On the future infima of some transient processes (Q1333573)

Let \((X(t)\), \(t \in S)\) be a real-valued stochastic process with \(\mathbb{P} (X(0) = 0) = 1\) and \(\mathbb{P} ({\displaystyle {\lim_{t \to \infty}}} X(t) = \infty) = 1\). We are interested in the reluctance of such a process to tend to infinity. This entails determining the rate of escape of the associated process \((\inf_{s \geq t} X(s), t \in S)\), the so-called future infima process.