  We propose a stochastic representation for a simple class of transport PDEs
based on Ito representations. We detail an algorithm using an estimator
stemming for the representation that, unlike regularization by noise
estimators, is unbiased. We rely on recent developments on branching
diffusions, regime switching processes and their representations of PDEs.
  There is a loose relation between our technique and regularization by noise,
but contrary to the latter, we add a perturbation and immediately its
correction. The method is only possible through a judicious choice of the
diffusion coefficient $\sigma$. A key feature is that our approach does not
rely on the smallness of $\sigma$, in fact, our $\sigma$ is strictly bounded
from below which is in stark contrast with standard perturbation techniques.
This is critical for extending this method to non-toy PDEs which have nonlinear
terms in the first derivative where the usual perturbation technique breaks
down.
  The examples presented show the algorithm outperforming alternative
approaches. Moreover, the examples point toward a potential algorithm for the
fully nonlinear case where the method of characteristics breaks down.
