Abstract: Let \(T_{c,\beta}\) denote the smallest \(t\ge1\) that a continuous, self-similar Gaussian process with self-similarity index \(\alpha>0\) moves at least \(\pm c t^\beta\) units. We prove that: (i) If \(\beta>\alpha\), then \(T_{c,\beta}=\infty\) with positive probability; (ii) If \(\beta<\alpha\) then \(T_{c,\beta}\) has moments of all order; and (iii) If \(\beta=\alpha\) and \(X\) is strongly locally nondeterministic in the sense of Pitt (1978), then there exists a continuous, strictly decreasing function \(\lambda:(0\,,\infty)\to(0\,,\infty)\) such that \({\textnormal{E}} (T_{c,\beta}^\mu)\) is finite when \(0<\mu<\lambda(c)\) and infinite when \(\mu>\lambda(c)\). Together these results extend a celebrated theorem of Breiman (1967) and Shepp (1967) for passage times of a Brownian motion on the critical square-root boundary. We briefly discuss two examples: One about fractional Brownian motion, and another about a family of linear stochastic partial differential equations.
This is based on joint work with Cheuk Yin Lee (CUHK-Shenzhen).