Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

Abstract

In this work, we construct compositions of vector processes of the form , t > 0, , β ∈ (0, 1], , 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 S2βn whose random time is represented by the inverse , t > 0, of the superposition of independent positively skewed stable processes, , t > 0, (H2ν1, Hν2, independent stable subordinators). As special cases for n = 1, and β = 1, we examine the telegraph process T at Brownian time B ([14 Orsingher, E., and Beghin, L. 2004. Time-fractional telegraph equations and telegraph processes with Brownian time. Probab. Theory Related Fields 128(1):141–160.[Crossref], [Web of Science ®] , [Google Scholar]]) and establish the equality in distribution , t > 0. Furthermore the iterated Brownian motion ([2 Allouba, H., and Zheng, W. 2001. Brownian-time processes: The PDE connection and the half-derivative generator. Ann. Probab. 29:1780–1795.[Crossref], [Web of Science ®] , [Google Scholar]]) 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 stable-distributed time.