<p>The extensive integration of the Internet and the rapid expansion of social media have significantly changed how rumors spread. In this study, a stochastic SIS rumor spreading model is considered, which combines the spreading dynamics with population dynamics. A key improvement lies in adopting a logarithmic Ornstein-Uhlenbeck (OU) process to model environmental noise, ensuring a relatively small fluctuation range in a short period of time. Utilizing Markov semigroup theory and constructing control functions, we demonstrate that the stochastic system possesses a unique invariant probability measure, that is, the stationary distribution exists and is unique. This forms one of our main contributions. Furthermore, by solving the Fokker-Planck equation, one derives an explicit expression for the probability density near the quasi-stationary state. Meanwhile, we investigate the extinction properties and obtain the conditions for rumors to disappear. Finally, we provide four instances to simulate the impact of changes in environmental noise, contact rate and recovery rate on the dynamic behavior and verify our theoretical results.</p>