An algebraic vanishing-noise rate for kinetic mean-field games
Fix $d \ge 1$, $T > 0$ and a smooth, nonnegative, compactly supported probability density $m_{\mathrm{in}}$ on $Q = \mathbb{T}^d \times \mathbb{R}^d$. For $0 \le \nu \le 1$ let $m_\nu$ be the density component of the pair $(m, a)$ minimizing $$\mathcal{J}(m, a) = \int_0^T \!\! \int_Q \Bigl(\frac{|a|^2}{2m} + \frac{m^2}{2}\Bigr)\,dx\,dv\,dt + \frac12 \int_Q m(T)^2\,dx\,dv$$ subject to $$\partial_t m + v \cdot \nabla_x m - \nu \Delta_v m + \nabla_v \cdot a = 0, \qquad m(0) = m_{\mathrm{in}}.$$ Admissible pairs have narrowly continuous probability densities with finite second velocity moment, satisfy the equation in the sense of distributions, and have $\mathcal{J} < \infty$; here $|a|^2/m = 0$ where $(m, a) = (0, 0)$ and $|a|^2/m = +\infty$ where $m = 0$ and $a \ne 0$. First-order kinetic mean-field games describe Nash equilibria of games in which agents control their acceleration; for those with local couplings ($\nu = 0$), Griffin-Pickering and Mészáros constructed weak solutions by characterizing them through two convex optimization problems in duality. For mean-field games without kinetic transport, Tang and Zhang proved polynomial rates of convergence as the viscosity vanishes.
Problem. Is there an exponent $\eta = \eta(d) > 0$ such that for every such datum and every horizon $T$ some $C < \infty$, independent of $\nu$, satisfies $$\|m_\nu - m_0\|_{L^2((0,T) \times Q)} \le C \nu^\eta \qquad (0 < \nu \le 1)?$$
Source: Tang and Zhang (2025), Section 7, Problem 4(b), which asks for the convergence rate of vanishing viscosity approximations for kinetic mean-field games. Stated by AIM — Open Applied Problems for a variational kinetic game with quadratic acceleration cost and local congestion.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, kinetic mean-field games, vanishing viscosity, variational methods
References
@article{tang2025,
author = {Tang, Wenpin and Zhang, Yuming Paul},
title = {The convergence rate of vanishing viscosity approximations for mean field games},
journal = {SIAM J. Math. Anal.},
volume = {57},
number = {3},
year = {2025},
pages = {3217--3254},
doi = {10.1137/24M1640008},
eprint = {2303.14560},
archivePrefix = {arXiv},
}
@article{griffinpickering2022,
author = {Griffin-Pickering, Megan and M{\'e}sz{\'a}ros, Alp{\'a}r R.},
title = {A variational approach to first order kinetic mean field games with local couplings},
journal = {Comm. Partial Differential Equations},
volume = {47},
number = {10},
year = {2022},
pages = {1945--2022},
doi = {10.1080/03605302.2022.2101003},
eprint = {2112.03141},
archivePrefix = {arXiv},
}
@misc{aim-kinetic-mfg-vanishing-viscosity-rate,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{An algebraic vanishing-noise rate for kinetic mean-field games}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/405-kinetic-mfg-vanishing-viscosity-rate.md},
note = {Entry 405 at commit 24edaad, 2026-10-15},
}
Resolution claims
We consider a variational kinetic mean-field game with quadratic acceleration cost, local quadratic congestion, and quadratic terminal congestion on $\T^d\times\R^d$. For a smooth, compactly supported probability datum, the density minimizers at velocity diffusivity $\nu$ and at zero diffusivity are at distance $O(\nu^{1/2})$ in spacetime $L^2$ and terminal $L^2$, in every dimension. The proof compares the two minimum values using kinetic …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00805, "An algebraic vanishing-noise rate for kinetic mean-field games", clopen.solutions.