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Einar Hille and Ralph S. Phillips, “Functional Analysis and SemiGroups”, revised edition, American Math. Soc. Colloquium Publications, vol. XXXI, 1957. Chapter 6 Gaussian processes Gaussian random variables were discussed by Gauss in 1809 and the central limit theorem was stated by Laplace in 1812. Laplace had already considered Gaussian random variables around 1780, and for this reason Frenchmen call Gaussian random variables “Laplacian”. However, the Gaussian measure and an important special case of the central limit theorem were discovered by de Moivre in 1733.

N−1 ] |ξ(0) − η(0)| + αn ξ − η . 6) A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS 39 If x0 is in ❘ , we shall also let x0 denote the constant map x0 (s) = x0 , and we shall let x be the fixed point of T with x(0) = x0 , so that x = lim T n x0 , n→∞ and similarly for y0 in ❘ . 6), x − y ≤ β|x0 − y0 |, where β = 1/(1 − α). Therefore, |x(t) − y(t)| ≤ eλt β|x0 − y0 |. Now let f be any Lipschitz function on ❘ with Lipschitz constant K. Then f x(t) − f y(t) ≤ Keλt β|x0 − y0 |. Since this is true for each fixed w path, the estimate remains true when we take expectations, so that |P t f (x0 ) − P t f (y0 )| ≤ Keλt β|x0 − y0 |.

This is not true. Even at the lowest pressure used, an enormous number of collisions takes place per period, and the irregularities in the curves are due to chance fluctuations in the sum of enormous numbers of individually negligible events. 58 CHAPTER 10 It is not correct to think simply that the jiggles in a Brownian trajectory are due to kicks from molecules. Brownian motion is unbelievably gentle. Each collision has an entirely negligible effect on the position of the Brownian particle, and it is only fluctuations in the accumulation of an enormous number of very slight changes in the particle’s velocity that give the trajectory its irregular appearance.