Harmonic analysis of random fractional diffusion-wave equations. (Q1408297)

The authors consider the fractional diffusion-wave equations with random initial conditions \[ {\partial^\beta u\over\partial t^\beta}= -\mu(I- \Delta)^{\gamma/2}(-\Delta)^{\alpha/2} u,\quad\mu> 0, \] where \(u= u(t,x)\), \(t\in\mathbb{R}^1\), \(x\in\mathbb{R}^n\), is assumed to be a function in time, for \(0<\beta\leq 1\), \(u(t,x)|_{t=0}= u_0(x)= \xi(x)\), \(x\in\mathbb{R}^n\) while for \(1< \beta\leq 2\), \[ u(t,x)|_{t=0}= u_0(x)= \xi(x),\quad{\partial\over\partial t} u(t,x)|_{t=0}= u_1(x)= \eta(x),\;x\in\mathbb{R}^n, \] where \(\xi(x)\) and \(\eta(x)\) are real measurable random fields defined on a complete probability space \((\Omega, F, P)\). The authors present the Green function and spectral representations of the mean-square solutions.