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