Harnack inequalities with power $$\pmb {p\in (1,+\infty )}$$ for transition semigroups in Hilbert spaces

Angiuli, Luciana; Bignamini, Davide A.; Ferrari, S.

doi:10.1007/s00030-022-00812-0

Cited by 4 publications

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References 30 publications

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“…The interest in weighted Gaussian measures ν in infinite dimension increases in last years, since in general they represent a class of infinite dimensional measures which are not decomposable along directions of H. Further, by means of the Malliavin derivative D H , on L p (X, ν) it is possible to define a strongly continuous semigroup (T p (t)) t≥0 whose infinitesimal generator L p is a perturbation of the Ornstein-Uhlenbeck operator. Features of (T p (t)) t≥0 and of L p , such as maximal Sobolev regularity for solution to the elliptic problem λu − L p u = f , with λ > 0 and f ∈ L 2 (X, ν), smoothness properties of the semigroup (T (t)) t≥0 and some functional inequalities are investigated both from an analytic and a probabilistic point of view also in more general setting, see [2,1,5,4,8,9,10,13,14,16,17]. We also refer to [20] for an in-depth analysis of the Sobolev spaces W k,p (X, ν).…”

Section: Introductionmentioning

confidence: 99%

Equivalence of Sobolev norms with respect to weighted Gaussian measures

Addona

2023

Preprint

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We consider the spaces Lp(X, ν; V), where X is a separable Banach space, μ is a centred non-degenerate Gaussian measure, ν := Ke−U μ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions F ∈ W 1,p(X, ν; V), which allows us to show that for every p ∈ (1,+∞) and every k ∈ N the norm in W k,p(X, ν) is equivalent to the graph norm of DkH (the k-th Malliavin derivative) in Lp(X, ν). To conclude, we show exponential decay estimates for (TV (t))t≥0 as t → +∞. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck (T(t))t≥0, and pointwise estimates for |DHT(t)f |pH by means both of T(t)|DHf |pH and of T(t)|f |p. SubjClass[2010]: 28C20; 46E35; 47D06

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Section: Introductionmentioning

confidence: 99%

Equivalence of Sobolev norms with respect to weighted Gaussian measures

Addona

2023

Preprint

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Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

Addona

2024

Potential Anal

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We consider the spaces $${\text {L}}^p(X,\nu ;V)$$ L p ( X , ν ; V ) , where X is a separable Banach space, $$\mu $$ μ is a centred non-degenerate Gaussian measure, $$\nu :=Ke^{-U}\mu $$ ν : = K e - U μ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions $$F\in W^{1,p}(X,\nu ;V)$$ F ∈ W 1 , p ( X , ν ; V ) , which allows us to show that for every $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and every $$k\in \mathbb {N}$$ k ∈ N the norm in $$W^{k,p}(X,\nu )$$ W k , p ( X , ν ) is equivalent to the graph norm of $$D_H^{k}$$ D H k (the k-th Malliavin derivative) in $${\text {L}}^p(X,\nu )$$ L p ( X , ν ) . To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup $$(T^V(t))_{t\ge 0}$$ ( T V ( t ) ) t ≥ 0 , defined in Section 2.6, as t goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $$(T(t))_{t\ge 0}$$ ( T ( t ) ) t ≥ 0 , and pointwise estimates for $$|D_HT(t)f|_H^p$$ | D H T ( t ) f | H p by means of both $$T(t)|D_Hf|^p_H$$ T ( t ) | D H f | H p and $$T(t)|f|^p$$ T ( t ) | f | p .

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Schauder regularity results in separable Hilbert spaces

Bignamini

Ferrari

2023

Journal of Differential Equations

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Harnack inequalities with power $$\pmb {p\in (1,+\infty )}$$ for transition semigroups in Hilbert spaces

Cited by 4 publications

References 30 publications

Equivalence of Sobolev norms with respect to weighted Gaussian measures

Equivalence of Sobolev norms with respect to weighted Gaussian measures

Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

Schauder regularity results in separable Hilbert spaces

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