Calculus of Variations and Geometric Measure Theory

A. Davini - A. Siconolfi - M. Zavidovique

Random Lax-Oleinik semigroups for Hamilton-Jacobi systems

created by davini on 12 Jul 2023


Published Paper

Inserted: 12 jul 2023
Last Updated: 12 jul 2023

Journal: J. Math. Pures Appl.
Volume: 9
Number: 120
Pages: 294–333
Year: 2018


Following the random approach of 1, 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.