Stochastic Processes Seminar

Date: April 4, 2024

Time: 10:00AM - 11:00AM

Location: ZOOM

Speaker: Tianxu Wang, University of Alberta

Title: Stochastic generalized Kolmogorov systems with small diffusion: I. Explicit approximations for invariant probability density function

Abstract: This paper focuses on studying the long-term coexistence states of stochastic generalized Kolmogorov systems with small diffusion. We establish a mathematical framework for approximating the invariant probability measures (IPMs) and density functions (IPDFs) of these systems. Compared with the existing approximation methods available only for systems with non-degenerate linear diffusion, this paper introduces two new and easily implementable approximation methods, the log-normal approximation (LNA) and updated normal approximation (uNA), which can be used for systems with not only non-degenerate but also degenerate diffusion. Moreover, we utilize the Kolmogorov-Fokker-Planck (KFP) operator and matrix algebra to develop algorithms for calculating the associated covariance matrix and verifying its positive definiteness. Our new approximation method exhibits good accuracy in approximating the IPM and IPDF at both local and global levels, and significantly relaxes the minimal criteria for positive definiteness of the solution of the continuous-type Lyapunov equation. We demonstrate the utility of our method in several application examples from biology and ecology.