The space-fractional telegraph equation and the related fractional telegraph process (Q1812233)

The authors consider the fractional telegraph equation \[ {\partial^2 u\over\partial t^2}+ 2\lambda{\partial u\over\partial t}= c^2{\partial^2 u\over\partial| x|^\alpha},\;1<\alpha< 2,\quad u(x,0)= \delta(x),\quad u_t(x, 0)= 0,\tag{1} \] where \({\partial^\alpha u\over\partial| x|^\alpha}\) is the Riesz fractional derivative. The authors obtain the Fourier transform \(U(\gamma, t)\) of the solution of (1). It is presented a symmetric process with discontinuous trajectories, whose characteristic function coincides with \(U(\gamma, t)\) and whose transition function satisfies (1) (the fractional telegraph process). It is also studied the convergence of this process to symmetric stable process as \(c\to\infty\), \(\lambda\to\infty\), in such a way that \(c^2/\lambda\to 1\). This result corresponds to the fact that the equation (1) converges, as \(c\to\infty\), \(\lambda\to\infty\), to the fractional heat-wave equation \[ {\partial u\over\partial t}= {1\over 2} {\partial^\alpha u\over\partial| x|^\alpha},\quad u(x,0)= \delta(x),\quad u_t(x, 0)= 0.\tag{2} \] {}