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We give various conditions for Hermite pseudo-multipliers to be bounded on L 2 (R n). As a by-product we also give new results for pseudo-multipliers in the Gaussian measure setting. One of our key tools is a new integration by-parts formula for Hermite expansions. [ABSTRACT FROM AUTHOR]

We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings μ : Ω → (P (D) , W 2) defined over a subset Ω of R p and valued in the space P (D) of probability measures on a compact convex subset D of R q endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Korevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetnyak of Sobolev spaces valued in metric spaces. We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary ∂Ω are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for Ω a segment of R. As the Wasserstein space (P (D) , W 2) is positively curved in the sense of Alexandrov we cannot apply the theory of Korevaar, Schoen and Jost and we use instead arguments based on optimal transport. We manage to get existence of harmonic mappings provided that the boundary values are Lipschitz on ∂Ω, uniqueness is an open question. If Ω is a segment of R , it is known that a curve valued in the Wasserstein space P (D) can be seen as a superposition of curves valued in D. We show that it is no longer the case in higher dimensions: a generic mapping Ω → P (D) cannot be represented as the superposition of mappings Ω → D. We are able to show the validity of a maximum principle: the composition F ∘ μ of a function F : P (D) → R convex along generalized geodesics and a harmonic mapping μ : Ω → P (D) is a subharmonic real-valued function. We also study the special case where we restrict ourselves to a given family of elliptically contoured distributions (a finite-dimensional and geodesically convex submanifold of (P (D) , W 2) which generalizes the case of Gaussian measures) and show that it boils down to harmonic mappings valued in the Riemannian manifold of symmetric matrices endowed with the distance coming from optimal transport. [ABSTRACT FROM AUTHOR]

Abstract Let γ d be the d -dimensional standard Gaussian measure and Q = { Q t } t ≥ 0 the Ornstein–Uhlenbeck semigroup acting on L 1 (γ d). The semigroup Q enjoys the hypercontractivity, which is also known to be equivalent to the exponential hypercontractivity. In this paper, by employing stochastic analysis, we derive a family of inequalities that unifies the exponential and original hypercontractivities; a generalization of the Gaussian logarithmic Sobolev inequality is obtained as a corollary. We then discuss a connection of those results with Φ-entropy inequalities in a general framework of Markov semigroups. A unification of the exponential hypercontractivity and the reverse hypercontractivity of the Ornstein–Uhlenbeck semigroup Q is also provided. [ABSTRACT FROM AUTHOR]

We consider a family of gradient Gaussian vector fields on the torus ( Z / L N Z ) d . Adams, Kotecký, Müller and independently Bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators, i.e., the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter L k . We improve this result and show that the decay behaviour of the kernels in Fourier space can be controlled. Then we show the regularity of the integration map that convolves functionals with the partial measures of the finite range decomposition. In particular the new finite range decomposition avoids the loss of regularity which arises in the renormalisation group approach to anisotropic problems in statistical mechanics. [ABSTRACT FROM AUTHOR]

For a complete connected Riemannian manifold M let V ∈ C 2 (M) be such that μ (d x) = e − V (x) vol (d x) is a probability measure on M. Taking μ as reference measure, we derive inequalities for probability measures on M linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We propose a new definition of the Gaussian multiplicative chaos and an approach based on the relation of subcritical Gaussian multiplicative chaos to randomized shifts of a Gaussian measure. Using this relation we prove general results on uniqueness and convergence for subcritical Gaussian multiplicative chaos that hold for Gaussian fields with arbitrary covariance kernels. [ABSTRACT FROM AUTHOR]

We investigate the problem of the equivalence of L q -Sobolev norms in Malliavin spaces for q ∈ [ 1 , ∞) , focusing on the graph norm of the k -th Malliavin derivative operator and the full Sobolev norm involving all derivatives up to order k , where k is any positive integer. The case q = 1 in the infinite-dimensional setting is challenging, since at such extreme the standard approach involving Meyer's inequalities fails. In this direction, we are able to establish the mentioned equivalence for q = 1 and k = 2 relying on a vector-valued Poincaré inequality that we prove independently and that turns out to be new at this level of generality, while for q = 1 and k > 2 the equivalence issue remains open, even if we obtain some functional estimates of independent interest. With our argument (that also resorts to the Wiener chaos) we are able to recover the case q ∈ (1 , ∞) in the infinite-dimensional setting; the latter is known since the eighties, however our proof is more direct than those existing in the literature, and allows to give explicit bounds on all the multiplying constants involved in the functional inequalities. Finally, we also deal with the finite-dimensional case for all q ∈ [ 1 , ∞) (where the equivalence, without explicit constants, follows from standard compactness arguments): our proof in such setting is much simpler, relying on Gaussian integration-by-parts formulas and an adaptation of Sobolev inequalities in Euclidean spaces, and it provides again quantitative bounds on the multiplying constants, which however blow up when the dimension diverges to ∞ (whence the need for a different approach in the infinite-dimensional setting). [ABSTRACT FROM AUTHOR]

