A Markov property for multicomponent Euclidean covariant Gaussian generalized stochastic fields

Abstract

In Chapter 1 to Section 2.3, pertinent probabilistic concepts, as well as notions of Markov property, are briefly discussed. Section 2.4 is an account of the theory of boundary value problems for elliptic systems of linear partial differential operators of arbitrary orders. Particular attention is paid to the solution of the Dirichlet problem for such a system because it intervenes in the analysis of Chapter 4.

In Chapter 3, a spectral representation (THEOREM (3.2.13)) is provided for an arbitrary Euclidean covariant (see (3.2.11)) multicomponent generalized stochastic field. This result, obtained group-theoretically, is then applied to the special case of a three component generalized stochastic field (see (3.2.25)), also needed in Chapter 4.

In Section 4.1, Wong's notion of Markov property is formulated. The rest of Chapter 4 is then concerned with the complete characterization of the class of all three dimensional Euclidean covariant Gaussian generalized stochastic fields which are Markov in the sense of Wong. It is also shown here that some of the latter are not also Markov in the sense of Nelson. Readers familiar with the work of Wong [94] will readily recognize the various results of Chapter 4 as extensions of those of Wong.

In Chapter 5, Wong's notion of Markov property is given abstract formulation (THEOREM (5.2)). Then it is demonstrated that, like Nelson's notion of Markov property, Wong's notion of Markov property is implied by the so-called pre-Markov property.

In Chapter 6, ways of extending the investigations of Chapter 4 to arbitrary multicomponent Euclidean covariant Gaussian generalized stochastic fields are indicated. It is then observed that the required extensions present no new problems or difficulties. Finally, Wong's notion of Markov property is formulated much more generally and sufficient conditions (THEOREM (6.2.1)) for Markovicity in this extended formulation are furnished. These indicate that quite a large class of mathematically and physically interesting multicomponent Gaussian generalized stochastic fields are Markov in the sense of Wong.