First passage in infinite paraboloidal domains

We study first-passage properties for a particle that diffuses either inside or outside of generalized paraboloids, defined by y = a(x1 + · · ·+ xd−1) where p > 1, with absorbing boundaries. When the particle is inside the paraboloid, the survival probability S(t) generically decays as a stretched exponential, lnS ∼ −t(p−1)/(p+1), independent of the spatial dimension. For a particle outside the paraboloid, the dimensionality governs the asymptotic decay, while the exponent p specifying the paraboloid is irrelevant. In two and three dimensions, S ∼ t−1/4 and S ∼ (ln t)−1, respectively, while in higher dimensions the particle survives with a finite probability. We also investigate the situation where the interior of a paraboloid is uniformly filled with non-interacting diffusing particles and estimate the distance between the closest surviving particle and the apex of the paraboloid.