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1 , ????2 , . . , ???????? ; ????1 , ????2 , . . 13) for any ????, ????1 , ????2 , . . , ???????? and ???? . e. ???? (????, ????) = ???? (????) cannot depend upon ????); all 2-dimensional distribution functions can only depend upon ????1 − ????2 ; and so on. It follows that the autocorrelation ???? depends upon time difference only, ⟨????(????)????(????)⟩ = ????(???? − ????) = ????(???? ). It turns out that there exists a whole class of phenomena where the underlying stochastic process is completely characterized by the mean ⟨????(????)⟩ = ???? = constant, and by the correlation function ????(???? ).

Fill in the steps that connect Eq. 28 to Eq. 30. It may be helpful to treat ????1 (????1 −????2 ) in Eq. 2). 2. Harmonically-bound particle: Repeat Ornstein and Uhlenbeck’s analysis to calculate ⟨????(????)⟩ and ⟨????2 (????)⟩ for the harmonically-bound particle, Eq. 45. 3. Benveniste: Read the paper “Human basophil degranulation triggered by very dilute antiserum against IgE,” by Davenas et al. Notice the disclaimer appended to the end of the article by the editor. Briefly (one or two sentences) summarize what the authors claim.

Fix ????, then ????(????, ????) = ???????? (????) is a family of random variables depending upon the parameter ????. In that way, a stochastic process can be regarded as either a family of realizations ???? (????) (????), or as a family of random variables ???????? (????). This may seem a pedantic distinction, but notice that Einstein’s point of view was to treat Brownian motion as a distribution of a random variable describing position (???????? (????)), while Langevin took the point of view that Newton’s law’s of motion apply to an individual realization (???? (????) (????)).

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