Vol. 15, No. 2, 1965

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ISSN: 0030-8730
Sample functions of certain differential processes on symmetric spaces

Ramesh Anand Gangolli

Vol. 15 (1965), No. 2, 477–496

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.

Mathematical Subject Classification
Primary: 60.08
Secondary: 60.60
Received: 10 January 1964
Published: 1 June 1965
Ramesh Anand Gangolli