On the Ornstein–Uhlenbeck operator in convex subsets of Banach spaces

Cappa, Gianluca

doi:10.4064/sm8229-3-2018

Cited by 5 publications

(8 citation statements)

References 7 publications

(18 reference statements)

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“…[7] and [11], we are not able to prove that the boundedness of (5), with T O t instead of T t , is equivalent to f ∈ BV (O, γ). Indeed, |∇ H T O t u| H ≤ e −t T O t |∇ H u| has been proved in [11] when O is an open convex domain, but it is not true for general open domains.…”

mentioning

confidence: 85%

“…Remark 3. By approximation it is possible to prove that the first equality in (11) holds true for any f ∈ W 1,p (X, γ), any g ∈ W 1,q (X, γ), with 1 < p < +∞ and q = p being its conjugate exponent. Further, the second equality in (11) holds true for any f ∈ W 1,p (X, γ) and any Φ ∈ Lip b (X, H), with 1 ≤ p < +∞ (see [8,Proposition 5.8.8]).…”

Section: Sobolev Spaces Andmentioning

confidence: 99%

“…In this direction, in [25] the authors study BV functions on X restricted to an open convex set Ω ⊆ X in terms of the Ornstein-Uhlenbeck semigroup on Ω (see [11] for a first analysis of the Ornstein-Uhlenbeck and its properties on convex domains). However, in this case the convexity of Ω plays an essential role and it is not possible to generalize the techniques in [11] for a general open domain. Further, as in [4,Proposition 3.103] and [5, Proposition 3.9] we describe the connections between the one dimensional sections of BV functions and directional derivatives.…”

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confidence: 99%

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<inline-formula><tex-math id="M1">\begin{document}$ BV $\end{document}</tex-math></inline-formula> functions on open domains: the Wiener case and a Fomin differentiable case

Addona¹,

Menegatti²,

Miranda³

2020

Communications on Pure &Amp; Applied Analysis

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We provide three different characterizations of the space BV (O, γ) of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure γ on open domains O in Wiener spaces. Throughout these different characterizations we deduce a sufficient condition in order to belong to BV (O, γ) by means of the Ornstein-Uhlenbeck semigroup and we provide an explicit formula for one-dimensional sections of functions of bounded variation. Finally, we apply our techniques to Fomin differentiable probability measures ν on a Hilbert space X, and we infer a characterization of the space BV (O, ν) of the functions of bounded variation with respect to ν on open domains O ⊆ X.

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confidence: 85%

Section: Sobolev Spaces Andmentioning

confidence: 99%

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confidence: 99%

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<inline-formula><tex-math id="M1">\begin{document}$ BV $\end{document}</tex-math></inline-formula> functions on open domains: the Wiener case and a Fomin differentiable case

Addona¹,

Menegatti²,

Miranda³

2020

Communications on Pure &Amp; Applied Analysis

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“…Logarithmic Sobolev inequalities are important tools in the study of Gaussian Sobolev spaces since they represent the counterpart of the Sobolev embeddings which in general fail to hold when the Lebesgue measure is replaced by other measures, as for example the Gaussian one. In infinite dimension such inequalities are known for the Gaussian measure on the whole space (see [10,Theorem 5.5.1]) and on convex domains (see [11,Proposition 3.5]). In the weighted Gaussian case the inequality is known in the whole space (see [20,Proposition 11.2.19]), for Fréchet differentiable functions.…”

Section: Logarithmic Sobolev Inequality and Other Consequencesmentioning

confidence: 99%

“…Introducing a different measure makes the finite dimensional approximation much more delicate and prevents to get explicit formulas even if the problem is studied in the whole space. Restricting to a domain, beside involving boundary conditions that have to be understood, makes still more difficult the infinite dimensional approximation, and in fact, to the best of our knowledge, the only case treated in the literature is that of convex domains, see [1,5,6,7,11,13,17,19,30].…”

Section: Introductionmentioning

confidence: 99%

Gradient estimates for perturbed Ornstein–Uhlenbeck semigroups on infinite-dimensional convex domains

