Stochastic
Processes| 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 statistics]; Gaveau et al JPA(06)
[geometry and observables]; Vallone mp/06-in
[and power-law distributions].
@ 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 de's]; Whitney 90 [and computation];
Moeschlin et al 03 [simulation]; > s.a. causality.
Stochastic Aspects of Geometry > see knot theory; quantum gravity phenomenology; statistical geometry.
Markov Chain or Process > s.a. formulations
of quantum mechanics; Master Equation; random
process [walk].
* Idea: A process in
which a system evolves through a sequence of steps in some set of possible
states, the probability of it going to a certain
state
in the next step depending only on the state it is in (no memory); It
is characterized by a transition matrix T such that Tij 
0
for all
i, j and 
i Tij =
1 for all j.
* History: Introduced by
Markov in 1906, who just wanted to show that independence was not needed for
the law of large numbers; An example he considered
was the
alternation of consonants and vowels in Pushkin's Eugene Onegin, which
he described as a two-state Markov chain; Soon Poincaré was
studying
Markov chains on finite groups to study card shuffling; Today they
are in all
applied sciences, from population biology to communication networks,
diffusion models, or social mobility.
@ General references: Revuz 84; Norris 97 [II]; Brémaud 99; Berg 04 [Monte
Carlo simulations]; Borovkov & Hordijk AAP(04)
[normed ergodicity]; Lecomte et al JSP(07)cm/06 [thermodynamic
formalism]; Kolokoltsov JSP(07)
[Markov semigroups]; Frank PLA(08) [non-linear].
@ Evolution, examples: Albeverio & Høegh-Krohn RPMP(84)
[fields]; Schächter FP(87);
Ibison CSF(99)qp/01 [1+1
Dirac equation]; Turova JSP(03)
[states = directed graphs]; Duchi & Schaeffer JCTA(05)
[jumping particles, and Catalan numbers]; Lecomte et al PRL(05)
[dynamic partition function, entropy]; Grone et al JPA(08)
[reversible, coarse-graining of stochastic matrix]; Hou et al a0805 [and
growing networks].
@ Quantum: Dynkin 82; Ghirardi et al PRA(90);
Marbeau & Gudder AIHP(90);
Gudder & Schindler JMP(91);
Accardi
et
al mp/04 [for
spin chains]; Tay & Petrosky PRA(07)-a0705 [thermal
symmetry]; Ibinson et al CMP(08)
[robustness]; Leifer & Poulin AP(08) [quantum graphical models of belief propagation].
Specific Types of Systems > s.a. differential
equations; fluctuations; 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].
@ Non-Markovian processes: Olla & Pignagnoli PLA(06)
[Fokker-Planck approach]; > s.a. 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].
> Examples: see brownian
motion, diffusion, fokker-planck
equation; hamiltonian systems; systems in statistical
mechanics; tilings.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
23 jun 2008