Global Dynamics of Granular Media Equations via Stochastic Order

TL;DR

This paper investigates the global behavior of one-dimensional granular media equations with attractive interactions and multiplicative noise, revealing ordered invariant measures and convergence properties in the Wasserstein space.

Contribution

It extends the monotone dynamical systems framework to include multiplicative noise, establishing order and convergence results for invariant measures in granular media equations.

Findings

Invariant measures are totally ordered under stochastic order.

Basins of attraction contain unbounded open sets in 2-Wasserstein space.

Global convergence to the order interval of invariant measures.

Abstract

This paper studies the rich dynamics of one-dimensional granular media equations with attractive quadratic interactions. Building on the monotone dynamical systems framework developed in an earlier work, we allow for multiplicative noise, in contrast to most existing results restricted to additive noise. Within this framework, we show that, in the one-dimensional setting, invariant measures are totally ordered with respect to the stochastic order. The basins of attraction of the minimal and maximal invariant measures contain unbounded open sets in the 2-Wasserstein space, which is vacant in previous research even for additive noises. Also, our main results address the global convergence to the order interval enclosed by the minimal and maximal invariant measures, and an alternating arrangement of invariant measures in terms of stability (locally attracting) and instability (as the backward limit of a connecting orbit). Our theorems cover a wide range of classical granular media equations, such as double-well and multi-well landscapes. Specific values for the parameter ranges, explicit descriptions of attracting sets and phase diagrams are provided.