L2 convergence rates for singular stochastic wave dynamics
Abstract
We establish quantitative hypocoercivity for singular stochastic damped wave equations arising in stochastic quantization. Building on the finite-dimensional argument of Cao, Lu, and Wang [Arch. Rat. Mech. Anal. 2023], we derive a dimension-free criterion that replaces uniform semiconvexity by a log-Sobolev inequality together with exponential lower-tail bounds for the bottom of the Hessian spectrum. We verify this criterion for the two-dimensional Φ42 and sine-Gordon models and for the three-dimensional Φ43 model, through estimates on the lowest eigenvalue of associated singular random Schrödinger operators. Combined with the continuum limit established in a companion paper, this yields exponential L2 convergence to equilibrium for the Φ42 dynamics and for sine-Gordon with β2<2π. For Φ42, we verify that in a weak-coupling, small-mass regime, optimizing the damping yields a kinetic relaxation rate of at least order m. This yields a genuine acceleration over the order-m2 relaxation rate of the overdamped dynamics. The lattice estimates are uniform in the discretization for sine-Gordon throughout β2<4π, and for Φ43 at fixed volume, including regimes beyond those for which the continuum dynamics is vpresently constructed.
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BibTeX
@misc{barashkov2026convergence,
title = {{$L^2$ convergence rates for singular stochastic wave dynamics}},
author = {Nikolay Barashkov and Giacomo Di Gesù and Petri Laarne},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17815},
eprint = {2610.17815},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17815},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - L^2 convergence rates for singular stochastic wave dynamics
AU - Nikolay Barashkov
AU - Giacomo Di Gesù
AU - Petri Laarne
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17815
UR - https://arxiv.org/abs/2610.17815
AB - We establish quantitative hypocoercivity for singular stochastic damped wave equations arising in stochastic quantization. Building on the finite-dimensional argument of Cao, Lu, and Wang [Arch. Rat. Mech. Anal. 2023], we derive a dimension-free criterion that replaces uniform semiconvexity by a log-Sobolev inequality together with exponential lower-tail bounds for the bottom of the Hessian spectrum. We verify this criterion for the two-dimensional Φ^4_2 and sine-Gordon models and for the three-dimensional Φ^4_3 model, through estimates on the lowest eigenvalue of associated singular random Schrödinger operators. Combined with the continuum limit established in a companion paper, this yields exponential L^2 convergence to equilibrium for the Φ^4_2 dynamics and for sine-Gordon with β^2<2π. For Φ^4_2, we verify that in a weak-coupling, small-mass regime, optimizing the damping yields a kinetic relaxation rate of at least order m. This yields a genuine acceleration over the order-m^2 relaxation rate of the overdamped dynamics. The lattice estimates are uniform in the discretization for sine-Gordon throughout β^2<4π, and for Φ^4_3 at fixed volume, including regimes beyond those for which the continuum dynamics is vpresently constructed.
ER -