Diffusion of brownian particles and Liouville field theory

In this work we review a recently proposed transformation which is useful in order to simplify non-polynomial potential given in the form of an exponential. As an application, it is shown that the Liouville field theory may be mapped into a field theory with a polynomial interaction between two scalar fields and a massive vector field. The used methodology is illustrated with the help of the simple case of a particle performing a random walk in a delta function potentials.