Contents -- I: STOCHASTIC CALCULUS -- 1 Heat equation and Brownian motion -- 1.1 Einstein's 1905 derivation -- 1.2 Brownian motion -- 2 Ito calculus -- 2.1 Ito stochastic integral -- 2.2 Ito differential -- 3 Stochastic differential geometry -- 3.1 Diffusions on manifolds -- 3.2 Kinematics of diffusion -- 4 Examples of stochastic differential equations -- 4.1 Geometric Brownian motion -- 4.2 Bessel process -- 4.3 Solution to Laplace's equation -- II: DYNAMICS OF THE SCATTERING PROCESS -- 5 Diffusion models of scattering -- 5.1 Non-Gaussian statistical models -- 5.2 Dynamics of the vector scattering process -- 5.3 Correlation in the vector scattering process -- 6 Rayleigh scattering -- 6.1 Quadrature components -- 6.2 Random walk model -- 7 Population dynamics -- 7.1 Master equations and the Kramers-Moyal expansion -- 7.2 Birth-death-immigration processes -- 7.3 Continuum diffusion limit -- 8 Dynamics of K-scattering -- 8.1 Stochastic dynamics of K-amplitude process -- 8.2 Geometry of K-amplitude fluctuations -- 8.3 Asymptotic behaviour -- 8.4 Correlation and spectra -- 8.5 Interpretation and implications -- 9 Models of weak scattering -- 9.1 Weak scattering amplitudes -- 9.2 Stochastic dynamics -- 9.3 Geometry of amplitude fluctuations -- 9.4 Asymptotic behaviour -- 10 Scattering from general populations -- 10.1 Extended random walk model -- 10.2 Generalized dynamics -- III: SIMULATION AND EXPERIMENT -- 11 Simulation of K-scattering -- 11.1 Iterative schemes -- 11.2 Rayleigh and gamma processes -- 11.3 Compound K-model -- 11.4 Influence of Doppler on volatilities -- 11.5 Coherent clutter -- 11.6 Second-order algorithms -- 12 Experimental tests -- 12.1 Scattering at optical wavelength -- 12.2 Scattering at radar wavelength -- 13 Non-linear dynamics of sea clutter -- 13.1 Hybrid AM/FM model of sea clutter -- 13.2 Non-linear dynamics from SDE theory.

13.3 Radar parameters -- 14 Observability of scattering cross-section -- 14.1 Simulated data -- 14.2 Experimental applications -- A: Stability and infinite divisibility -- B: Ito versus Stratonovich stochastic integrals -- C: Filtrations, conditional probability, and Markov property -- D: Girsanov's theorem -- D.0.1 Relation to mathematical finance -- E: Partition function solution to BDI model -- F: Summary of K-scattering -- F.1 Rayleigh scattering -- F.2 K-distributed noise -- F.2.1 Amplitude -- F.2.2 Intensity -- F.2.3 Phase -- F.2.4 Geometry of fluctuations -- G: Iterative solution for vector processes -- H: Open problems -- I: Suggested further reading -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- V -- W -- X.