On Global and Sharp Markov Properties of Random Functional Series in Sobolev Spaces

Abstract

For a generalized random function $\xi = \sum\nolimits_{n = 1}^\infty {u_n \xi _n } $, where $\{ {u_n } \}_{n = 1}^\infty $ is a complete orthonormal system in a Sobolev space $W_2^p (T)$ with a regular domain $T \subseteq {\bf R}^d $ and $\{ {\xi _n } \}_{n = 1}^\infty $ is a sequence of independent $N(0,1)$ random variables, we establish the global Markov property of $\xi $. We also characterize the splitting $\sigma $-algebras $\sigma ^ + (\partial G) = : \cap _{\varepsilon > 0} \sigma ((\varphi ,\xi );\varphi \in C_0^\infty (\partial G^\varepsilon ))$ for any $G \subseteq T$ as $\sigma ((\varphi ,\xi );\varphi \in W_2^p (T)',{\operatorname{supp}}\varphi \subseteq \partial G)$. For a regular subdomain $G \subseteq T$, this characterization reduces to $\sigma ^ + (\partial G) = \sigma (\sum\nolimits_{n = 1}^\infty {(\varphi ,u_n^{(k)} )} _{L^2 } \xi _n ;\varphi \in L^2 (\partial G),u_n^{(k)} \,{\text{is the }}k{\text{th trace of }}u_k {\text{ on }}\partial G{\text{ for }}k = 1, \ldots ,p - 1\,)$ for if p is isotropic. An example of nondeterministic generalized random function satisfying the sharp Markov property is also given.