Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure µ. The associated Cameron-Martin space is denoted by H. Let ν = e −U µ, where U : X → R is a sufficiently regular convex and continuous function. In this paper we are interested in the W 2,2 regularity of the weak solutions of elliptic equations of the type λu − Lν u = f, where λ > 0, f ∈ L 2 (X, ν) and Lν is the self-adjoint operator associated with the quadratic form (ψ, ϕ) → X
Abstract. Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure µ. The associated Cameron-Martin space is denoted by H. Consider two sufficiently regular convex functions U : X → R and G : X → R. We let ν = e −U µ and Ω = G −1 (−∞, 0]. In this paper we are interested in the W 2,2 regularity of the weak solutions of elliptic equations of the typeand L ν,Ω is the self-adjoint operator associated with the quadratic form (ψ, ϕ) → Ω
Let [Formula: see text] be a separable Banach space endowed with a non-degenerate centered Gaussian measure [Formula: see text]. The associated Cameron–Martin space is denoted by [Formula: see text]. Consider two sufficiently regular convex functions [Formula: see text] and [Formula: see text]. We let [Formula: see text] and [Formula: see text]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form [Formula: see text] and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with [Formula: see text] and on half-spaces, namely if [Formula: see text] and [Formula: see text] is an affine function, then the domain of the operator defined via (0.1) is the space [Formula: see text] where [Formula: see text] is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.
We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space X. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincaré inequalities, and thanks to these inequalities we deduce spectral properties of the Ornstein-Uhlenbeck operator.
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