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

Let X be a separable Banach space and let X be its topological dual. Let Q : X → X be a linear, bounded, non-negative and symmetric operator and let A : D(A) ⊆ X → X be the infinitesimal generator of a strongly continuous semigroup of contractions on X. We consider the abstract Wiener space (X, μ∞,H∞) where μ∞ is a centred non-degenerate Gaussian measure on X with covariance operator defined, at least formally, as

[1]  A. Lunardi,et al.  On the Ornstein-Uhlenbeck Operator in Spaces of Continuous Functions , 1995 .

[2]  I. Shigekawa.  Sobolev spaces over the Wiener space based on an Ornstein-Uhlenbeck operator , 1992 .

[3]  J. Maas,et al.  ON THE DOMAIN OF NONSYMMETRIC ORNSTEIN–UHLENBECK OPERATORS IN BANACH SPACES , 2008 .

[4]  D. Pallara,et al.  The Ornstein–Uhlenbeck semigroup in finite dimension , 2020, Philosophical Transactions of the Royal Society A.

[5]  L. Lorenzi.  Schauder estimates for degenerate elliptic and parabolic problems with unbounded coefficients in ${\Bbb R}^N$ , 2005, Differential and Integral Equations.

[6]  On the infinite dimensional Laplacian operator , 1965 .

[7]  Petr Hájek,et al.  Banach Space Theory , 2011 .

[8]  Transition Semigroups of Banach Space-Valued Ornstein–Uhlenbeck Processes , 2003, math/0606785.

[9]  M. Piech.  The Ornstein-Uhlenbeck semigroup in an infinite dimensional L2 setting , 1975 .

[10]  B. Goldys,et al.  On Closability of Directional Gradients , 2003 .

[11]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[12]  L. Lorenzi.  Estimates of the derivatives for a class of parabolic degenerate operators with unbounded coefficients in R N , 2011 .

[13]  Invariant measures and maximal L2 regularity for nonautonomous Ornstein–Uhlenbeck equations , 2008, 0801.3224.

[14]  D. Pallara,et al.  Gradient estimates for perturbed Ornstein–Uhlenbeck semigroups on infinite-dimensional convex domains , 2018, Journal of Evolution Equations.

[15]  E. Priola,et al.  Global Lp estimates for degenerate Ornstein–Uhlenbeck operators , 2008, 1209.0387.

[16]  B. Goldys,et al.  Generalized Ornstein–Uhlenbeck Semigroups: Littlewood–Paley–Stein Inequalities and the P. A. Meyer Equivalence of Norms , 2001 .

[17]  Diego Pallara,et al.  Spectrum of Ornstein-Uhlenbeck Operators in Lp Spaces with Respect to Invariant Measures , 2002 .

[18]  Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions , 2013, 1607.05439.

[19]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[20]  G. Cappa,et al.  Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: the convex subset case , 2016, 1609.07337.

[21]  D. Addona,et al.  Analyticity of Nonsymmetric Ornstein-Uhlenbeck Semigroup with Respect to a Weighted Gaussian Measure , 2019, Potential Analysis.

[22]  S. Ferrari.  Sobolev spaces with respect to a weighted Gaussian measure in infinite dimensions , 2015, 1510.08283.

[23]  B. Farkas,et al.  Maximal regularity for Kolmogorov operators in L2 spaces with respect to invariant measures , 2006 .

[24]  M. A. Kaashoek.  Closed Linear Operators on Banach Spaces , 1965 .

[25]  D. A. Bignamini,et al.  Regularizing Properties of (Non-Gaussian) Transition Semigroups in Hilbert Spaces , 2020, Potential Analysis.

[26]  G. Cappa,et al.  Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure , 2016, 1603.09474.

[27]  The Sector of Analyticity of the Ornstein–Uhlenbeck Semigroup on Lp Spaces with Respect to Invariant Measure , 2005 .

[28]  A. Lunardi,et al.  Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in R , 2018 .

[29]  A. Rhandi,et al.  The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure , 2002 .

[30]  L. Gross.  Potential theory on Hilbert space , 1967 .

[31]  D. Addona,et al.  Domains of elliptic operators on sets in Wiener space , 2020 .

[32]  G. Prato,et al.  Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension , 2012, 1208.0437.

[33]  A. Pascucci.  PDE and Martingale Methods in Option Pricing , 2010 .

[34]  Giuseppe Da Prato,et al.  Second Order Partial Differential Equations in Hilbert Spaces: Bibliography , 2002 .

[35]  A. Lunardi,et al.  Optimal L ∞ and Schauder Estimates for Elliptic and Parabolic Operators with Unbounded Coefficients , 2020 .

[36]  G. Metafune.  $L^p$-spectrum of Ornstein-Uhlenbeck operators , 2001 .

[37]  J. van Neerven.  Nonsymmetric Ornstein–Uhlenbeck Semigroups in Banach Spaces☆ , 1998 .

[38]  V. Grusin.  ON A CLASS OF HYPOELLIPTIC OPERATORS , 1970 .

[39]  Robert E. Megginson.  An Introduction to Banach Space Theory , 1998 .

[40]  J. Maas,et al.  Gradient Estimates and Domain Identification for Analytic Ornstein-Uhlenbeck Operators , 2009, 0911.4336.

[41]  G. Prato.  Kolmogorov Equations For Stochastic Pdes , 2004 .

[42]  Ornstein–Uhlenbeck operators with time periodic coefficients , 2007 .

[43]  A. Lunardi.  On the Ornstein-Uhlenbeck operator in ² spaces with respect to invariant measures , 1997 .

[44]  V. Bogachev.  Gaussian Measures on a , 2022 .