On global and sharp Markov properties of random functional series in Sobolev spaces

T. S. Chiang   ; Y. Chow

Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 2, pp. 464-471

Résumé

For a generalized random function $\xi=\sum_{n=1}^\infty u_n\xi_n$, where $\{u_n\}_{n=1}^\infty$ is a complete orthonormal system in a Sobolev space $W_2^p(T)$ with a regular domain $T\subseteq\mathbf{R}^d$ 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 $\sigma^+(\partial G)=:\bigcap_{\varepsilon>0}\sigma((\varphi,\xi);\varphi\in C_0^\infty(\partial G^\varepsilon ))$ 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 $\sigma^+(\partial G)=\sigma(\sum_{n=1}^\infty(\varphi,u_n^{(k)})_{L^2}\xi_n;\varphi\in L^2(\partial G),u_n^{(k)}$ is the $k$th trace of $u_k$ on $\partial G$ for $k=1,\dots,p-1)$ for if $p$ is isotropic. An example of nondeterministic generalized random function satisfying the sharp Markov property is also given.

Détail

Keywords: a generalized random function, random functional series, the Sobolev space, the Hilbert space, the global Markov property, the sharp Markov property.