Sample path properties of Banach space-valued stable stochastic processes

The first kind of results which are of interest to us are the zero-one laws for the regularity properties of sample paths of real-valued Gaussian and real-valued symmetric stable stochastic processes. One result of this type is obtained by Rajput and Cambanis [22] who using a general zero-one law of Kallianpur [18] showed that most regularity properties of sample paths of real-valued Gaussian stochastic processes hold with probability zero or one; that this result also holds for real-valued symmetric stable stochastic processes has been pointed out by Miller and Cambanis [19].

The second kind of results which are related to the above and of interest to us are those which give verifiable necessary and sufficient conditions for the two alternatives in the zero-one laws mentioned above. Only few results are known in this direction; Cambanis [8], [9], and Miller and Cambanis [19] have obtained necessary and sufficient conditions for almost sure absolutely continuity of sample paths of realvalued separable Gaussian and real-valued separable symmetric stable stochastic processes, respectively; and Dudley [15] and Fernique [17] have obtained necessary and sufficient conditions for almost sure continuity of sample paths of real-valued stationary Gaussian stochastic processes. The third type of results which have relevance to our work deals with (a) the existence of measurable modification of real-valued stochastic processes and (b) conditions under which the sample paths of measurable stochastic processes belong to L_p. With regards to (a), Cambanis [7] and Miller and Cambanis [19] have obtained several equivalent conditions for second and p-th order real-valued stochastic processes to have a measurable modification, respectively. With regards to (b), Rajput in [2l] showed that there is a one-to-one correspondence between Gaussian measures on L_p , 1 ≤ p < ∞, and measurable Gaussian stochastic processes with almost all sample paths in L_p; he further showed that the sample paths of such a stochastic process belong to L_p with probability zero or one and also obtained conditions for the two alternatives in terms of the mean and covariance of the stochastic process. Byczkowski extended all the above three results by showing that these results hold as well when L_p , 1 ≤ p < ∞ , is replaced by L_p ,0 < p < ∞ ; the last result above is extended for real-valued symmetric stable stochastic processes by Miller and Cambanis [19].

Our main purpose in this dissertation is to obtain results similar to the above for Banach space-valued general (not necessarily symmetric) stable stochastic processes and we thus unify and generalize most of the above results (except the one due to Dudley and Fernique) in two directions: (i) our state space is an infinite dimensional Banach space and (ii) we have no restrictive conditions (like symmetry) on our stochastic processes. We now give a summary of the main results. The first chapter includes basic definitions, notations, and known results needed in the sequel. In the second chapter we obtain three main results: First, we find conditions for a p-th order Banach space-valued stochastic process to have a measurable modification. Second, using ideas of Byczkowski [4], [5], we show that there is a one-to-one correspondence between Banach space-valued measurable stable stochastic processes ξ = {ξ_t |t ∈ T} with almost all sample paths belonging to L_p(T, 𝔹) and stable measures on L_p(T, 𝔹). Third, we show that sample paths of Banach space-valued measurable stable stochastic processes ξ belong to L_p with probability zero or one and find conditions for the two alternatives. In chapter three we first obtain conditions under which almost all sample paths of ξ are absolutely continuous and then we establish that, just as in the real case, most regularity properties of sample paths of Banach space-valued stable stochastic processes hold with probability zero or one.