  We study representations of positive definite kernels $K$ in a general
setting, but with view to applications to harmonic analysis, to metric
geometry, and to realizations of certain stochastic processes. Our initial
results are stated for the most general given positive definite kernel, but are
then subsequently specialized to the above mentioned applications. Given a
positive definite kernel $K$ on $S\times S$ where $S$ is a fixed set, we first
study families of factorizations of $K$. By a factorization (or representation)
we mean a probability space $\left(B,\mu\right)$ and an associated stochastic
process indexed by $S$ which has $K$ as its covariance kernel. For each
realization we identify a co-isometric transform from $L^{2}\left(\mu\right)$
onto $\mathscr{H}\left(K\right)$, where $\mathscr{H}\left(K\right)$ denotes the
reproducing kernel Hilbert space of $K$. In some cases, this entails a certain
renormalization of $K$. Our emphasis is on such realizations which are minimal
in a sense we make precise. By minimal we mean roughly that $B$ may be realized
as a certain $K$-boundary of the given set $S$. We prove existence of minimal
realizations in a general setting.
