Generalized Wright Noise Stochastic Calculus and Applications

This paper primarily aims to derive combinatorial properties of generalized Wright falling operators and establish their connections with a generalization of Appell polynomials associated with infinite-dimensional generalized Wright measures. Building on these results, we investigate the stochastic counterpart of generalized Wright analysis. We introduce a novel class of non-Gaussian processes, termed the two-parameter generalized Wright process (tpgWp) and we analyze their joint probability density and covariance. As applications of our results, we first prove the existence of local times for the generalized Wright process. Next, we characterize the two-parameter generalized Wright Ornstein–Uhlenbeck process as the unique solution to a Langevin equation driven by the (tpgWp), as well as a linear Wick-type stochastic differential equation driven by multi-two-parameter generalized Wright processes (mtpgWp). We establish that these solutions lie within a suitable distribution space.