Stability of a class of stochastic distributed parameter systems with random boundary conditions

In this article we consider the question of stability of a class of stochastic systems governed by elliptic and parabolic second order partial differential equations with Neumann boundary conditions. Results on the “stability in the mean” are given in Theorems 1 and 2, and those on “almost sure stability” are presented in Theorems 3 and 4. These results are proved under the assumption that the perturbing forces are measurable stochastic processes defined on $\text{I × Ω}$. In Theorem 5 it is shown that the proofs require only minor modification to admit progressively measurable (predictable or optional) processes.