Following the random approach of , we define a Lax-Oleinik formula adapted to evolutive weakly coupled systems of Hamilton-Jacobi equations. It is reminiscent of the corresponding scalar formula, with the relevant difference that it has a stochastic character since it involves, loosely speaking, random switchings between the various associated Lagrangians. We prove that the related value functions are viscosity solutions to the system, and establish existence of minimal random curves under fairly general hypotheses. Adding Tonelli like assumptions on the Hamiltonians, we show differentiability properties of such minimizers, and existence of adjoint random curves. Minimizers and adjoint curves are trajectories of a twisted generalized Hamiltonian dynamics.
Random Lax–Oleinik semigroups for Hamilton–Jacobi systems / Siconolfi, Antonio; Davini, Andrea; Zavidovique, Maxime. - In: JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES. - ISSN 1776-3371. - ELETTRONICO. - (2017). [10.1016/j.matpur.2017.12.005]
Random Lax–Oleinik semigroups for Hamilton–Jacobi systems
Antonio, Siconolfi;Andrea, Davini;
2017
Abstract
Following the random approach of , we define a Lax-Oleinik formula adapted to evolutive weakly coupled systems of Hamilton-Jacobi equations. It is reminiscent of the corresponding scalar formula, with the relevant difference that it has a stochastic character since it involves, loosely speaking, random switchings between the various associated Lagrangians. We prove that the related value functions are viscosity solutions to the system, and establish existence of minimal random curves under fairly general hypotheses. Adding Tonelli like assumptions on the Hamiltonians, we show differentiability properties of such minimizers, and existence of adjoint random curves. Minimizers and adjoint curves are trajectories of a twisted generalized Hamiltonian dynamics.File | Dimensione | Formato | |
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