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

In this paper, we consider the general space–time fractional equation of the form $$\sum _{j=1}^m \lambda _j \frac{\partial ^{\nu _j}}{\partial t^{\nu _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)$$ ∑ j = 1 m λ j ∂ ν j ∂ t ν j w ( x 1 , … , x n ; t ) = - c 2 - Δ β w ( x 1 , … , x n ; t ) , for $$\nu _j \in \left( 0,1 \right] $$ ν j ∈ 0 , 1 and $$\beta \in \left( 0,1 \right] $$ β ∈ 0 , 1 with initial condition $$w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)$$ w ( x 1 , … , x n ; 0 ) = ∏ j = 1 n δ ( x j ) . We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) $$ S n 2 β c 2 L ν 1 , … , ν m ( t ) , $$t>0$$ t > 0 , where $$\varvec{S}_n^{2\beta }$$ S n 2 β is an isotropic stable process independent from $$\mathcal {L}^{\nu _1, \ldots , \nu _m}(t)$$ L ν 1 , … , ν m ( t ) , which is the inverse of $$\mathcal {H}^{\nu _1, \ldots , \nu _m} (t) = \sum _{j=1}^m \lambda _j^{1/\nu _j} H^{\nu _j} (t)$$ H ν 1 , … , ν m ( t ) = ∑ j = 1 m λ j 1 / ν j H ν j ( t ) , $$t>0$$ t > 0 , with $$H^{\nu _j}(t)$$ H ν j ( t ) independent, positively skewed stable random variables of order $$\nu _j$$ ν j . The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) $$ S n 2 β c 2 L ν 1 , … , ν m ( t ) , $$t>0$$ t > 0 , supplies a probabilistic representation for the solutions of the fractional equations above and coincides for $$\beta = 1$$ β = 1 with the n-dimensional Brownian motion at the random time $$\mathcal {L}^{\nu _1, \ldots , \nu _m} (t)$$ L ν 1 , … , ν m ( t ) , $$t>0$$ t > 0 . The iterated process $$\mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t)$$ L r ν 1 , … , ν m ( t ) , $$t>0$$ t > 0 , inverse to $$\mathfrak {H}^{\nu _1, \ldots , \nu _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/\nu _j} \, _1H^{\nu _j} \left( \, _2H^{\nu _j} \left( \, _3H^{\nu _j} \left( \ldots \, _{r}H^{\nu _j} (t) \ldots \right) \right) \right) $$ H r ν 1 , … , ν m ( t ) = ∑ j = 1 m λ j 1 / ν j 1 H ν j 2 H ν j 3 H ν j … r H ν j ( t ) … , $$t>0$$ t > 0 , permits us to construct the process $$\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t) \right) $$ S n 2 β c 2 L r ν 1 , … , ν m ( t ) , $$t>0$$ t > 0 , the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For $$r \rightarrow \infty $$ r → ∞ and $$\beta = 1$$ β = 1 , we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation $$\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)$$ ∑ j = 1 m λ j w ( x 1 , … , x n ) = c 2 ∑ j = 1 n ∂ 2 ∂ x j 2 w ( x 1 , … , 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.