  We quantify the asymptotic behaviour of multidimensional drifltess diffusions
in domains unbounded in a single direction, with asymptotically normal
reflections from the boundary. We identify the critical growth/contraction
rates of the domain that separate stability, null recurrence and transience. In
the stable case we prove existence and uniqueness of the invariant distribution
and establish the polynomial rate of decay of its tail. We also establish
matching polynomial upper and lower bounds on the rate of convergence to
stationarity in total variation. All exponents are explicit in the model
parameters that determine the asymptotics of the growth rate of the domain, the
interior covariance, and the reflection vector field. Proofs are probabilistic,
and use upper and lower tail bounds for additive functionals up to return times
to compact sets, for which we develop novel sub/supermartingale criteria,
applicable to general continuous semimartingales. Narrowing domains fall
outside of the standard literature, in part because boundary local time can
accumulate arbitrarily rapidly. Establishing Feller continuity (essential for
characterizing stability) thus requires an extension of the usual approach. Our
recurrence/transience classification extends previous work on strictly normal
reflections, and expands the range of phenomena observed across all dimensions.
For all recurrent cases, we provide quantitative information through upper and
lower bounds on tails of return times to compact sets (see
https://www.youtube.com/watch?v=oDDDWdbPx74 for a short YouTube video
describing the results).
