I'm working through an old Diaconis paper on Bayesian numerical analysis, and am currently calculating the details behind his brief comments on using $n$-times integrated Brownian motion as a function prior for quadratures.
In short, that means working out the covariance, and then the expectation and covariance conditional on a later value of the same $n$-integrated motion, i.e. the equivalent to a Brownian bridge.
I've been working these out from scratch, mostly by heavy use of recurrence relations, and so far I've tentatively got the statistics for $n$-integrated Brownian motion $B_t^{(n)},$ $n\geq0,$ as being
$$\begin{aligned}\textrm{Cov}(B_s^{(n)}, B_t^{(n)}) &= \mathbb{E}(B_s^{(n)} B_t^{(n)}) \\&= \frac{1} {(2n+1)!} \sum_{k=0}^n (-1)^{n-k} \binom{2n+1}{k} \min(s,t)^{2n+1-k} \max(s,t)^k,\end{aligned}$$ $$\mathrm{Var}(B_t^{(n)}) = \frac{1} {(2n+1)!} \binom{2n}{n} t^{2n+1},$$ $$\mathbb{E}(B_s^{(n)} | B_t^{(n)}) = \binom{2n}{n}^{-1} \sum_{k=0}^n (-1)^{n-k} \binom{2n+1}{k} \left(\frac{s}{t}\right)^{2n+1-k} B_t^{(n)},$$ $$\textrm{e.g. } \mathbb{E}(B_s^{(1)} | B_t^{(1)}) = \frac{1} {2} \left(\frac{s}{t}\right)^2 \left(3-\frac{s}{t}\right) B_t^{(1)},$$ $$\mathbb{E}(B_r^{(n)} B_s^{(n)} | B_t^{(n)})\textrm{ pending}.$$
Feedback on whether I'm correct so far would be nice, but what I'd really like are pointers to good references on the subject, preferably books. Thanks in advance.