  We present a proof for the existence and uniqueness of weak solutions for a
cut-off and non cut-off model of non-linear diffusion equation in
finite-dimensional space RD useful for modelling flows on porous medium with
saturation, turbulent advection, etc. - and subject to deterministic or
stochastic (white noise) stirrings. In order to achieve such goal, we use the
powerful results of compacity on functional Lp spaces (the Aubin-Lion Theorem).
We use such results to write a path-integral solution for this problem.
Additionally, we present the rigourous functional integral solutions for the
Linear Diffussion equation defined in Infinite-Dimensional Spaces (Separable
Hilbert Spaces). These further results are presented in order to be useful to
understand Polymer cylindrical surfaces probability distributions and
functionals on String theory.
