Abstract

The Dickman function, originally appeared in Ramanujan’s unpublished notes and first explored by Karl Dickman in the field of number theory, has garnered interesting recognition in recent years. This function has given rise to the Dickman distribution [1,2] and, subsequently, the Dickman subordinator [3]. The Dickman distribution also appears in various limiting schemes [1]. It also appears as a special case of Vervaat perpetuities. We introduce the stationary solution of the Ornstein-Uhlenbeck SDE driven by Poisson backward driving Levy process follows a Dickman marginal distribution [2]. Additionally we investigate superpositions of such processes which has a long-range dependent properties and marginal Dichman distribution, limit theorems for such a processes and intermittency in the spirit of the papers [4,5,6,7]. The distribution of the first passage time process for Dickman subordinator is discussed  as well as the corresponding convolution type non-local operator (or Dickman fractional derivatives)[2]. The nonlocal Poisson processes are introduced as time-change Poisson process and independent inverse Dichman subordinator in the spirit of fractional Poisson process obtained  as time-change Poisson process and independent inverse stable subordinator[8,9]. The first part is joint work with D. Grahovac (Osijek University, Croatia), A.Kovtun and A.Pepelyshev (Cardiff University, UK). The second part is based on the joint work with N.Gupta (Indian Statistical Institute, New Delhi, India), A.Kumar(Indian Institute of Technology, Ropar, India) and J.Vaz (Unicamp, Campinas, Brazil).