In this paper, the numerical positivity and almost surely exponential stability of the stochastic heat equation are discussed. The finite difference method and the split-step backward Euler are considered for spatial and temporal, respectively. Motivated from physical applications such as temperature, finance and so on, positivity has real significance, which is volatilized by some common numerical treatments. To this end the numerical positivity is obtained by the properties of M-matrix and the truncated random variables, which overcomes the unboundedness of the random variables. For the investigation of the almost surely exponential stability, a stochastic stability matrix is introduced, and then the stability analysis reduces to the estimation of the eigenvalues and martingales thorough a family of matrices and perturbation theorems. The stabilization ability of the multiplicative noise is verified again from a generalization of the stochastic heat equation. Finally, some numerical experiments are given to validate our numerical results.

在本文中，讨论了随机热方程的数值正性和几乎肯定的指数稳定性。空间和时间分别考虑了有限差分法和分步后向欧拉法。受温度、金融等物理应用的启发，正性具有真正的意义，它被一些常见的数值处理所挥发。为此，数值正性由M的性质获得-矩阵和截断的随机变量，克服了随机变量的无界性。为了研究几乎可以肯定的指数稳定性，引入了随机稳定性矩阵，然后通过一系列矩阵和摄动定理将稳定性分析简化为特征值和鞅的估计。通过对随机热方程的推广，再次验证了乘性噪声的稳定能力。最后，给出了一些数值实验来验证我们的数值结果。