# On the convergence of stochastic transport equations to a deterministic parabolic one

## TL;DR

This paper investigates how solutions of a stochastic transport linear equation with multiplicative noise converge to a deterministic parabolic equation under certain conditions, extending analysis to higher dimensions.

## Contribution

It demonstrates convergence of stochastic solutions to a deterministic PDE via multiplicative renormalization, and discusses existence and uniqueness in multiple dimensions.

## Key findings

- Solutions converge to a deterministic parabolic equation under noise assumptions.
- Existence and uniqueness of solutions for the stochastic transport linear equation are established.
- The method applies in dimensions d ≥ 2; the one-dimensional case remains inconclusive.

## Abstract

A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed. Our method works in dimension $d\geq 2$; the case $d=1$ is also investigated but no conclusive answer is obtained.