Decorrelation of total mass via energy

TL;DR

This paper demonstrates that small initial mutual energy between solutions of nonlinear stochastic heat equations leads to near-uncorrelation of their total masses over time, linking initial conditions to long-term independence.

Contribution

It provides a quantified relationship between initial mutual energy and the correlation of total masses in stochastic heat equations, highlighting conditions for finite systems.

Findings

Small initial mutual energy implies near-uncorrelation of total masses over time.

A stochastic heat equation with regular coefficients is finite iff the initial state is integrable.

The result quantifies the decay of correlation based on initial conditions.

Abstract

The main result of this small note is a quantified version of the assertion that if u and v solve two nonlinear stochastic heat equations, and if the mutual energy between the initial states of the two stochastic PDEs is small, then the total masses of the two systems are nearly uncorrelated for a very long time. One of the consequences of this fact is that a stochastic heat equation with regular coefficients is a finite system if and only if the initial state is integrable.