{#glob-proc}
================

Let $X$ be a separable complete metric space with metric $\rho$, which we assume to be bounded: $\rho(x, y)\le 1$ for all $x, y\in X$. Denote by ${\cal B}_X$ the Borel $\sigma$-algebra of $X$. Let $\Omega$ be the space of all continuous infinite one-side paths in $X$ equipped with the compact-open topology. The elements of $\Omega$ are continuous maps $u: [0, +\infty)\to X$, and convergence in $\Omega$ is the uniform convergence on every finite time interval. The space $\Omega$ is Polish; we fix a complete bounded metric on $\Omega$ by setting \[met-Om\] d(u, v) = \_[= 1]{}\^2\^[-]{}\_[t]{}(u(t), v(t))\^[-1]{}. Sometimes it is convenient to view the paths parametrized by $s\in [\tau, +\infty)$. We denote by $\Omega^\tau$ the space of continuous maps from $[\tau, +\infty)$ into $X$. It can be identified with the image of $\Omega$ under the (past erasing) map $\sigma_\tau$: if $u(t)$, $t\ge 0$, is a path in $\Omega$, then the path $\sigma_\tau u : [\tau, +\infty)\to X$ is defined as $\sigma_\tau u(t) = u(t)$ for $t\ge \tau$. On the other hand, $\Omega^\tau$ can be viewed as a subset of $\Omega$ if we extend the paths $v(t)$ in $\Omega^\tau$ to $[0, \tau)$ as staying at $v(\tau)$: $$\Game_\tau : 