Spectral gap for the three-dimensional damped cubic wave equation with degenerate noise
Abstract
We establish a weighted Wasserstein spectral gap for the three-dimensional damped cubic wave equation with genuinely finite rank Brownian forcing. Under a saturation condition, the gap holds with respect to the negative phase topology ℰs=H−s× H−1−s for every 0<s<1/2, from which we deduce unique ergodicity and exponential mixing in the energy topology. Sharp geometric characterizations of the saturation condition are also obtained. The method we develop is a stable--compact asymptotic coupling mechanism for hypoelliptic dissipative SPDEs beyond the parabolic setting. Instead of relying on positive time smoothing or asymptotic gradient estimates, it reduces the infinite-dimensional obstruction to contraction to a compact defect in a weaker coupling topology. Dense Malliavin range then permits this defect to be compensated by a finite dimensional Cameron--Martin shift, producing a finite distance contraction on bounded Lyapunov cores. Together with a separate high energy contraction from dissipation, this yields a global weighted Wasserstein spectral gap.
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BibTeX
@misc{liu2026spectral,
title = {{Spectral gap for the three-dimensional damped cubic wave equation with degenerate noise}},
author = {Rongchang Liu and Kening Lu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28459},
eprint = {2608.28459},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28459},
}
RIS
TY - UNPB
TI - Spectral gap for the three-dimensional damped cubic wave equation with degenerate noise
AU - Rongchang Liu
AU - Kening Lu
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28459
UR - https://arxiv.org/abs/2608.28459
AB - We establish a weighted Wasserstein spectral gap for the three-dimensional damped cubic wave equation with genuinely finite rank Brownian forcing. Under a saturation condition, the gap holds with respect to the negative phase topology ℰ_s=H^−s× H^−1−s for every 0<s<1/2, from which we deduce unique ergodicity and exponential mixing in the energy topology. Sharp geometric characterizations of the saturation condition are also obtained. The method we develop is a stable--compact asymptotic coupling mechanism for hypoelliptic dissipative SPDEs beyond the parabolic setting. Instead of relying on positive time smoothing or asymptotic gradient estimates, it reduces the infinite-dimensional obstruction to contraction to a compact defect in a weaker coupling topology. Dense Malliavin range then permits this defect to be compensated by a finite dimensional Cameron--Martin shift, producing a finite distance contraction on bounded Lyapunov cores. Together with a separate high energy contraction from dissipation, this yields a global weighted Wasserstein spectral gap.
ER -