On Two Parameter Time-Changed Poisson Random Fields with Drifts
Pradeep Vishwakarma, Manisha Dhillon, Kuldeep Kumar Kataria
TL;DR
This work develops a comprehensive framework for time-changing two-parameter Lévy dynamics via bivariate subordinators and their inverses, deriving the double Laplace transforms and fractional governing equations for the resulting processes. It introduces time-changed Poisson random fields (PRFs) with and without drifts, obtaining explicit distributional forms in terms of Mittag-Leffler and Wright functions and establishing associated fractional differential equations. The authors also characterize the mean, dispersion, and auto-covariance structures of time-changed bivariate and two-parameter Lévy processes, and they define coordinatewise two-parameter semigroup operators that preserve semigroup composition. By extending PRFs to random and deterministic drifts and to drifted PRFs, the paper provides tools for modeling anisotropic subdiffusion and complex spatio-temporal dynamics in a multiparameter setting, with potential applications to anomalous diffusion and related stochastic systems.
Abstract
We study the composition of bivariate Lévy process with bivariate inverse subordinator. The explicit expressions for its dispersion and auto correlation matrices are obtained. Also, the time-changed two parameter Lévy processes with rectangular increments are studied. We introduce some time-changed variants of the Poisson random field in plane with and without drift, and derive the associated fractional differential equations for their distributions. Later, we consider some time-changed Lévy processes where the time-changing components are two parameter Poisson random fields with drifts. Moreover, two parameter coordinatewise semigroup operators associated with some of the introduced processes are discussed.