Quantitative estimates for SPDEs on the full space with transport noise and $L^p$-initial data

TL;DR

This paper provides quantitative estimates comparing solutions of stochastic linear transport equations and stochastic 2D Euler equations with deterministic heat equations, focusing on $L^p$-initial data and negative Sobolev norms.

Contribution

It introduces new quantitative estimates for SPDEs with transport noise on the full space, extending understanding of solution behavior with $L^p$-initial data.

Findings

Quantitative estimates in negative Sobolev norms for stochastic transport equations.

Extension of estimates to stochastic 2D Euler equations with transport noise.

Results applicable to $L^p$-initial data with $1<p<2$.

Abstract

For the stochastic linear transport equation with $L^p$-initial data ($1<p<2$) on the full space $\mathbb{R}^d$, we provide quantitative estimates, in negative Sobolev norms, between its solutions and that of the deterministic heat equation. Similar results are proved for the stochastic 2D Euler equations with transport noise.