> for random fields

I. V. Evstigneev

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 575-581

Résumé

Let $\{\mathscr F_V\}$ be a family of $\sigma$-algebras parametrized by closed subsets $V$ in an $n$-dimensional Euclidean space $X$. Assume that $\{\mathscr F_V\}$ possesses the following properties: (I) if $V'\subseteq V''$, then $\mathscr F_{V'}\subseteq\mathscr F_{V''}$; (II) $\displaystyle\bigcap_{\varepsilon>0}\mathscr F_{V_{\varepsilon}}=\mathscr F_V$, $V_{\varepsilon}$ being the $\varepsilon$-vicinity of $V$. For any random field (usual or generalized), the family of $\sigma$-algebras $\mathscr F_V$ describing the behaviour of the field in the infinitesimal vicinity of $V$ has the above properties. A random closed set $T(\omega)$ is called optional with respect to the family $\{\mathscr F_V\}$ if $\{\omega\colon T(\omega)\subseteq V\}\in\mathscr F_V$ for all $V$. Such random sets are analogous to optional times in the one-dimensional case. In particular, if the field is Markov, we can prove a version of the strong Markov property with respect to such sets. The result is formulated in terms of $\{\mathscr F_V\}$ only and requires no additional information about the field. Given a usual random field with continous sample functions, a connected component of a level set is an example of a multidimensional «optional time».