  Suppose that $\gamma$ and $\sigma$ are two continuous bounded variation paths
which take values in a finite-dimensional inner product space $V$. Recent
papers have introduced the truncated and the untruncated signature kernel of
$\gamma$ and $\sigma$, and showed how these concepts can be used in
classification and prediction tasks involving multivariate time series. In this
paper, we introduce general signature kernels and show how they can be
interpreted, in many examples, as an average of PDE solutions, and hence how
they can be computed using suitable quadrature rules. We extend this analysis
to derive closed-form formulae for expressions involving the expected
(Stratonovich) signature of Brownian motion. In doing so, we articulate a novel
connection between signature kernels and the hyperbolic development map, the
latter of which has been a broadly useful tool in the analysis of the
signature. As an application we evaluate the use of different general signature
kernels as the basis for non-parametric goodness-of-fit tests to Wiener measure
on path space.
