Homogenization of some viscous Hamilton--Jacobi equations using variational arguments on probability spaces
Abstract
This work studies the asymptotic behavior of a class of viscous Hamilton--Jacobi equations with oscillatory periodic potentials, already studied by other authors in the PDE framework, and described in terms of an effective Hamiltonian. A novel variational approach is introduced, based on the classical stochastic control representation involving a Brownian motion, which allows the problem to be reformulated in terms of functionals defined on probability spaces. The core result is the identification of a homogenized Lagrangian obtained via a Γ-convergence procedure adapted to random processes, where filtrations and Brownian motions are treated as variables. The effective Hamiltonian is then recovered as the Legendre transform of this homogenized Lagrangian. The analysis provides both lower and upper bounds by means of blow-up arguments and the construction of stochastic correctors, yielding a complete characterization of the homogenized limit. This approach allows for a stability result with respect to the addition of some localized defects embedded in the periodic background.
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BibTeX
@misc{braides2026homogenization,
title = {{Homogenization of some viscous Hamilton--Jacobi equations using variational arguments on probability spaces}},
author = {Andrea Braides and Gianni Dal Maso and Claude Le Bris},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05423},
eprint = {2610.05423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05423},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Homogenization of some viscous Hamilton--Jacobi equations using variational arguments on probability spaces
AU - Andrea Braides
AU - Gianni Dal Maso
AU - Claude Le Bris
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05423
UR - https://arxiv.org/abs/2610.05423
AB - This work studies the asymptotic behavior of a class of viscous Hamilton--Jacobi equations with oscillatory periodic potentials, already studied by other authors in the PDE framework, and described in terms of an effective Hamiltonian. A novel variational approach is introduced, based on the classical stochastic control representation involving a Brownian motion, which allows the problem to be reformulated in terms of functionals defined on probability spaces. The core result is the identification of a homogenized Lagrangian obtained via a Γ-convergence procedure adapted to random processes, where filtrations and Brownian motions are treated as variables. The effective Hamiltonian is then recovered as the Legendre transform of this homogenized Lagrangian. The analysis provides both lower and upper bounds by means of blow-up arguments and the construction of stochastic correctors, yielding a complete characterization of the homogenized limit. This approach allows for a stability result with respect to the addition of some localized defects embedded in the periodic background.
ER -