Scaling limit solution of a fractional Burgers equation

A fractional version of the heat equation, involving fractional powers of the negative Laplacian operator, with random initial conditions of exponential type, is introduced. Two cases are considered, depending on whether the Hopf-Cole transformation of such random initial conditions coincides, in the mean-square sense, with the gradient of the fractional Riesz-Bessel motion introduced in Anh et al. (J. Statist. Plann. Inference 80 (1999) 95-110), or with a quadratic function of such a random field. The scaling limits of the random fields defined by the Hopf-Cole transformation of the solutions to the fractional heat equation introduced in the two cases considered are then calculated via their spectral representations.