Space–Time Fractional Equations and the Related Stable Processes at Random Time

Abstract

In this paper, we consider the general space–time fractional equation of the form $$\sum _{j=1}^m \lambda _j \frac{\partial ^{\nu _j}}{\partial t^{\nu _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)$$ , for $$\nu _j \in \left( 0,1 \right] $$ and $$\beta \in \left( 0,1 \right] $$ with initial condition $$w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)$$ . We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) $$ , $$t>0$$ , where $$\varvec{S}_n^{2\beta }$$ is an isotropic stable process independent from $$\mathcal {L}^{\nu _1, \ldots , \nu _m}(t)$$ , which is the inverse of $$\mathcal {H}^{\nu _1, \ldots , \nu _m} (t) = \sum _{j=1}^m \lambda _j^{1/\nu _j} H^{\nu _j} (t)$$ , $$t>0$$ , with $$H^{\nu _j}(t)$$ independent, positively skewed stable random variables of order $$\nu _j$$ . The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) $$ , $$t>0$$ , supplies a probabilistic representation for the solutions of the fractional equations above and coincides for $$\beta = 1$$ with the n-dimensional Brownian motion at the random time $$\mathcal {L}^{\nu _1, \ldots , \nu _m} (t)$$ , $$t>0$$ . The iterated process $$\mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t)$$ , $$t>0$$ , inverse to $$\mathfrak {H}^{\nu _1, \ldots , \nu _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/\nu _j} \, _1H^{\nu _j} \left( \, _2H^{\nu _j} \left( \, _3H^{\nu _j} \left( \ldots \, _{r}H^{\nu _j} (t) \ldots \right) \right) \right) $$ , $$t>0$$ , permits us to construct the process $$\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t) \right) $$ , $$t>0$$ , the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For $$r \rightarrow \infty $$ and $$\beta = 1$$ , we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation $$\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)$$ . Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.