  We consider global pseudodifferential operators on symmetric spaces of
noncompact type, defined using spherical functions. The associated symbols have
a natural probabilistic form that extend the notion of the characteristic
exponent appearing in Gangolli's L\'evy-Khinchine formula to a function of two
variables. The Hille-Yosida-Ray theorem is used to obtain conditions on such a
symbol so that the corresponding pseudodifferential operator has an extension
that generates a sub-Feller semigroup, generalising existing results for
Euclidean space.
