Some probabilistic properties of fractional point processes

ABSTRACT

In this article, the first hitting times of generalized Poisson processes N^f(t), related to Bernštein functions f are studied. For the space-fractional Poisson processes, N^α(t), t > 0 (corresponding to f = x^α), the hitting probabilities P{T^α_k < ∞} are explicitly obtained and analyzed. The processes N^f(t) are time-changed Poisson processes N(H^f(t)) with subordinators H^f(t) and here we study and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form where are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space–time Poisson process is no longer a renewal process.

KEYWORDS:

• Fractional point processes
• Berns̆tein functions
• space–time fractional Poisson processes
• grey Brownian motion

MATHEMATICS SUBJECT CLASSIFICATION:

• 60G55
• 26A33