Anomalous Dissipation in a Stochastically Forced Infinite-Dimensional System of Coupled Oscillators

TL;DR

This paper investigates a stochastically forced infinite-dimensional oscillator system that, despite conserving energy, exhibits anomalous dissipation leading to a unique invariant measure, revealing phenomena absent in finite dimensions.

Contribution

It demonstrates the existence of an invariant measure in an infinite-dimensional system through anomalous dissipation, a phenomenon not present in finite-dimensional analogs.

Findings

Existence of a nontrivial invariant measure.

Anomalous dissipation transports energy to infinity.

Explicit construction and analysis of the invariant measure.

Abstract

We study a system of stochastically forced infinite-dimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has a nontrivial invariant probability measure. This phenomenon, which has no finite dimensional equivalent, is due to the appearance of some anomalous dissipation mechanism which transports energy to infinity. This prevents the energy from building up locally and allows the system to converge to the invariant measure. The invariant measure is constructed explicitly and some of its properties are analyzed.