Abstract
For a suitable scaling of the solution to the one-dimensional heat equation with spatial-dependent coefficients and weakly dependent random initial conditions, the convergence to the Gaussian limiting distribution is proved. The scaling proposed and methodology followed allow us to obtain Gaussian scenarios for related equations such as the one-dimensional Burgers equation as well as for the multidimensional formulation of both the heat and Burgers equations. Furthermore, the investigation of non-Gaussian scenarios is opened with a different proposed scaling, proving the convergence of the second-order moments.
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