Stochastic Rumor Propagation Dynamics with Log-Normal Ornstein–Uhlenbeck Process

Considering the inevitable impact of environmental interference on rumor propagation, this paper constructs and studies a stochastic rumor propagation model in which the diffusion rate is described by the Log-normal Ornstein–Uhlenbeck (OU) process. Firstly, we rigorously prove the existence and uniqueness of global positive solutions for the model, which serves as a fundamental prerequisite for further dynamical analysis. Subsequently, by constructing an appropriate Lyapunov function and combining it with the ergodic property of the OU process, sufficient criteria for the existence of a stationary distribution of the system are given. Finally, we demonstrate that when R0e≤1, rumors will tend to become extinct at an exponential rate.