Synchronization by noise for order-preserving random dynamical systems

TL;DR

This paper establishes conditions under which noise induces weak synchronization in order-preserving random dynamical systems on general Polish spaces, extending previous results and applying to stochastic porous media equations.

Contribution

It generalizes weak synchronization results to broader spaces and order structures, and demonstrates the phenomenon in stochastic porous media equations.

Findings

Existence of a weak point attractor under certain conditions

Weak synchronization by noise in stochastic porous media equations

Extension beyond Banach spaces and normal admissible orders

Abstract

We provide sufficient conditions for weak synchronization by noise for order-preserving random dynamical systems on Polish spaces. That is, under these conditions we prove the existence of a weak point attractor consisting of a single random point. This generalizes previous results in two directions: First, we do not restrict to Banach spaces and second, we do not require the partial order to be admissible nor normal. As a second main result and application we prove weak synchronization by noise for stochastic porous media equations with additive noise.