Multi-subject collaborative participation is the key to building a secure and trustworthy Generative Artificial Intelligence（GAI） service ecosystem. This paper explores the strategic behaviors and influencing factors of relevant subjects in the context of GAI service user privacy protection， broadens the research scenarios of stochastic evolutionary game theory， deepens the understanding of the dynamic mechanism of privacy leakage for GAI service users， enriches the theories and methods in the field of GAI privacy protection， and provides enlightenment for promoting the practice of privacy protection for GAI service users.

This paper， based on evolutionary game theory and stochastic processes， constructed a time-varying user privacy leakage risk function， and built a “user-service provider-government” three-party stochastic evolutionary game model with the particularity of the GAI service scenario. By applying Itô stochastic differential equation theory and numerical simulation， this paper analyzed the stability and evolution of the behavior strategies of these three parties.

The findings indicate that： ①The initial willingness of the government， service providers， and users significantly influences the direction and speed of subsequent system evolution； only when both users and the government exhibit high initial willingness could GAI service providers adopt proactive protection strategies， leading the system toward an optimal state. ②The greater the intensity of random disturbances， the slower the convergence speed of the three parties to a stable strategy， and GAI service providers are more sensitive to uncertain factors. ③Under conditions of high initial willingness from all three parties， market winners and participants converge to the ideal state， whereas market survivors find it challenging to reach this state； vertical GAI service providers demonstrate faster forward convergence compared to general GAI service providers. ④When the government increases the fines for GAI service providers and provides moderate rewards， it would increase the probability of their active privacy protection and thereby reduce the probability of privacy leakage. However， if the government reduces the punishment for users’ false reports， although it increases the probability of users disclosing privacy， if the fines for service providers are too low， it would not prompt them to adopt an active protection strategy. Increasing user rights protection compensation or reducing the cost of rights protection by the government could both encourage users to actively disclose privacy.