Stochastic heat equation: Numerical positivity and almost surely exponential stability

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.