Spatial Scalings for Randomly Initialized Heat and Burgers Equations with Quadratic Potentials

Abstract

A general functional class of spatial scalings is introduced, jointly with the logarithmic transformation of the temporal component, to get the convergence to the Gaussian limit distribution of the solution to the heat and Burgers equations with quadratic external potentials, considering weakly dependent Gaussian random initial conditions. The results derived extend the ones obtained in Leonenko and Ruiz-Medina [31] to a more general spatial scaling setting.

Keywords:

• Burgers equation
• Heat equation
• Quadratic potentials
• Scaling laws
• Spatiotemporal random fields
• Weak-dependent random initial conditions

Mathematics Subject Classification:

• 60G60
• 60G15
• 62M15
• 60H15

N. N. Leonenko and M. D. Ruiz-Medina were partially supported by grant of the European commition PIRSES-GA-2008-230804 (Maric Curie), projects MTM2008-03903 of the DGI, and P06-FQM-02271 of the Andalousian CICE, Spain, and the Australian Research Council grants A10024117 and DP 0345577.