Gaussian Processes and Brownian Motion
Symmetries of Gaussian distribution; existence and path properties of Brownian motion; strong Markov and reflection properties; arcsine and uniform laws; law of the iterated logarithm; Wiener integrals and isonormal Gaussian processes; multiple Wiener-Itô integrals; chaos expansion of Brownian functionals
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- The Gaussian density function first appeared in the work of De Moiv-Re (1733–56), and the corresponding distribution became explicit through the work of Laplace (1774, 1812–20). The Gaussian law was popularized by Gauss (1809) in his theory of errors and so became named after him. Maxwell derived the Gaussian law as the velocity distribution for the molecules in a gas, assuming the hypotheses of Proposition 13.2. Theorem 13.3 was originally stated by Schoenberg (1938) as a relation between positive definite and completely monotone functions; the probabilistic interpretation was later noted by Freedman (1962–63). Isonormal Gaussian processes were introduced by Segal (1954).Google Scholar
- The process of Brownian motion was introduced by Bachelier (1900, 1901) to model fluctuations on the stock market. Bachelier discovered some basic properties of the process, such as the relation Mt=d|Bt|. Einstein (1905, 1906) later introduced the same process as a model for the physical phenomenon of Brownian motion—the irregular movement of microscopic particles suspended in a liquid. The latter phenomenon, first noted by VAN Leeuwenhoek in the seventeenth century, is named after the botanist Brown (1828) for his systematic observations of pollen grains. Einstein’s theory was forwarded in support of the still-controversial molecular theory of matter. A more refined model for the physical Brownian motion was proposed by Langevin (1909) and Ornstein and Uhlenbeck (1930).Google Scholar
- The mathematical theory of Brownian motion was put on a rigorous basis by Wiener (1923), who constructed the associated distribution as a measure on the space of continuous paths. The significance of Wiener’s revolutionary paper was not fully recognized until after the pioneering work of Kolmogorov (1931a, 1933), Levy (1934–35), and Feller (1936). Wiener also introduced stochastic integrals of deterministic L 2-functions, which were later studied in further detail by Paley et al. (1933). The spectral representation of stationary processes, originally deduced from Bochner’s (1932) theorem by Cramer (1942), was later recognized as equivalent to a general Hilbert space result due to M.H. Stone (1932). The chaos expansion of Brownian functionals was discovered by Wiener (1938), and the theory of multiple integrals with respect to Brownian motion was developed in a seminal paper of Ito (1951c).Google Scholar
- The law of the iterated logarithm was discovered by Khinchin, first (1923, 1924) for Bernoulli sequences, and later (1933) for Brownian motion. A systematic study of the Brownian paths was initiated by Lévy (1954, 1965), who proved the existence of the quadratic variation in (1940) and the arcsine laws in (1939, 1965). Though many proofs of the latter have since been given, the present deduction from basic symmetry properties may be new. The strong Markov property was used implicitly in the work of Levy and others, but the result was not carefully stated and proved until Hunt (1956).Google Scholar
- Many modern probability texts contain detailed introductions to Brownian motion. The books by Itô and McKean (1965), Freedman (1971b), Karatzas and Shreve (1991), and Revuz and Yor (1999) provide a wealth of further information on the subject. Further information on multiple Wiener-Itô integrals is given by Kallianpur (1980), Dellacherie et al. (1992), and Nualart (1995). The advanced theory of Gaussian distributions is nicely surveyed by Adler (1990).Google Scholar