How can we understand the complex dynamics of stochastic systems? This paper delves into the geometric conditions underpinning the decomposition of stochastic flows on manifolds, a critical area in probability and mathematical statistics. Focusing on Liao's factorization, the research explores scenarios where such decomposition holds, particularly on simply connected manifolds with constant curvature. The findings reveal a link between the manifold's geometry and the factorization properties of stochastic flows. Going beyond Liao's initial work, the study investigates further factorization possibilities, expressing flows as affine transformations. It also investigates the asymptotic behavior of the isometric component, using a rotation matrix and providing a Furstenberg–Khasminskii formula for this skew-symmetric matrix. This rigorous analysis contributes to a deeper understanding of stochastic flows, providing insights that may be applicable in various fields where such flows are used to model dynamic systems, such as in physics, engineering, and mathematical modeling. The study offers a valuable framework for future research into the behavior of complex stochastic systems.
Published in Stochastics and Dynamics, this paper aligns with the journal's focus on probability and mathematical statistics. The study's exploration of stochastic flows and their geometric conditions contributes to the journal's ongoing discourse on mathematical models and dynamical systems. By delving into the Liao's factorization and asymptotic behavior via rotation matrix, it offers valuable insights for researchers in mathematics.