Invariant measures and maximal L ² regularity for nonautonomous Ornstein-Uhlenbeck equations

Abstract

We characterize the domain of the realizations of the linear parabolic operator G defined by (1.4) in L 2 spaces with respect to a suitable measure, that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L 2 regularity results for evolution equations with time-depending Ornstein-Uhlenbeck operators.2000 Mathematics Subject Classification. 47D06, 47F05, 35B65.

Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

“…See, e.g., the book [4] and the references therein. In the case of timedependent Ornstein-Uhlenbeck operators, the use of the evolution semigroup was essential in establishing optimal regularity results for evolution equations and also in getting precise asymptotic behavior estimates for G(t, s); see [10,11]. However, the general theory of evolution semigroups is well established only for evolution families acting on a fixed Banach space X, which is not our case.…”

On the Navier–Stokes equations with rotating effect and prescribed outflow velocity

“…Such a backward problem was considered by Da Prato and Lunardi [5] and Geissert and Lunardi [10], since their main motivation came from stochastics. In our case, with the application to problem (1.5) in mind, it is more convenient to work with the forward problem.…”

“…In Sect. 2, we review and prove results on time-dependent Ornstein-Uhlenbeck operators, studied recently by Da Prato and Lunardi [5] and Geissert and Lunardi [10]. By using these results in Sect.…”

Cores for parabolic operators with unbounded coefficients

“…The counterpart to the paper [16] in the nonautonomous case are the papers [17,25,26] all of them concerned with the nonautonomous Ornstein-Uhlenbeck operator which is given by (1.3) when the diffusion coefficients are independent of x and b i (s, x) = N j=1 b ij (s)x j , under the assumptions that the matrix-valued functions s → Q (s) = (q ij (s)) and s → B(s) = (b ij (s)) are continuous and periodic in R (see [17]) or, more generally, bounded and continuous in R, Q (s) is symmetric for any s and its minimum eigenvalue is bounded from below by a positive constant, independent of s (see [25,26]). In both these cases explicit formulas for the evolution family (P (s, r)) solving (1.2) and for the corresponding families {μ s : s ∈ R} of invariant measures are available.…”

“…In both these cases explicit formulas for the evolution family (P (s, r)) solving (1.2) and for the corresponding families {μ s : s ∈ R} of invariant measures are available. Such a family of (probability) measures, which is called evolution system of invariant measures in [19] and entrance laws at −∞ in [21], are characterized by the property that Starting from the evolution family (P (s, r)) the authors of [25] introduce the evolution semigroup (T (t)), defined by (1.4) in the space of all bounded and continuous functions f : R N+1 → R (say C b (R N+1 )) and, then, they extend the restriction of (T (t)) to C ∞ c (R N+1 ) with a strongly continuous semigroup (T p (t)) in the space L p (R N+1 , ν), 1 p < +∞, where ν is the only positive measure such that ν(I × B) = I μ s (B) ds, (1.5) for all the Borel sets I ⊂ R and B ⊂ R N . Of course, ν is not a finite measure but to some extent we can still call it invariant measure of (T (t)).…”