In a low-carbon supply chain consisting of a single manufacturer and a single retailer, the different power structures between the manufacturer and the retailer determine their emission reduction strategies and the game ways. At the same time, the goodwill and demand for low carbon products are influenced by the long-term dynamics of the manufacturer's and the retailer's abatement strategies and stochastic factors. Stochastic differential game analysis of the manufacturer's abatement effort strategies and the retailer's promotional effort strategies under different power structures is conducted. On the basis of centralized decision-making, the manufacturer-dominated and the retailer-dominated Stackelberg game and non-dominant Nash game models are constructed respectively, and the models are solved and the results are analyzed theoretically and numerically. The study finds that the other party's dominant role will prompt the manufacturer or the retailer to make greater emission reduction effort or promotion effort than their own as the dominant player. A supply chain without a leader may achieve greater goodwill for low-carbon products than a supply chain with a leader. The manufacturer as the leader can increase the goodwill and demand for low carbon products and total profit of the supply chain. Random factors drive the manufacturer and the retailer to increase their emission reduction effort and promotion effort, and low-carbon product goodwill.