06080622

j

06080622 Smii, Boubaker A linked cluster theorem of the solution of the generalized Burger equation. Appl. Math. Sci., Ruse 6, No. 1-4, 21-38 (2012). 2012 Hikari Ltd, Ruse EN stochastic partial differential equations trees Borel summability L\'evy noise linked cluster theorem

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Summary: 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 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 L\'evy type gives a simplified representations of the solution of the generalized Burger equation and hence a Linked Cluster theorem is recalled.