In a recent paper, we have
proved a formula characterizing the abstract Fourier-Stieltjes transform of an
isotropic infinitely divisible probability measures on a symmetric space. The
formula is the full analogue of the classical Lévy-Khinchine formula for the
Fourier-Stieltjes transform of infinitely divisible probability measures on the real
line.
Now, just as in the case of the line, an isotropic, infinitely divisible probability
measure on a symmetric space gives rise in a natural way to a continuous
one parameter convolution semigroup of such measures; and thence to a
stochastic process with stationary independent “increments”. It is the purpose of
this paper to construct the sample functions of such a process. We shall
exhibit the sample functions of such a process as limits with probability one
(uniformly on compact subsets of the parameter set) of sequences of continuous
Brownian trajectories interlaced with finitely many isotropic Poissonian
jumps.
