By A. T. Bharucha-Reid

ISBN-10: 0120956012

ISBN-13: 9780120956012

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The physical model consists of a particle mov­ ing along a line with constant speed and reversing its direction at random times. It was then shown by Goldstein [40] that, under a certain limit, the mean concentration satisfies the Telegrapher's equation. Many years later, Kac [49] recovered this result by introducing a random walk model. Let v{t) denote the velocity of a marked particle moving along a line at time /, and let \v(t)\ =V0,SL constant. 1) where ξ0 =±ν0 with equal probability. Let φ(χ) be a smooth real function on IR.

I > * = 1,2,.... 15) with respect to λ, and then set λ = 0, we obtain the first moment equation dtTx = vA^ + iKVtr^. 19) where (Δ Θ Δ) Γ2 (t, x, x') = (Δ + Δ') Γ2 (ί, χ, χ'). 19), the first two moment equations, similar to Kraichnan's direct interaction equation, are closed. 15) becomes a linear hyperbolic equation of first order, similar to Eq. 11), which can be solved exactly. 15) is the Gaussian characteristic functional with mean Γ^ and covariance (Γ2 — Tt ® Γ^). 19) have equilibrium solutions, then they determine the Gaussian invariance measure.

56 (1971), 405-418. 42. " MIT Press, Cambridge, Massachusetts, 1964, pp. 51-68. 43. , Abstract Wiener spaces, Proc. 5th Berkeley Symp. Math. , Vol. II, 1967, 31-42. 44. Herring, J. , and Kraichnan, R. , Comparison of some approximations for isotropie turbulence,/« "Statistical Models and Turbulence," Springer Pubi. New York, 1972, pp. 148-177. 42 PAO-LIU CHOW 45. , "Random evolutions: A survey of results and problems, Rocky Mountain J. Math. 4 (1974), 443-496. 46. , Statistical hydromechanics and functional calculus,/.

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Probabilistic Analysis and Related Topics. Volume 1 by A. T. Bharucha-Reid

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