Dyson Brownian motion
In mathematics, the Dyson Brownian motion is a real-valued continuous-time stochastic process named for Freeman Dyson.[1] Dyson studied this process in the context of random matrix theory.
There are several equivalent definitions:[2][3]
Definition by stochastic differential equation:
where are different and independent Wiener processes.
Start with a Hermitian matrix with eigenvalues , then let it perform Brownian motion in the space of Hermitian matrices. Its eigenvalues constitute a Dyson Brownian motion.
Start with independent Wiener processes started at different locations , then condition on those processes to be non-intersecting for all time. The resulting process is a Dyson Brownian motion starting at the same .[4]
References
- ^ Dyson, Freeman J. (1962-11-01). "A Brownian-Motion Model for the Eigenvalues of a Random Matrix". Journal of Mathematical Physics. 3 (6): 1191–1198. doi:10.1063/1.1703862. ISSN 0022-2488.
- ^ Bouchaud, Jean-Philippe; Potters, Marc, eds. (2020), "Dyson Brownian Motion", A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists, Cambridge: Cambridge University Press, pp. 121–135, ISBN 978-1-108-48808-2, retrieved 2023-11-25
- ^ Tao, Terence (2010-01-19). "254A, Notes 3b: Brownian motion and Dyson Brownian motion". What's new. Retrieved 2023-11-25.
- ^ Grabiner, David J. (1999). "Brownian motion in a Weyl chamber, non-colliding particles, and random matrices". Annales de l'I.H.P. Probabilités et statistiques. 35 (2): 177–204. ISSN 1778-7017.
- v
- t
- e
Stochastic processes
- Bernoulli process
- Branching process
- Chinese restaurant process
- Galton–Watson process
- Independent and identically distributed random variables
- Markov chain
- Moran process
- Random walk
- Loop-erased
- Self-avoiding
- Biased
- Maximal entropy
- Additive process
- Bessel process
- Birth–death process
- Brownian motion
- Cauchy process
- Contact process
- Continuous-time random walk
- Cox process
- Diffusion process
- Dyson Brownian motion
- Empirical process
- Feller process
- Fleming–Viot process
- Gamma process
- Geometric process
- Hawkes process
- Hunt process
- Interacting particle systems
- Itô diffusion
- Itô process
- Jump diffusion
- Jump process
- Lévy process
- Local time
- Markov additive process
- McKean–Vlasov process
- Ornstein–Uhlenbeck process
- Poisson process
- Schramm–Loewner evolution
- Semimartingale
- Sigma-martingale
- Stable process
- Superprocess
- Telegraph process
- Variance gamma process
- Wiener process
- Wiener sausage
- Binomial options pricing model
- Black–Derman–Toy
- Black–Karasinski
- Black–Scholes
- Chan–Karolyi–Longstaff–Sanders (CKLS)
- Chen
- Constant elasticity of variance (CEV)
- Cox–Ingersoll–Ross (CIR)
- Garman–Kohlhagen
- Heath–Jarrow–Morton (HJM)
- Heston
- Ho–Lee
- Hull–White
- Korn-Kreer-Lenssen
- LIBOR market
- Rendleman–Bartter
- SABR volatility
- Vašíček
- Wilkie
- Central limit theorem
- Donsker's theorem
- Doob's martingale convergence theorems
- Ergodic theorem
- Fisher–Tippett–Gnedenko theorem
- Large deviation principle
- Law of large numbers (weak/strong)
- Law of the iterated logarithm
- Maximal ergodic theorem
- Sanov's theorem
- Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
- Cameron–Martin formula
- Convergence of random variables
- Doléans-Dade exponential
- Doob decomposition theorem
- Doob–Meyer decomposition theorem
- Doob's optional stopping theorem
- Dynkin's formula
- Feynman–Kac formula
- Filtration
- Girsanov theorem
- Infinitesimal generator
- Itô integral
- Itô's lemma
- Karhunen–Loève theorem
- Kolmogorov continuity theorem
- Kolmogorov extension theorem
- Lévy–Prokhorov metric
- Malliavin calculus
- Martingale representation theorem
- Optional stopping theorem
- Prokhorov's theorem
- Quadratic variation
- Reflection principle
- Skorokhod integral
- Skorokhod's representation theorem
- Skorokhod space
- Snell envelope
- Stochastic differential equation
- Stopping time
- Stratonovich integral
- Uniform integrability
- Usual hypotheses
- Wiener space
- Actuarial mathematics
- Control theory
- Econometrics
- Ergodic theory
- Extreme value theory (EVT)
- Large deviations theory
- Mathematical finance
- Mathematical statistics
- Probability theory
- Queueing theory
- Renewal theory
- Ruin theory
- Signal processing
- Statistics
- Stochastic analysis
- Time series analysis
- Machine learning
- List of topics
- Category