Hypercontractivity and Asymptotic Behavior in Nonautonomous Kolmogorov Equations

Angiuli, Luciana; Lorenzi, Luca; Lunardi, Alessandra

doi:10.1080/03605302.2013.840790

Cited by 22 publications

(64 citation statements)

References 16 publications

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“…We also show a Poincaré inequality in L p (Ω, ν) for p ∈ [2, ∞) that together with the hypercontractivity estimate T Ω (t)f L p (Ω,ν) ≤ c p,q,Ω f L q (Ω,ν) which holds for any t > 0, f ∈ L q (Ω, ν) and some p > q, allows us to study the asymptotic behaviour of T Ω (t)f as t → +∞ for f ∈ L p (Ω, ν), p > 1, and to relate it to the behaviour of the derivative |D H T Ω (t)f | as t → +∞. This last result was already known in the finite dimensional setting for evolution operators associated to non-autonomous elliptic operator (see [3]). These estimates are drawn in a more or less standard way: we have presented sketches of proofs (or even complete proofs) for the convenience of the reader.Further consequences can be deduced, but these will be hopefully matter of other works.…”

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confidence: 83%

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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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confidence: 83%

“…Step 1. We use an idea of [37] (see also [3,Theorem 5.2]). Let f ∈ FC 1 b (Ω), η > 0 and consider the function f η = 1 + η(f − m Ω (f )).…”

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confidence: 99%

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

2019

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“…for any t > s, f ∈ C 1 b (R d ) and q ∈ (1, +∞). 3) to hold are given in [3]. In particular, (4.3) holds true when (2.3) is satisfied with p = 1 (see Remark 2.2).…”

Section: Hypercontractivity Throughout This and The Forthcoming Subsmentioning

confidence: 97%

Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

Addona

Angiuli

Lorenzi

2017

Advances in Nonlinear Analysis

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Abstract. We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over R d and in L p -spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous secondorder elliptic operator with unbounded coefficients defined in I × R d , (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.

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“…Comparatively, in [12] the Euclidean distance is used as reference distance and then a space only Lyapunov condition is sufficient for existence and uniqueness of an evolution system of measures. In [1,12] the coefficients in the Lyapunov condition are uniformly bounded, and as consequence a time-homogeneous process can be used for comparison with the original process. In our setting, the coefficients in the Lyapunov conditions need to be time-dependent to preserve the information about the varying space.…”

Section: Introductionmentioning

confidence: 99%

“…for every s ∈ I, f ∈ H 1 (M, µ s ) and some positive decreasing function β s . Note that the function β s may depend on the current time s which generalizes the notion of super-log-Sobolev inequalities for non-autonomous systems on R d in [1]. Moreover, combining the super-log-Sobolev inequalities and dimension-free Harnack inequalities, we prove that the exponential integrability of radial function with respect to (µ t ) t∈I or (P s,t ) (s,t)∈Λ is equivalent to supercontractivity or ultraboundedness of the corresponding semigroup.…”

Section: Introductionmentioning

confidence: 99%

Evolution systems of measures and semigroup properties on evolving manifolds

Cheng¹,

Thalmaier²

2018

Electron. J. Probab.

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An evolving Riemannian manifold (M, g t ) t∈I consists of a smooth d-dimensional manifold M, equipped with a geometric flow g t of complete Riemannian metrics, parametrized by I = (−∞, T ). Given an additional C 1,1 family of vector fields (Z t ) t∈I on M. We study the family of operators L t = ∆ t + Z t where ∆ t denotes the Laplacian with respect to the metric g t . We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by L t , and for existence of evolution systems of probability measures associated to it. Coupling methods are used to establish uniqueness of the evolution systems under suitable curvature conditions. Adopting such a unique system of probability measures as reference measures, we characterize supercontractivity, hypercontractivity and ultraboundedness of the corresponding time-inhomogeneous semigroup. To this end, gradient estimates and a family of (super-)logarithmic Sobolev inequalities are established.We investigate this problem from a probabilistic point of view. Let X t be the diffusion process generated by L t (called L t -diffusion) which is assumed to be non-explosive before time T (see [2,8,14] for details). As in the time-homogeneous case, we construct L t -diffusions X t via horizontal diffusions u t above X t .

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Hypercontractivity and Asymptotic Behavior in Nonautonomous Kolmogorov Equations

Cited by 22 publications

References 16 publications

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

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

Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

Evolution systems of measures and semigroup properties on evolving manifolds

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