Diffusions and Waves

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.