Abstract. In this work we construct compositions of vector processes of the form S 2β n c 2 L ν (t) , t > 0, ν ∈ 0, 1 2 , β ∈ (0, 1], n ∈ N, whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes S 2β n whose random time is represented by the inverse L ν (t), t > 0, of the superposition of independent positively-skewed stable processes,, independent stable subordinators). As special cases for n = 1, ν = ) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes we present their counterparts as Brownian motion at delayed stabledistributed time.
We present the stochastic solution to a generalized fractional partial differential equation involving a regularized operator related to the so-called Prabhakar operator and admitting, amongst others, as specific cases the fractional diffusion equation and the fractional telegraph equation. The stochastic solution is expressed as a Lévy process time-changed with the inverse process to a linear combination of (possibly subordinated) independent stable subordinators of different indices. Furthermore a related SDE is derived and discussed.
We study the connection between PDEs and L\'{e}vy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenius-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup
a b s t r a c tIn this paper we study the solutions of different forms of fractional equations on the unit sphere S 2 1 ⊂ R 3 possessing the structure of time-dependent random fields. We study the correlation structures of the random fields emerging in the analysis of the solutions of two kinds of fractional equations displaying (Theorem 1) a long-range behaviour and (Theorem 2) a short-range behaviour.
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