On the passage times of self-similar Gaussian processes on curved boundaries
Davar Khoshnevisan, Cheuk Yin Lee
TL;DR
This work extends the classical Brownian boundary-crossing results to general self-similar Gaussian processes on curved boundaries $ct^β$, providing a complete tripartite regime (supercritical, subcritical, critical) determined by the relation between $β$ and the self-similarity index $α$. The authors develop a cohesive framework leveraging LIL-type bounds, Borell concentration, Gaussian correlation inequalities, and the Lamperti transform to an Ornstein–Uhlenbeck process, together with subadditive techniques (Fekete’s lemma) to define a global critical boundary crossing exponent $λ(c)$ in the critical case. They establish precise moment and tail behaviors in each regime, prove convexity and monotonicity properties of $λ$, and connect to Brownian and fractional Brownian motion limits. The paper also provides concrete examples via fractional Brownian motion and linear SPDEs, along with a comparison principle that links the critical exponents to those of fBm, and discusses simulation strategies for estimating $λ$ in practice. The results yield a quantitative, model-agnostic view of boundary-crossing phenomena for non-Markov Gaussian processes with potential applications in SPDE analysis and statistical testing frameworks.
Abstract
Let $T_{c,β}$ denote the smallest $t\ge1$ that a continuous, self-similar Gaussian process with self-similarity index $α>0$ moves at least $\pm c t^β$ units. We prove that: (i) If $β>α$, then $T_{c,β}=\infty$ with positive probability; (ii) If $β<α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then $T_{c,β}$ has moments of all order; and (iii) If $β=α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then there exists a continuous, strictly decreasing function $λ:(0\,,\infty)\to(0\,,\infty)$ such that $\mathrm{E}(T_{c,β}^μ)$ is finite when $0<μ<λ(c)$ and infinite when $μ>λ(c)$. 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.