## 1 Introduction  
The current global climate situation is dire and demands urgent attention and governance [1]. China has placed great emphasis on carbon emission reduction, signing the Paris Agreement in 2016 and pledging a \"dual-carbon\" goal at the 75 th United Nations General Assembly [2]. As a key pillar and high-energy-consuming sector of the national economy, the construction industry is a primary target for carbon emission reduction [3]. It actively promotes the development of prefabricated buildings and supply chain construction to meet industry requirements for reducing carbon emissions. However, PBSC carbon reduction still faces numerous challenges.  
Firstly, PBSC coordination difficulties. PBSC faces problems due to the large number of participants and the complex coordination process of each link [4]. For example, suppliers and contractors struggle with variable transportation distances and component sizes [5]. Insufficient on-site inventory and unchangeable assembly processes cause stagnation in the construction process [6]. These difficulties result in uncertainty in PBSC participants' various links, leading to progress, material and energy waste, and an inevitable increase in carbon emissions. The participants involved in the prefabricated building supply chain (PBSC) operation are the basic units to achieve carbon emission reduction [7]. Coordination within PBSC plays a vital role in its carbon reduction [8].  
In addition, PBSC participants at all stages have limited rationality and different interests. In actual decision-making, PBSC participants have problems with insufficient information sharing and insufficient data utilization [9]. This leads them to make decisions based on limited resources, which increases the complexity of making decisions. PBSCs have different demands for benefit distribution [10], and all want to benefit more from carbon emission reduction. For example, prefabricated component suppliers aim to increase profits from raw materials, while constructors seek to reduce construction costs to boost revenue. This results in inconsistent emission reduction goals and benefit disputes among PBSC participants, hindering cooperation.  
Finally, with regard to the complexity of the environment outside PBSC, policies, technologies, and markets all impact PBSC carbon reduction [8; 11]. For example, carbon prices and subsidy imperfections create uncertainty about carbon reduction effects [11]. The uncertainty of market fluctuations and technological changes [12] make it difficult for participants to predict the future benefits of emission reduction, affecting emission reduction decisions.  
To address the above issues, this paper develops a stochastic evolutionary game model to explore how PBSC participants interact under numerous challenges to achieve carbon reduction goals. The model parameters include supply chain coordination, participants' interests, and interventions from external entities. This study aims to provide a decision-making basis for PBSC practitioners and suggest how external entities, such as governments, financial institutions, and research institutes, can support PBSC carbon emission reduction. This study enriches the scope of PBSC research and is significant for the long-term development of carbon emission reduction in the construction industry.  
The structure of the rest of this study is as follows: Section 2 summarizes the literature on PBSC, carbon reduction in PBSC, and evolutionary games. Section 3 illustrates the game relationship between the agents in PBSC. An evolutionary game model is established, then Gaussian white noise is introduced to construct a stochastic evolutionary game model, and finally, the model is analyzed. In Section 4, numerical simulation experiments are performed to model the strategy evolution of the agents under different conditions. Section 5 summarizes and analyzes the simulation results and makes recommendations. Finally, Section 6 concludes this study.