Recent research highlights significant advancements in optimization techniques on Riemannian manifolds, with a particular focus on the Wasserstein probabilistic space. These developments are notable for their potential applications in practical sampling processes, an area that has garnered increasing attention within the optimization community. The continuous-time Riemannian stochastic gradient descent (SGD) and stochastic variance reduced gradient (SVRG) flows represent promising methods emerging from this line of inquiry. Such approaches leverage the geometric structure of Wasserstein space to improve optimization efficiency and accuracy. The growing interest in this domain reflects its relevance to both theoretical and applied aspects of machine learning and statistical inference. As reported, these advancements contribute positively to the broader field of optimization, offering new tools for handling complex probabilistic models. This progress aligns with ongoing efforts to integrate geometric insights into algorithmic design, underscoring the dynamic nature of research in optimization on manifolds.