  We study first-passage properties for a particle that diffuses either inside
or outside of generalized paraboloids, defined by
y=a(x_1^2+...+x_{d-1}^2)^{p/2} where p>1, with absorbing boundaries. When the
particle is inside the paraboloid, the survival probability S(t) generically
decays as a stretched exponential, ln(S) ~ -t^{(p-1)/(p+1)}, independent of the
spatial dimensional. 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.
