Research Projects

Multiparameter random processes naturally generalize the traditional random processes indexed by the real line and are useful in the modeling of spatial data. Many real-life processes have complex temporal patterns and long-range dependence structure. It has been observed that standard stochastic models often fall short in capturing these properties. This project aims to overcome this limitation by studying some novel time-changed random processes, specifically, some time-changed variants of the Poisson random field (PRF) in the plane and on the sphere, time-changed multiparameter Markov and Lévy processes. These processes provide a strong mathematical framework for modeling the complex spatial data exhibiting the long-range dependence properties in various disciplines. To enhance the understanding and use of these processes, in this project we will: (i) develop a precise mathematical model for analyzing the multiparameter inverse subordinator driven time-changed Poisson random fields on plane, (ii) construct and investigate a family of semi-Markov processes, including their practical applications in key areas, such as mathematical statistics and statistical mechanics, (iii) investigate some time-changed multiparameter Markov and Lévy processes, and associated non-local equations, and (iv) analyze the scaling limit of PRF driven two parameter continuous-time random walk. In conclusion, this study aims to provide essential theoretical understanding and practi cal methods for more precisely modeling and forecasting complex temporal patterns in spatial data that exhibit long-range dependence. Additionally, some semi-Markov models with both one-dimensional and multi-dimensional index sets will be developed and investigated. These models exhibit long memory properties and have applications in various fields, such as statistical physics for analyzing anomalous diffusion, financial analysis, biology, and more.