Zero-one laws and the minimum of a Markov process

Abstract:

If $\{ {X_t},t > 0\}$ is a real strong Markov process whose paths assume a (last) minimum at some time M strictly before the lifetime, then conditional on I, the value of this minimum, the process $\{ X(M + t),t > 0\}$ is shown to be Markov with stationary transitions which depend on I. For a wide class of Markov processes, including those obtained from Lévy processes via time change and multiplicative functional, a zero-one law is shown to hold at M in the sense that ${ \cap _{t > 0}}\sigma \{ X(M + s),s \leqslant t\} = \sigma \{ X(M)\}$, modulo null sets. When such a law holds, the evolution of $\{ X(M + t),t \geqslant 0\}$ depends on events before M only through $X(M)$ and I.