A linked cluster theorem of the solution of the generalized burger equation

In this paper we consider a stochastic partial differential equation defined on a Lattice L 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 given 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 Lévy type gives a simplified representations of the solution of the generalized Burger equation and hence a Linked Cluster theorem is recalled.