2019

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Let X be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure γ and let λ 1 be the maximum eigenvalue of the covariance operator associated with γ. The associated Cameron-Martin space is denoted by H. For a sufficiently regular convex function U : X → R and a convex set Ω ⊆ X, we set ν := e −U γ and we consider the semigroup (T Ω (t)) t≥0 generated by the self-adjoint operator defined via the quadratic formwhere ϕ, ψ belong to D 1,2 (Ω, ν), the Sobolev space defined as the domain of the closure in L 2 (Ω, ν) of D H , the gradient operator along the directions of H.A suitable approximation procedure allows us to prove some pointwise gradient estimates for (T Ω (t)) t≥0 . In particular, we show thatfor any p ∈ [1, +∞) and f ∈ D 1,p (Ω, ν). We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in Ω for the measure ν and some improving summability properties for (T Ω (t)) t≥0 . In addition we prove that if f belongs to L p (Ω, ν) for some p ∈ (1, ∞), thenwhere Kp is a positive constant depending only on p. Finally we investigate on the asymptotic behaviour of the semigroup (T Ω (t)) t≥0 as t goes to infinity. GRADIENT ESTIMATES ON INFINITE DIMENSIONAL CONVEX DOMAINS3 therein) and coupling methods (see for example [15,16,39]). On the other hand, in infinite dimensional Wiener spaces some partial results are also available. In the case of a Gaussian measure γ and Ω = X, the classical Mehler's representation formulawhere the equality has to be meant componentwise, (see [10, Proposition 1.5.6]). Again for the Gaussian measure γ on a convex subset Ω, in [11, Theorem 3.1] it is proved that |D H T (t)f | H ≤ e −t T (t)|D H f | H for any smooth function f . In this case, the idea consists in approximating the parabolic problem with a sequence of finite dimensional parabolic problems and using the factorisation of the Gaussian measure. Clearly, this approach does not work in our case since our measure in general does not decompose as a product of measures on orthogonal subspaces. Finally, the case of a weighted Gaussian measure is also considered in [20] where a version of (3) is proved when Ω = X and the H-derivative is replaced by the Fréchet one. We point out that, in this latter case, the proof of the gradient estimate is based on purely stochastic techniques. Hence, taking account of the existing literature, estimate (3) represents a generalisation of all the above results and the purely analytical proof we proposed, inspired by an idea due to Bakry andÉmery (see [4] and [38]), is a novelty in the proofs of gradient estimates.As announced, the pointwise gradient estimate (3) has several interesting consequences. First of all it yields that the semigroup T Ω (t) is smoothing, in the sense that it is bounded from L p (Ω, ν) into D 1,p (Ω, ν), for any p ∈ (1, ∞) and t > 0 as the estimatereveals. Due to the fact that the Sobolev embedding theorems fail to hold when we replace the Lebesgue measure with another general m...

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On integration by parts formula on open convex sets in Wiener spaces

2021

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In Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a measure P( , •) which is equivalent to the one-codimensional Hausdorff measure restricted to the reduced boundary of . The same result has been proved in an abstract Wiener space, typically an infinite-dimensional space, where the surface measure considered is the one-codimensional spherical Hausdorff-Gauss measure S ∞−1 restricted to the measuretheoretic boundary of . In this paper, we consider an open convex set and we provide an explicit formula for the density of P( , •) with respect to S ∞−1 . In particular, the density can be written in terms of the Minkowski functional p of with respect to an inner point of . As a consequence, we obtain an integration by parts formula for open convex sets in Wiener spaces.

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On the Ornstein–Uhlenbeck operator in convex subsets of Banach spaces

Cited by 5 publications

References 7 publications

<inline-formula><tex-math id="M1">\begin{document}$ BV $\end{document}</tex-math></inline-formula> functions on open domains: the Wiener case and a Fomin differentiable case

<inline-formula><tex-math id="M1">\begin{document}$ BV $\end{document}</tex-math></inline-formula> functions on open domains: the Wiener case and a Fomin differentiable case

Gradient estimates for perturbed Ornstein–Uhlenbeck semigroups on infinite-dimensional convex domains

On integration by parts formula on open convex sets in Wiener spaces

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