Prove Doob’s theorem. More specifically, for any Gaussian-Markov random process, show that P₂(y₂, t|y₁) is given by Eqs. (6.18a,b).

(a) Show that the Gaussian process y_new has probability distributions

and show that the constant C₂₁ that appears here is the correlation function C₂₁ = C_y(t₂ − t₁).

(b) From the relationship between absolute and conditional probabilities [Eq. (6.4)], show that

(c) Show that for any three times t₃ > t₂ > t₁,

To show this, you could 

(i) Use the relationship between absolute and conditional probabilities and the Markov nature of the random process to infer that

then

(ii) Compute the last expression explicitly, getting

(iii) Then using this expression, evaluate

The result should be C₃₁ = C₃₂C₂₁.

(d) Argue that the unique solution to this equation, with the “initial condition” that C_y(0) = σy² = 1, is C_y(τ ) = e^−τ/τ_r, where τ_r is a constant (which we identify as the relaxation time). Correspondingly, C₂₁ = e^{−(t₂−t₁)/τ_r}.

(e) By inserting this expression into Eq. (6.22c), complete the proof for y_new(t), and thence conclude that Doob’s theorem is also true for our original, unrescaled y_old(t).

Data from Equation 6.4

2(2, y1) = 1 [2 2]1 exp "]| ( ,)2 20, 2 (6.18a)