Exponential Ergodicity and Invariant Distribution of a Degenerate Reflected SDE with Stable Noise

We study a two-dimensional degenerate stochastic differential equation with oblique reflection and spectrally positive strictly stable noise of index in $(1,2)$, motivated by join-the-shortest-queue systems with heavy-tailed interarrival times. A polynomial-exponential Lyapunov function and fixed-time minorization establish positive Harris recurrence and exponential ergodicity in weighted total variation. We construct the invariant law from a stable path and prove its absolute continuity. Regulator identities describe stationary dependence and recover the joint Laplace transform from the queue marginal. Idleness inherits the classical stable Ornstein-Uhlenbeck tail; the queue has a sharp exponential tail with a polynomial prefactor whose power is half the stable index and a positive constant represented through regeneration. Conditional on small idleness, idleness divided by the conditioning threshold and the queue converge to independent beta and size-biased limits. Under large-queue conditioning, normalized idleness and queue excess converge to independent Mittag-Leffler and exponential limits.