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We consider the stochastic evolution equation with continuous one-sided linearly bounded “drift” F that is of at most polynomial growth. The “diffusion” is globally Lipschitz continuous, while the driving term dW is either a space-time Gaussian white noise or correlated in space and white in time. The spatial domain is allowed to be unbounded. A covers a general class of operators including uniformly elliptic differential operators. We prove existence of a solution and uniqueness in the case when dW is space-time white and the spatial domain is bounded
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Manthey et al. (1999) studied this question.
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