Emanuel József Godfried

arXiv2026

Geodesic Lévy flights on Zoll surfaces

We study the mean first capture time of isotropic Lévy flights on Zoll surfaces, namely the expected time for a geodesic Lévy process to reach a shrinking geodesic ball. While the leading-order asymptotics are universal, we prove that the first correction term encodes subtle geometric information. More precisely, it is completely determined by the local singularity type of the conjugate locus, quantified by the degree of the conjugate point. This yields a hierarchy of asymptotic regimes governed by the Lévy exponent.

math.DG↗

arXiv2026

Mean first escape times of Brownian motion on asymptotically hyperbolic and gas giant metric surfaces

This paper deals with the mean first escape time of Brownian motion on asymptotically hyperbolic and gas giant surfaces. We show that for a boundary defining function $ρ$, the mean first escape time $u_ε(x)$ from the truncated Riemannian surface with an asymptotically hyperbolic metric $(M_ε,\bar{g}/ρ^2) = (\{x\in M:ρ(x)\geq ε\},\bar{g}/ρ^2) \subset (M,\bar{g}/ρ^2)$ satisfies the asymptotic expansion $u_ε(x) = -\log ε+ \mathcal{O}(1)$ as $ε\to 0 $. Furthermore, we show that in the case of a gas giant metric $g = \bar{g}/ρ^α$, where $α\in (0,2)$, the mean first escape time from the surface $(M_ε,\bar{g}/ρ^α)$ satisfies $u_ε(x) = \mathcal{O}(1)$ as $ε\to 0 $. Using techniques from the theory of polyhomogeneous conormal functions we explain this difference between in the mean first escape time on gas giant metric surfaces and asymptotically hyperbolic surfaces on the unit disc. Finally, we confirm these results using Monte Carlo simulations and finite difference methods on the disc.

math.AP↗