Appl Math Optim 44:203–225 (2001)
2001 Springer-Verlag New York Inc.
Asymptotic Behavior of Inﬁnite Dimensional Stochastic Differential
Equations by Anticipative Variation of Constants Formula
and G. Tessitore
Dipartimento di Matematica, Universit`a di Trento,
via Sommarive 14, 38050 Povo (TN), Italy
Dipartimento di Matematica (DIMA), Universit`a di Genova,
via Dodecaneso 35, 16146 Genova, Italy
Communicated by M. Rockner
Abstract. We consider the long time behavior of aninﬁnite dimensional stochastic
evolution equation with respect to a cylindrical Wiener process. New estimates on
the disturbance operator related to the problem are proved using a “variation of
constants”-type formula. Such estimates, under the natural assumption of mean-
squarestabilityfor thelinearpart of theequation, lead directlyto sufﬁcientconditions
for the exponential stability of the problem. In the last part of the paper we prove
that, under suitable conditions, the equation admits a unique invariant measure
that is strongly mixing. To complete the paper, we present examples of interesting
situations where our construction applies.
Key Words. Stochastic evolution equations in inﬁnite dimensional spaces, Malli-
avin calculus, Skorohod anticipative integral, Mean-square stability, Existence and
uniqueness of invariant measure.
AMS Classiﬁcation. 60H15, 35B35, 60H30, 93E15.
We are concerned here with the asymptotic behavior of an inﬁnite dimensional stochastic
differential equation of the following kind:
y(t, s, x) = Ay(t, s, x) dt + By(t, s, x) dW(t) + D(y(t, s, x)) dW(t),
y(s, s, x) = x,