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

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 $dgeq 2$; the case $d=1$ is also investigated but no conclusive answer is obtained.