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; Stirzaker 05; Lawler 06 [intro]; Prabhu 07 [mathematical]; Jacobs 10 [noisy systems, r JSP(12)]; Bass 11 [r CP(12)]; Wergen JPA(13) [statistics of record-breaking events]; Castañeda et al 12 [with applications]; Chaumont & Yor 12 [problems, r CP(13)]; Klebaner 12 [stochastic calculus].
@ Stochastic dynamical systems: Casati & Ford ed-79; Honerkamp 94; Cercignani et al ed-97; Crauel & Gundlach ed-99; Arseniev & Moss 01 [kinetic equations]; Lemons 02; Gardiner 04; Kharchenko & Kharchenko PhyA(05) [in Tsallis non-extensive statistics]; Gaveau et al JPA(06) [geometry and observables]; Vallone mp/06-conf [and power-law distributions]; Jorgensen & Song a0903 [spectral analysis]; Costanza PhyA(09) [continuum stochastic evolution equations from discrete ones]; Meroz et al PRL(11) [inequivalent systems with the same probability distribution function].
@ Auto-correlation functions: Franosch JPA(14) [long-time limit]; > s.a. correlations.
@ 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]; Bauer & Bernard JSM(01)-a1101 [detailed study of simple example]; Karimipour & Memarzadeh a1105 [maps relating classical and quantum stochastic processes]; Gudder a1106 [discrete quantum processes].

Noise and Other Related Topics > s.a. Feynman-Kac Formula.
* Types of noise: Stationary/non-stationary; Ergodic; Gaussian.
* Gaussian: The auto-correlation 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 point processes].

Specific Types of Systems > s.a. critical phenomena; differential equations; fluctuations; markov chain / process; probability in physics.
* Wiener processes: A type of continuous-time stochastic process; Brownian motion is the most common example; > s.a. MathWorld page; Wikipedia page.
* Birth-and-death processes: Continuous-time Markov procesess in which the state transitions either increase or decrease the number of state variables by one; The name comes from a common application, to represent the size of a population where the transitions are literal births and deaths; > s.a. Wikipedia page.
* 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]; > s.a. Vlasov-Poisson Equations.
@ 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; Wio 13 [path integrals for stochastic processes, r CP(13)]; > s.a. path integrals.
@ Growth processes: Marsili et al RMP(96) [surfaces]; Johansson CMP(03)m.PR/02 [polynuclear, Airy process]; Ferrari & Spohn a1003 [rev]; Alekseev & Mineev-Weinstein a1611 [statistical mechanics]; > s.a. Triangulations.
@ Birth-and-death processes: Flajolet & Guillemin AAP(00) [and continued fractions]; Canessa PhyA(07) [in curved spacetime]; Sasaki JMP(09)-a0903 [exactly solvable, examples]; Finkelshtein et al JFA(12)-a1109 [semigroup approach]; Bücher a1408 [quantum, invariant states and applications]; Friesen & Kondratiev a1702 [stochastic averaging principle].
@ Non-Markovian processes: Olla & Pignagnoli PLA(06) [Fokker-Planck approach]; Wolf et al PRL(08)-a0711; Madsen PRL(11)-a1012 [observation, single quantum dot in a micropillar cavity]; Costanza PhyA(11) [and deterministic evolution equations]; Mazzola et al PRA(12)-a1203 [dynamical role of system-environment correlations]; Chruściński & Kossakowski IJGMP(12) [geometric perspective]; Drummond JPA(14) [higher-order stochastic equations]; Garrido et al PRA(16)-a1507 [transition to Markovian dynamics]; > 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]; Wang & Roberts a0903 [slow-fast system]; Addis et al PRA(14)-a1402; Frasca & Farina a1403 [existence of fractional Wiener processes]; Frasca a1412 [and non-commutative geometry]; > s.a. locality [and non-locality in time].
> Examples: see brownian motion; diffusion; fokker-planck equation; hamiltonian systems; random tilings; systems in statistical mechanics.

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