Unique Ergodicity for Strongly Damped Stochastic Wave Equations under Finite-Rank Pure-Jump Lévy Forcing
Abstract
We study a strongly damped stochastic wave equation on the three-dimensional torus driven by finite-rank pure-jump Lévy noise obtained from subordinated Brownian motion. Lyapunov estimates and an asymptotic compactness decomposition yield invariant measures, and a small-noise event gives weak irreducibility. To prove uniqueness, we combine conditional Malliavin calculus with terminal observability and estimates for the full variational flow starting from high modes. A control constructed on alternating stopping blocks yields the e-property under a fixed-rank balance condition involving high-mode decay, spectral leakage, and the random clock. Consequently, the invariant measure is unique and ergodic, and the Cesàro averages of the transition laws from every initial state converge weakly to it. We verify this condition for sufficiently small positive cubic coefficients at each fixed forcing operator of sufficiently large rank. Under a separate explicit smallness condition, synchronous coupling proves the e-property and weak convergence of the transition laws, without a lower bound on the forcing rank.
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BibTeX
@misc{zhang2026unique,
title = {{Unique Ergodicity for Strongly Damped Stochastic Wave Equations under Finite-Rank Pure-Jump Lévy Forcing}},
author = {Jiangwei Zhang and Jianhua Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09521},
eprint = {2610.09521},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09521},
}
RIS
TY - UNPB
TI - Unique Ergodicity for Strongly Damped Stochastic Wave Equations under Finite-Rank Pure-Jump Lévy Forcing
AU - Jiangwei Zhang
AU - Jianhua Huang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09521
UR - https://arxiv.org/abs/2610.09521
AB - We study a strongly damped stochastic wave equation on the three-dimensional torus driven by finite-rank pure-jump Lévy noise obtained from subordinated Brownian motion. Lyapunov estimates and an asymptotic compactness decomposition yield invariant measures, and a small-noise event gives weak irreducibility. To prove uniqueness, we combine conditional Malliavin calculus with terminal observability and estimates for the full variational flow starting from high modes. A control constructed on alternating stopping blocks yields the e-property under a fixed-rank balance condition involving high-mode decay, spectral leakage, and the random clock. Consequently, the invariant measure is unique and ergodic, and the Cesàro averages of the transition laws from every initial state converge weakly to it. We verify this condition for sufficiently small positive cubic coefficients at each fixed forcing operator of sufficiently large rank. Under a separate explicit smallness condition, synchronous coupling proves the e-property and weak convergence of the transition laws, without a lower bound on the forcing rank.
ER -