We propose a stochastic energy balance differential equation with multiplicative Gaussian noise, which better captures the influence of the atmosphere as external noise on climate change. Meanwhile, we apply the Milstein method with stronger convergence to approximate the stochastic climate change trajectory. Further, the numerical approach can efficiently calculate two exit concepts: the mean first exit time and the first escape probability. These two quantities are used to examine the transition of the climate from one state to another. For global warming of $1.5^{\circ }$C and $1.0^{\circ }$C, we find that the enhanced noise intensity is more likely to induce a shorter mean first exit time or a higher first escape probability. These results provide important insights into the underlying mechanisms of climate change for global warming.

我们提出了一种带有乘性高斯噪声的随机能量平衡微分方程，它可以更好地捕捉大气作为外部噪声对气候变化的影响。同时，我们应用收敛性更强的米尔斯坦方法来近似随机气候变化轨迹。此外，数值方法可以有效地计算两个退出概念：平均首次退出时间和首次逃脱概率。这两个量用于检查气候从一种状态到另一种状态的转变。对于全球变暖$1。5^{\circ }$C 和$1。0^{\circ }$C，我们发现增强的噪声强度更有可能导致更短的平均首次退出时间或更高的首次逃逸概率。这些结果为了解气候变化导致全球变暖的根本机制提供了重要见解。