Abstract
We develop several aspects of the infinite-dimensional Weak KAM theory using a random variables' approach. We prove that the infinite-dimensional cell problem admits a viscosity solution that is a fixed point of the Lax-Oleinik semigroup. Furthermore, we show the existence of invariant minimizing measures and calibrated curves defined on $\mathbb{R}$.
Highlights
We study dynamical systems with an infinite number indistinguishable particles on a d-dimensional torus Td
We extend and apply methods from the Weak KAM theory [16, 20, 18, 17, 19, 21, 22, 23, 24, 6, 7, 5, 29, 30, 31, 32] to the infinite-dimensional setting using a random variables’ approach
We prove that the infinitedimensional cell problem admits a viscosity solution that is a fixed point of the Lax-Oleinik semigroup
Summary
We study dynamical systems with an infinite number indistinguishable particles on a d-dimensional torus Td. In the seminal papers [26, 27], Gangbo and Tudorascu introduced and developed the weak KAM theory for infinite systems They considered an infinite system of particles on the torus T1 and modeled it using L2([0, 1]) functions as random variables. They introduced the infinite-dimensional torus and proved a Weak KAM theorem on it. In the one-dimensional case, the existence of optimal trajectories is proved for monotone and square integrable initial configurations of particles. This technique is used to overcome the fact that L2([0, 1]) is not locally compact. We present the statements of our main results and give a detailed description of our methods
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