  In this paper we consider a stochastic partial differential equation defined
on a Lattice $L_\delta$ with coefficients of non-linearity with degree $p$. An
analytic solution in the sense of formal power series is given. The obtained
series can be re-expressed in terms of rooted trees with two types of leaves.
Under the use of the so-called Cole-Hopf transformation and for the particular
case $p=2$, one thus get the generalized Burger equation. A graphical
representation of the solution and its logarithm is done in this paper. A
discussion of the summability of the previous formal solutions is done in this
paper using Borel sum. A graphical calculus of the correlation function is
given. The special case when the noise is of L\'evy type gives a simplified
representations of the solution of the generalized Burger equation. From the
previous results we recall a graphical representation of the grand canonical
partition function.
