Stochastic Processes  

In General > s.a. computation; modified classical mechanics; probability and statistics in physics; random process.
* Idea: Stochastic dynamics ideas can be used directly to model physical processes, or applied to derive kinetic equations, such as the Boltzmann, Vlasov, Fokker-Planck, Landau, and quantum Neumann-Liouville equations.
@ General references: Papoulis 65; Lamperti 77; Van Kampen 81; Chung 82; Wong 83; Emery 89 [on manifolds]; Helstrom 91; Reif 98.
@ Stochastic dynamical systems: Casati & Ford ed-79; Honercamp 94; Gardiner 97; Crauel & Gundlach 99; Arseniev & Moss 01 [kinetic equations]; Lemons 02; Kharchenko & Kharchenko PhyA(05) [in Tsallis non-extensive statistics]; Gaveau et al JPA(06) [geometry and observables]; Vallone mp/06-in [and power-law distributions]; Jorgensen & Song a0903 [spectral analysis]; Costanza PhyA(09) [continuum stochastic evolution equations from discrete ones].
@ Quantum stochastic process: Hall & Collins JMP(71) [representation in Hilbert space]; Belavkin TMP(85) [reconstruction theorem], & Kolokol'tsov TMP(91) [semiclassial asymptotics]; Nicrosini & Rimini FP(90) [continuous vs discontinuous].

Noise and Other Related Topics > s.a. Feynman-Kac Formula.
* Types of noise: Stationary/non-stationary; Ergodic; Gaussian.
* Gaussian: The autocorrelation function contains all the information on the noise.
* Noise level: Defined as a function of frequency by hn(f):= [f Sn(f)]1/2, where Sn(f) is the spectral density.
@ Noise: Helstrom 94; Ghanem & Doostan JCP(06) [limited data and propagation of errors].
@ Related topics: Arnold 74 [stochastic differential equations]; Whitney 90 [and computation]; Moeschlin et al 03 [simulation]; > s.a. causality.

Stochastic Aspects of Geometry > see knot theory; quantum-gravity phenomenology; statistical geometry [including Poisson and other processes].

Specific Types of Systems > s.a. differential equations; fluctuations; markov chain / process; probability in physics.
* Non-Markovian processes: Two broad types are history-dependent processes (which may be formally turned into Markovian processes by redefining the configurations to include the relevant part of the history), and open systems (whose dynamics depends on their environment).
@ Vlasov-Maxwell: Rein CMS(04)mp [relativistic]; Yang & Zhao CMP(06) [Vlasov-Poisson-Boltzmann, existence].
@ Einstein-Vlasov: Lee AHP(05)gq/04 [+ scalar field]; > s.a. gravitating matter.
@ Gravity: Kandrup PRP(80) [self-gravitating, mean-field]; > s.a. Induced Gravity [stochastic gravity].
@ Stochastic methods in quantum mechanics: de la Peña-Auerbach JMP(69); Zaslavsky PRP(81); Mitter & Pittner ed-84; > s.a. path integrals.
@ Growth processes: Marsili et al RMP(96) [surfaces]; Johansson m.PR/02 [polynuclear, Airy process].
@ Birth-and-death processes: Flajolet & Guillemin AAP(00) [and continued fractions]; Canessa PhyA(07) [in curved spacetime]; Sasaki a0903 [exactly solvable, examples].
@ Non-Markovian processes: Olla & Pignagnoli PLA(06) [Fokker-Planck approach]; Wolf et al PRL(08)-a0711; > s.a. brownian motion, diffusion, dissipation, Master Equation, Open Systems, spin models.
@ Related topics: Escande PRP(85) [classical Hamiltonian systems]; Zimmer PRL(95) [Monte Carlo based on probabilities for histories]; Horbacz et al JSP(05) [generalization of Markov process]; Freidlin & Wentzell JSP(06) [from long-time behavior of many deterministic degrees of freedom]; > s.a. locality [and non-locality in time]; Wang & Roberts a0903 [slow-fast system].
> Examples: see brownian motion, diffusion, fokker-planck equation; hamiltonian systems; systems in statistical mechanics; tilings.

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