Appl Math Optim 53:221–258 (2006)
2006 Springer Science+Business Media, Inc.
A Stochastic Tikhonov Theorem in Inﬁnite Dimensions
and Giuseppina Guatteri
Laboratoire de Math´ematiques, Universit´e de Bretagne Occidentale,
6 avenue Le Gorgeu - CS93837, 29238 Brest Cedex 3, France
Dipartimento di Matematica, Politecnico di Milano,
Piazza Leonardo da Vinci 32, 20133 Milano, Italia
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 C
-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 un-
natural in the inﬁnite-dimensional framework.
Key Words. Stochastic differential equations in inﬁnite dimensions, Two-scale
stochastic systems, Singular perturbations.
AMS Classiﬁcation. 60H30, 35R15, 35B25.
Since the celebrated paper by Prandtl  to which the origin of the theory of singular
perturbation is usually attributed, many famous mathematicians have contributed to
this theory, attracted by the numerous important applications. At the end of the 1940s,
Tikhonov ,  developed the theory of nonlinear two-scale systems in which the
fast variables can almost reach their equilibrium states while the slow variables remain
near their initial values. Motivated by the needs of applications essential progress has
been achieved in the study of stochastic models of two-scale systems in the last 25 years.