Fractional Poisson Random Fields on \(\mathbb {R}^2_+\)

We consider a fractional Poisson random field (FPRF) on positive plane. It is defined as a process whose one dimensional distribution is the solution of a system of fractional partial differential equations. A time-changed representation for the FPRF is given in terms of the composition of Poisson random field with a bivariate random process. Some integrals of the FPRF are introduced and studied. Using the Adomian decomposition method, a closed form expression for its probability mass function is obtained in terms of the generalized Wright function. Some results related to the order statistics of random numbers of random variables are presented. Also, we introduce a generalization of Poisson random field on \(\mathbb {R}^d_+\), \(d\ge 1\) which reduces to the Poisson random field in a special case. Later, we define the compound fractional Poisson random field via FPRF. Moreover, a generalized version of it on \(\mathbb {R}^d_+\) is discussed.