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The space-fractional telegraph equation and the related fractional telegraph process. (English) Zbl 1033.60077
The authors consider the fractional telegraph equation $${\partial^2 u\over\partial t^2}+ 2\lambda{\partial u\over\partial t}= c^2{\partial^2 u\over\partial\vert x\vert^\alpha},\ 1<\alpha< 2,\quad u(x,0)= \delta(x),\quad u_t(x, 0)= 0,\tag1$$ where ${\partial^\alpha u\over\partial\vert x\vert^\alpha}$ 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 $${\partial u\over\partial t}= {1\over 2} {\partial^\alpha u\over\partial\vert x\vert^\alpha},\quad u(x,0)= \delta(x),\quad u_t(x, 0)= 0.\tag2$${}

60H30Applications of stochastic analysis
35G05General theory of linear higher-order PDE
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