A nonlinear Hilbert-space-valued stochastic differential equation where L -1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L -1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L -1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable.
Applied Mathematics and Optimization – Springer Journals
Published: Apr 1, 2023
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