Stochastic processes with independent increments, taking values in a Hilbert space

AbstractLet H be a real separable Hilbert space; let X(t), tϵ[0, 1], be a separable, stochastically continuous, H-valued stochastic process with independent increments. Then a decomposition of X(t) into a uniformly convergent sum of independent processes is found. In this decomposition one of the processes is Gaussian with continuous sample functions, and the remainder of the processes have sample functions whose discontinuities correspond to those of certain real-valued Poisson processes. The decomposition of X(t) leads to a Levy-Khintchine representation of the characteristic functional of X(t). In addition, the case when X(t) has finite variance is explored, and, as a consequence of the above decomposition, a Kolmogorov-type representation of the characteristic functional of X(t) is derived