Let X be a separable Banach space and let X  be its topological dual. Let Q W X !X be a linear, bounded, non-negative, and symmetric operator and let A WD.A/ X !X be the infinitesimal generator of a strongly continuous semigroup of contractions on X. We consider the abstract Wiener space .X;  1;H1/, where  1 is a centered non-degenerate Gaussian measure on X with covariance operator defined, at least formally, as Q1 D Z C1 0 esAQesA  ds; and H1 is the Cameron–Martin space associated to  1. Let H be the reproducing kernel Hilbert space associated with Q with inner product OE ;   H. We assume that the operator Q1A  W D.A /   X  ! X extends to a bounded linear operator B 2 L.H/ which satisfies B C B  D 􀀀IdH, where IdH denotes the identity operator on H. Let D and D2 be the first and second order Fréchet derivative operators. We denote by DH and .DH;D2 H/ the closure in L2.X; 1/ of the operators QD and .QD;QD2/, respectively, defined on smooth cylindrical functions, and by W 1;2 H .X; 1/ and W 2;2 H .X; 1/, respectively, their domains in L2.X;  1/. Furthermore, we denote by DA1 the closure of the operator Q1A D in L2.X;  1/ defined on smooth cylindrical functions, and by W 1;2 A1 .X;  1/ the domain of DA1 in L2.X; 1/. We characterize the domain of the operator L, associated to the bilinear form .u; v/ 7! 􀀀 Z X OEBDHu;DHv H d 1; u; v 2 W 1;2 H .X; 1/; in L2.X; 1/. More precisely, we prove that D.L/ coincides, up to an equivalent renorming, with a subspace of W 2;2 H .X; 1/ \ W 1;2 A1 .X; 1/. We stress that we are able to treat the case when L is degenerate and non-symmetric.

On the domain of non-symmetric and, possibly, degenerate Ornstein-Uhlenbeck operators in separable Banach spaces / Addona, Davide; Cappa, Gianluca; Ferrari, Simone. - In: ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI. - ISSN 1120-6330. - (2022).

On the domain of non-symmetric and, possibly, degenerate Ornstein-Uhlenbeck operators in separable Banach spaces.

Davide Addona;Gianluca Cappa;Simone Ferrari
2022-01-01

Abstract

Let X be a separable Banach space and let X  be its topological dual. Let Q W X !X be a linear, bounded, non-negative, and symmetric operator and let A WD.A/ X !X be the infinitesimal generator of a strongly continuous semigroup of contractions on X. We consider the abstract Wiener space .X;  1;H1/, where  1 is a centered non-degenerate Gaussian measure on X with covariance operator defined, at least formally, as Q1 D Z C1 0 esAQesA  ds; and H1 is the Cameron–Martin space associated to  1. Let H be the reproducing kernel Hilbert space associated with Q with inner product OE ;   H. We assume that the operator Q1A  W D.A /   X  ! X extends to a bounded linear operator B 2 L.H/ which satisfies B C B  D 􀀀IdH, where IdH denotes the identity operator on H. Let D and D2 be the first and second order Fréchet derivative operators. We denote by DH and .DH;D2 H/ the closure in L2.X; 1/ of the operators QD and .QD;QD2/, respectively, defined on smooth cylindrical functions, and by W 1;2 H .X; 1/ and W 2;2 H .X; 1/, respectively, their domains in L2.X;  1/. Furthermore, we denote by DA1 the closure of the operator Q1A D in L2.X;  1/ defined on smooth cylindrical functions, and by W 1;2 A1 .X;  1/ the domain of DA1 in L2.X; 1/. We characterize the domain of the operator L, associated to the bilinear form .u; v/ 7! 􀀀 Z X OEBDHu;DHv H d 1; u; v 2 W 1;2 H .X; 1/; in L2.X; 1/. More precisely, we prove that D.L/ coincides, up to an equivalent renorming, with a subspace of W 2;2 H .X; 1/ \ W 1;2 A1 .X; 1/. We stress that we are able to treat the case when L is degenerate and non-symmetric.
On the domain of non-symmetric and, possibly, degenerate Ornstein-Uhlenbeck operators in separable Banach spaces / Addona, Davide; Cappa, Gianluca; Ferrari, Simone. - In: ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI. - ISSN 1120-6330. - (2022).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11381/2933352
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