Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions
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Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions Davide Addona
Received: 13 November 2012 / Accepted: 24 April 2013 © Springer Science+Business Media New York 2013
Abstract We consider the nonautonomous Ornstein-Uhlenbeck operator in some weighted spaces of continuous functions in RN . We prove sharp uniform estimates for the spatial derivatives of the associated evolution operator Ps,t , which we use to prove optimal Schauder estimates for the solution to some nonhomogeneous parabolic Cauchy problems associated with the Ornstein-Uhlenbeck operator. We also prove that, for any t > s, the evolution operator Ps,t is compact in the previous weighted spaces. Keywords Nonautonomous parabolic equations · Weighted spaces of continuous functions · Uniform estimates · Nonhomogeneous Cauchy problems · Optimal regularity results · Compactness
1 Introduction The nondegenerate nonautonomous Ornstein-Uhlenbeck operator L(t), defined on smooth functions ϕ : RN → R by L(t)ϕ(x) =
1 Tr Q(t)D 2 ϕ(x) + A(t)x + h(t), Dϕ(x) , 2
(1.1)
is the prototype of nonautonomous elliptic operators with unbounded coefficients. Here, A, Q : R → RN ×N , h : RN → R are continuous and bounded functions, and
Communicated by Abdelaziz Rhandi. D. Addona () Dipartimento di Matematica, Università degli Studi di Milano Bicocca, Via Cozzi, 53, 20155 Milan, Italy e-mail: [email protected]
D. Addona
Q(t)x, x > μ0 x2 for any x ∈ RN , any t ∈ R and some positive constant μ0 , Such an operator has wide applications in many fields of applied sciences. It naturally arises in the study of the backward stochastic equation dXt = (A(t)Xt + h(t))dt + (Q(t))1/2 dWt , t ≥ 0, (1.2) Xs = x, where x ∈ RN and Wt is a standard Brownian motion. The unique mild solution to (1.2) is given by t t 1/2 X(t, s, x) = U (t, s)x + U (t, r)h(r)dr + U (t, r) Q(r) dWr , s
s
where U (t, s) is the evolution operator associated to A(t), i.e., U (t, s) is the solution of the Cauchy problem ∂U t ∈ R, ∂t (t, s) = A(t)U (t, s), (1.3) U (s, s) = I. The family of bounded operators Ps,t , defined by Ps,t ϕ(x) = E ϕ X(t, s, x) = ϕ(y)Na(t,s),Q(t,s) (dy), RN
(1.4)
for any bounded and continuous function ϕ : RN → R (in short ϕ ∈ Cb (RN )), any s, t ∈ R, with s < t, and any x ∈ RN , is the so-called nonautonomous OrnsteinUhlenbeck evolution operator. Here, Na(t,s),Q(t,s) (dy) is the Gaussian measure with mean t U (t, r)h(r)dr, s < t, (1.5) a(s, t, x) = U (t, s)x + g(t, s) := U (t, s)x + s
and covariance operator
t
Q(t, s) =
U (t, r)Q(r)U ∗ (t, r)dr,
s < t.
s
As in the autonomous case (see e.g., [4, 7, 13, 17, 20–23]), nonautonomous Ornstein-Uhlenbeck operators have been studied in the last years both in Cb (RN ) (see [6, 14]) and in Lp -spaces related to families of probability measures {μt : t ∈ R} (the so called “evolution systems of invariant measures”, see [6, 8–11]), characterized by the following property: Ps,t f (y)μs (dy) = f (y)μt (dy), RN
RN
for any s < t, f ∈ Cb (RN ). These systems of inva
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