Scaling laws for fractional diffusion-wave equations with singular data
Let \(u(t,x,\beta)\), \(t>0,x\in \mathbb R,0<\beta <1\), be the solution of the Cauchy problem for the equation \[ {\partial^{2\beta }u}/{\partial t^{2\beta }}={\partial^2 u}/{\partial x^2} \] [see \textit{W. R. Schneider} and \textit{W. Wyss}, J. Math. Phys. 30, No.~1, 134-144 (1989; Zbl 0692.45004) and \textit{A. N. Kochubej} [Differ. Equations 26, No. 4, 660-670 (1990; Zbl 0712.35049)] with random Gaussian initial data. It is shown, under certain conditions upon the initial random function \(u(0,x,\beta)\), that the finite-dimensional distributions of the random field \[ X_\varepsilon (t,x,\beta)=\varepsilon^{-\gamma }\left[ u\left( \frac{t}{\varepsilon},\frac{x}{\varepsilon^\beta},\beta \right) - C_0\right] \] (the parameters \(\gamma >0\) and \(C_0\) are determined by \(u(0,x,\beta)\)) converge weakly as \(\varepsilon \to 0\). The limit field is found.
scientific article; zbMATH DE number 1486957
scientific article