  Our goal is to find classes of convolution semigroups on Lie groups $G$ that
give rise to interesting processes in symmetric spaces $G/K$. The
$K$-bi-invariant convolution semigroups are a well-studied example. An
appealing direction for the next step is to generalise to right $K$-invariant
convolution semigroups, but recent work of Liao has shown that these are in
one-to-one correspondence with $K$-bi-invariant convolution semigroups. We
investigate a weaker notion of right $K$-invariance, but show that this is, in
fact, the same as the usual notion. Another possible approach is to use
generalised notions of negative definite functions, but this also leads to
nothing new. We finally find an interesting class of convolution semigroups
that are obtained by making use of the Cartan decomposition of a semisimple Lie
group, and the solution of certain stochastic differential equations. Examples
suggest that these are well-suited for generating random motion along geodesics
in symmetric spaces.
