Scaling limit solution of a fractional Burgers equation.
The fractional version of the heat equation \[ \frac {\partial Y}{\partial t}(t,z)= -\mu (I-\Delta )^{\alpha /2}(-\Delta )^{\gamma /2}Y(t,z),\quad (t,z)\in R_+\times R^d, \] is considered with the initial conditions either of the form \[ Y(0,z)= \exp \left \{\frac {-\xi (z)}{2\widetilde \mu }\right \}\quad\text{or}\quad Y(0,z)=\exp \left \{-\frac {\xi ^2(z)-1}{2\mu \sigma ^2}\right \} \] where \(\xi \) represents the fractional Riesz-Bessel motion. Fractional scaling limits of the classes of random fields defined as the Hopf-Cole transformations, in the mean square sense, of the solutions to the equation are studied for both types of initial conditions. The scaling limits are calculated via their spectral representations. The spectral properties of the limiting random field are determined by the scaling asymptotic properties of the convolution kernel of the Lévy semigroup and the initial velocity potential random field.
scientific article; zbMATH DE number 2119500