A new proof of Liggett’s theorem for non-interacting Brownian motions

Xinxin Chen (1, 2) , Christophe Garban (2, 3) , Atul Shekhar (2)

1 PSPM - Probabilités, statistique, physique mathématique
2 ICJ - Institut Camille Jordan
3 IUF - Institut universitaire de France

In this note, we give a new proof of Liggett's theorem on the invariant measures of independent particle systems from [11] in the particular case of independent drifted Brownian motions. This particular case has received a lot of attention recently due to its applications for the analysis of the local extrema of discrete Gaussian free field. The novelty of our proof is that it identifies directly the expected Poisson Point Process with exponential intensity without relying on the Choquet-Deny convolution equation µ * P = µ ([6, 7]).

Mots clés: invariant measures; Brownian motion; Poisson point processes; Cox processes.

Domaines: Probabilités [math.PR]

Xinxin Chen, Christophe Garban, Atul Shekhar. A new proof of Liggett’s theorem for non-interacting Brownian motions. Electronic Communications in Probability, 2021, 26, ⟨10.1214/21-ecp435⟩. ⟨hal-03774724⟩