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Minorization and exponential ergodicity for degenerate diffusions with countable switching
Abstract
We establish a local-minorization criterion for diffusions with countably many state-dependent switching regimes, even when every frozen transition law is singular. One regular controlled path yields a common absolutely continuous component for nearby initial states, with rate positivity required only at the switching locations. The proof controls a family of Brownian perturbations and its derivatives while retaining a path restriction for the switching likelihood. Controlled entrance and a joint Lyapunov inequality then give weighted exponential ergodicity. A concrete coefficient and rank criterion and degenerate network examples illustrate the result.
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BibTeX
@misc{cheng2026minorization,
title = {{Minorization and exponential ergodicity for degenerate diffusions with countable switching}},
author = {Yuzhong Cheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27680},
eprint = {2610.27680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27680},
}
RIS
TY - UNPB
TI - Minorization and exponential ergodicity for degenerate diffusions with countable switching
AU - Yuzhong Cheng
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27680
UR - https://arxiv.org/abs/2610.27680
AB - We establish a local-minorization criterion for diffusions with countably many state-dependent switching regimes, even when every frozen transition law is singular. One regular controlled path yields a common absolutely continuous component for nearby initial states, with rate positivity required only at the switching locations. The proof controls a family of Brownian perturbations and its derivatives while retaining a path restriction for the switching likelihood. Controlled entrance and a joint Lyapunov inequality then give weighted exponential ergodicity. A concrete coefficient and rank criterion and degenerate network examples illustrate the result.
ER -