«Splitting times» for random fields

There are various results concerning the strong Markov property for one-dimensional processes. The essence of these results can be expressed as follows. For a «good» Markov process $x_t$ and a special class of random times $\tau$ (the so called optional times): a) the behaviour of the process $x_t$ before $\tau$ and its behaviour after $\tau$ are conditionally independent given $\tau$, $x_{\tau}$; b) the forecast of the process' behaviour after $\tau$ based on known $\tau$, $x_{\tau}$ is quite the same as if $\tau$ be non-random (this forecast is determined by the transition function $p(\tau,x_{\tau},s,\Gamma)$).
Williams and Jacobsen introduced a class of random times $\tau$, the so called splitting times, for which property a) only is valid. In this paper, the concept of splitting time is generalized for random fields.
A random field is defined as a system of $\sigma$-algebras $\{\mathscr F_V\}$, $V$ being a closed subset of a finite-dimensional Euclidean space $X$. The Markov property means that $\mathscr F_V$ and $\mathscr F_W$ are conditionally independent given $\mathscr F_{V\bigcap W}$ provided $V\bigcup W=X$. A random time is a pair of random closed sets $V$ and $W$ such that $V\bigcup W=X$. Given $\tau=(V,W)$, we introduce the $\sigma$-algebras of the «past» $\mathscr F_V$, the «future» $\mathscr F_W$ and the «present» $\mathscr F_{V\bigcap W}$. A random time $\tau$ is called a splitting time if the past and the future are conditionally independent given the present. We give necessary and sufficient conditions for a random time with countably many values to be splitting and find sufficient conditions in the case of uncountably many values.