are explored. The connection comes via the probabilistic (or path integral) representation of both the (fixed frequency) Green functions and of the propagators -operators mapping initial into present time data. The formalism includes both waves in continuous space and in discrete structures. One of the main applications of the formalism developed is to inverse problems in wave propagation. Using the probabilistic formalism, the parameters of the medium and the surfaces determining the region of propagation appear explicitly in the path integral representation of the Green functions and propagators. This fact is what provides a useful starting point for inverse problem formulation. Audience: The book is suitable for advanced graduate students in the mathematical, physical or in the engineering sciences. The presentation is quite self-contained, and not extremely rigorous.

Sommaire:
1. Basic Probabilistic Notions.- 1.1. Probability spaces.- 1.2. Conditional expectations.- 1.3. Examples of conditional expectations.- 1.4. Processes.- 1.5. Semigroups, generators and resolvents.- 1.6. Examples.- 1.7. Construction of Markov processes.- 1.8. Transformations of Markov processes.- 2. From Brownian Motion to Diffusions.- 2.1. Brownian motion.- 2.2. Diffusions.- 2.3. Diffusions as solutions of stochastic equations.- 2.4. Reflected diffusions.- 2.5. Killed diffusions and some fundamental identities.- 3. Waves.- 3.1. Waves of constant speed in ?d.- 3.2. Propagators and Green functions.- 3.3. Geometrical optics.- 3.4. General representation of solutions.- 4. Waves and Brownian Motions.- 4.1. Waves in full space.- 4.2. Dirichlet problems.- 4.3. Neumann type boundary conditions.- 4.4. Existence results.- 4.5. Problems of Dirichlet type in unbounded domains: from the Markov property to the Huygens condition and the Sommerfeld radiation condition.- 4.6. Extended Hadamard's construction.- 4.7. From resolvents to propagators.- 4.8. Reciprocity: A probabilistic approach.- 5. Waves and Diffusions.- 5.1. Waves in full space.- 5.2. Existence of solutions to the wave equations.- 5.3. An evaluation of some path integrals.- 5.4. Waves in stratified media.- 5.5. Maxentropic equivalent linearization and approximate solutions to the wave equations.- 6. Asymptotic Expansions.- 6.1. Digressive introduction.- 6.2. Probabilistic approach to geometrical optics.- 6.3. Geometrical optics and the Dirichlet problem.- 6.4. Two variations on a theme.- 6.5. Geometrical optics and the Neumann problem.- 6.6. Example.- 6.7. Long time asymptotics.- 7. Transmutation Operations.- 7.1. Basic transmutations.- 7.2. Probabilistic version of transmutation operations.- 7.3. Examples.- 7.4. Moreinversion techniques and simple examples.- 7.5. The ascent method.- 7.6. The closing of the circle. Some heuristics.- 8. More Connections.- 8.1. Waves in discrete structures and Markov chains.- 8.2. Approximate Laplacians, regular jump processes and random flights.- 8.3. Regular jump processes.- 8.4. Random flights.- 8.5. Random evolutions.- 8.6. First-order hyperbolic systems.- 8.7. Pseudo processes and Euler's equation.- 8.8. Damped waves: Playing with a simple model.- 9. Applications.- 9.1. An inverse source problem.- 9.2. Probabilistic approach to a discrete inverse problem.- 9.3. Dependence of boundary data on propagation velocity.- 9.4. The Born approximation.- 9.5. Scattering by a bounded object.