2010.12193
Weak KAM theory for action minimizing random walks
Kohei Soga
correcthigh confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves the uniform O(√h) convergence of the discrete effective Hamiltonian to the continuous one under hyperbolic scaling (Theorem 3.9: sup_{c∈P}|H̄_δ(c)−H̄(c)| ≤ b1√h) via a controlled random-walk/variational argument and Lipschitz interpolation of time-1 periodic discrete solutions , leveraging the discrete scheme (2.4) and its Lax–Oleinik map representation , . The candidate solution reaches the same rate using standard monotone-scheme PDE error bounds and a nonlinear spectral-perturbation comparison of time-1 maps. Both arguments are logically sound and compatible; they differ in technique (probabilistic/variational vs. PDE stability/error analysis).
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The work rigorously constructs discrete analogues of weak KAM objects for a staggered Lax–Friedrichs scheme and proves their convergence to the continuous counterparts under hyperbolic scaling, including a sharp O(√h) error on the effective Hamiltonian. The approach is technically sound and complements classical PDE monotone-scheme theory with a probabilistic/variational perspective. Some expository refinements would improve accessibility to numerical analysts.