Abstract
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 
useful for modeling 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 AubinLion Theorem). We use such results to write a path-integral solution for this problem.
Additionally, we present the rigorous functional integral solutions for the linear diffusion and wave equations 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.
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