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 S^2β_n whose random time is represented by the inverse , t > 0, of the superposition of independent positively skewed stable processes, , t > 0, (H^2ν₁, H^ν₂, independent stable subordinators). As special cases for n = 1, and β = 1, we examine the telegraph process T at Brownian time B ([14]) and establish the equality in distribution , t > 0. Furthermore the iterated Brownian motion ([2]) 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.

Keywords:

• Riemann-Liouville fractional calculus
• Telegraph processes
• Stable positively skewed r.v.’s
• Subordinators
• Fractional Laplacian
• Mittag-Leffler functions
• Time-changed processes
• Airy functions.

Mathematics Subject Classification:

• 60G51
• 60G52
• 35C05.