Physics – Mathematical Physics
Scientific paper
2010-01-07
Physics
Mathematical Physics
20 pages, 1 figure This paper has been withdrawn by the author due to a crucial sign error in equation 15
Scientific paper
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.
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