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Lyapunov functions for the Langevin equation with dissipative damping

  • Barkat Mian
  • Hung D. Nguyen
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Abstract

We study the well-posedness and the large time stability of the Langevin equation in the presence of potentials and dissipative damping together with additive white noise. While the confining potential is assumed to be smooth, the nonlinear damping is allowed to have either polynomial growth or singular behaviors. We construct novel Lyapunov functions and determine the corresponding parameter regimes that lead to global well-posedness or finite-time explosion. These Lyapunov functions also allow for deriving quantitative mixing rates toward equilibrium. Our results extend existing ergodic theory established for the instance of nonlinear damping satisfying suitable integrability conditions.

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Cite

BibTeXRIS
BibTeX
@misc{mian2026lyapunov,
  title = {{Lyapunov functions for the Langevin equation with dissipative damping}},
  author = {Barkat Mian and Hung D. Nguyen},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01187},
  eprint = {2611.01187},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01187},
}
RIS
TY  - UNPB
TI  - Lyapunov functions for the Langevin equation with dissipative damping
AU  - Barkat Mian
AU  - Hung D. Nguyen
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01187
UR  - https://arxiv.org/abs/2611.01187
AB  - We study the well-posedness and the large time stability of the Langevin equation in the presence of potentials and dissipative damping together with additive white noise. While the confining potential is assumed to be smooth, the nonlinear damping is allowed to have either polynomial growth or singular behaviors. We construct novel Lyapunov functions and determine the corresponding parameter regimes that lead to global well-posedness or finite-time explosion. These Lyapunov functions also allow for deriving quantitative mixing rates toward equilibrium. Our results extend existing ergodic theory established for the instance of nonlinear damping satisfying suitable integrability conditions.
ER  -