Calculus of Variations and Geometric Measure Theory

A. Davini - J. L. P. Garmendia - R. Iturriaga - J. C. Pardo - H. Sanchez Morgado

Discrete approximation of stochastic Mather measures

created by davini on 13 Jul 2023


Published Paper

Inserted: 13 jul 2023
Last Updated: 13 jul 2023

Journal: Proceedings of the American Mathematical Society
Year: 2023


In this paper, we construct measures which minimize a discrete version of the stochastic Mather problem associated to a Tonelli Lagrangian $L:\mathbb T^d\times\mathbb R^d\to\mathbb R$, where $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ is the flat $d$-dimensional torus. We show that the discrete variational problems approximate the stochastic Mather problem as the step of the discretization goes to zero, in the sense that the minima of the discrete problems converge to the minimum of the stochastic Mather problem and the discrete minimizing measures converge to the unique stochastic Mather measure.