Abstract
The present paper studies the problem of singular perturbation in the infinite-dimensional framework and gives a Hilbert-space-valued stochastic version of the Tikhonov theorem. We consider a nonlinear system of Hilbert-space-valued equations for a "slow" and a "fast" variable; the system is strongly coupled and driven by linear unbounded operators generating a C0-semigroup and independent cylindrical Brownian motions. Under well-established assumptions to guarantee the existence and uniqueness of mild solutions, we deduce the required stability of the system from a dissipativity condition on the drift of the fast variable. We avoid differentiability assumptions on the coefficients which would be unnatural in the infinite-dimensional framework.
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Buckdahn, R., Guatteri, G. A Stochastic Tikhonov Theorem in Infinite Dimensions. Appl Math Optim 53, 221–258 (2006). https://doi.org/10.1007/s00245-005-0845-y
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DOI: https://doi.org/10.1007/s00245-005-0845-y