随机变分方程解的存在、唯一性问题

On the Existence and Uniqueness of the Solution to Stochastic Variational Problems

研究了随机有限元方法的理论基础——随机变分方程解的合理性问题，证明了一般形式的随机变分方程解的存在与唯一性定性，明确了在随机扰动算子不足以破坏均值微分算子的强制性条件——不会导致方程类型变异的前提下，Ｆｒｅｄｈｏｌｍ择一定理的第二个结果成立，此时在概率１的意义上变分解存在而且唯一。

To provide an accurate description for randomness of a system, some uncertain internal parameters of the system are characterized by stochastic processes and random fields with finite second order moments. The control equations become stochastic differential ones with stochastic coefficients. In this paper, the theoretical basis of stochastic FEM—the rationality of the stochastic rariational equation is discussed. The existence and uniqueness of the stochastic variational solution to the system are discussed. When the mean operator is not destroyed by the stochastic pertubation of differential operators, the second conclusion of Fredholm alternative theorem will be kept, and a unique stochastic variational solution will exist.