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Additional info for A brief course in the calculus, by William Cain.
Suppose that t ≥ s. Then the covariance is given by Exi (t)xj (s) − Exi (t)Exj (s) t A(t−t1 ) = E e 0 k s eA(t−r) = 0 s = k,h ik dwk (t1 ) 2ckh eA(s−r) T (s−r) eA(t−r) 2ceA 0 = ik s s A(t−s) e ij dr T eAr 2ceA r dr 0 The case t ≤ s is analogous. ij QED. 0 eA(s−s1 ) h jh dr jh dwh (s1 ) 44 CHAPTER 8 Reference . , No 117, “Les ´equations aux d´eriv´ees partielles”, Editions Paris, 1962. ) Chapter 9 The Ornstein-Uhlenbeck theory of Brownian motion The theory of Brownian motion developed by Einstein and Smoluchowski, although in agreement with experiment, was clearly a highly idealized treatment.
Thus (using Newton’s law F = ma) we are considering the force on a free Brownian particle as made up of two parts, a frictional force F0 = −mβv with friction coeﬃcient mβ and a ﬂuctuating force F1 = mdB/dt which is (formally) a Gaussian stationary process with correlation function of the form a constant times δ, where the constant will be determined later. 2) v(s) ds. 0 For a free particle there is no loss of generality in considering only the case of one-dimensional motion. Let σ 2 be the variance parameter of B (inﬁnitesimal generator 12 σ 2 d2 /dv 2 , EdB(t)2 = σ 2 dt).
Proof. 1 by integration, t m(t) ˜ = x0 + t r˜(t, s) = 0 dt1 m(s) ds, 0 s 0 ds1 r(t1 , s1 ). The second integration is tedious but straightforward. In particular, the variance of x(t) is 2Dt + D (−3 + 4e−βt − e−2βt ). β QED. THE ORNSTEIN-UHLENBECK THEORY OF BROWNIAN MOTION 49 The variance in Einstein’s theory is 2Dt. By elementary calculus, the absolute value of the diﬀerence of the two variances is less than 3Dβ −1 . , we make a proportional error of less than 3 × 10−8 by adopting Einstein’s value for the variance.