Exponential Ergodicity and Invariant Distribution of a Degenerate Reflected SDE with Stable Noise
Abstract
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.
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BibTeX
@misc{pang2026exponential,
title = {{Exponential Ergodicity and Invariant Distribution of a Degenerate Reflected SDE with Stable Noise}},
author = {Guodong Pang and Wei You},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39471},
eprint = {2610.39471},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39471},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exponential Ergodicity and Invariant Distribution of a Degenerate Reflected SDE with Stable Noise
AU - Guodong Pang
AU - Wei You
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39471
UR - https://arxiv.org/abs/2610.39471
AB - 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.
ER -