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We find a random matrix to study a stochastic heat equation (SHE), and in doing so, we propose a method to discretize stochastic partial differential equations. Moreover, the convergence result helps to corroborate that standard partitions in the deterministic problem can also be considered in the stochastic case. In our study, we focus on the stochastic Schrödinger operator associated to the SHE, and prove a weak convergence of the random matrix to the stochastic operator. We do this by defining properly the space where the operators act, and by constructing a proper projection using the matrix.
Journal: Statistics & Probability Letters - Volume 113, June 2016, Pages 71–78