We construct an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using Wick ordered polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the Mittag-Leffler measures. Therefore we are able to introduce a test function and a distribution space. As an application we construct Donsker's delta in a non-Gaussian setting as a weak integral in the distribution space. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained. [Copyright &y& Elsevier]

Abstract: Let γ be a Gaussian measure on a locally convex space and H be the corresponding Cameron–Martin space. It has been recently shown by L. Ambrosio and A. Figalli that the linear first-order PDE where is a probability measure, admits a weak solution, in particular, under the following assumptions: Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures ν on , under the main assumption that for every , where is the logarithmic derivative of ν along the coordinate . We also show uniqueness of the solution for a wide class of measures. This class includes uniformly log-concave Gibbs measures and certain product measures. [Copyright &y& Elsevier]

Abstract: We generalize the Poincaré limit which asserts that the n-dimensional Gaussian measure is approximated by the projections of the uniform probability measure on the Euclidean sphere of appropriate radius to the first n-coordinates as the dimension diverges to infinity. The generalization is done by replacing the projections with certain maps. Using this generalization, we derive a Gaussian isoperimetric inequality for an absolutely continuous probability measure on Euclidean spaces with respect to the Lebesgue measure, whose density is a radial function. [Copyright &y& Elsevier]

Abstract: Let a family of gradient Gaussian vector fields on be given. We show the existence of a uniform finite range decomposition of the corresponding covariance operators, that is, the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of diameters . In addition we prove natural regularity for the subcovariance operators and we obtain regularity bounds as we vary within the given family of gradient Gaussian measures. [Copyright &y& Elsevier]

Abstract: We develop a Radon transform on Banach spaces using Gaussian measure and prove that if a bounded continuous function on a separable Banach space has zero Gaussian integral over all hyperplanes outside a closed bounded convex set in the Hilbert space corresponding to the Gaussian measure then the function is zero outside this set. [Copyright &y& Elsevier]

Abstract: Consider an infinite-dimensional linear space equipped with a Gaussian measure and the group of linear transformations that send the measure to equivalent one. Limit points of can be regarded as ‘spreading’ maps (polymorphisms). We show that the closure of in the semigroup of polymorphisms contains a certain semigroup of operator colligations and write explicit formulas for action of operator colligations by polymorphisms of the space with Gaussian measure. [Copyright &y& Elsevier]

Abstract: We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincaré inequality for the Gaussian measure. [Copyright &y& Elsevier]

Abstract: For , the Bargmann projection is the orthogonal projection from onto the holomorphic subspace , where is the standard Gaussian probability measure on with variance . The space is classically known as the Segal–Bargmann space. We show that extends to a bounded operator on , and calculate the exact norm of this scaled Bargmann projection. We use this to show that the dual space of the -Segal–Bargmann space is an Segal–Bargmann space, but with the Gaussian measure scaled differently: (this was shown originally by Janson, Peetre, and Rochberg). We show that the Bargmann projection controls this dual isomorphism, and gives a dimension-independent estimate on one of the two constants of equivalence of the norms. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context. [Copyright &y& Elsevier]

Abstract: We consider the Itô stochastic differential equation on . The diffusion coefficients are supposed to be in the Sobolev space with , and to have linear growth. For the drift coefficient , we distinguish two cases: (i) is a continuous vector field whose distributional divergence with respect to the Gaussian measure exists, (ii) has Sobolev regularity for some . Assume for some . In case (i), if the pathwise uniqueness of solutions holds, then the push-forward admits a density with respect to . In particular, if the coefficients are bounded Lipschitz continuous, then leaves the Lebesgue measure quasi-invariant. In case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish existence and uniqueness of stochastic flow of maps. [Copyright &y& Elsevier]

Abstract: We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group , G be the subgroup of X generated by all elements of order 2, and be the set of all topological automorphisms of X. Let , , , such that for all . Let be independent random variables with values in X and distributions with non-vanishing characteristic functions. If the conditional distribution of given is symmetric, then each , where are Gaussian measures, and are distributions supported in G. [Copyright &y& Elsevier]

Abstract: We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on and different classes of measures: Gaussian measures on , symmetric Bernoulli and symmetric uniform probability measures on , as well as their convolutions. Surprisingly, a slightly weaker strong hypercontractivity property holds for any symmetric measure on . A log-Sobolev inequality (LSI) is deduced from the (SHC) for compactly supported measures on , still for log-subharmonic functions. An analogous (LSI) is proved for Gaussian measures on and for other measures for which we know the (SHC) holds. Our log-Sobolev inequality holds in the log-subharmonic category with a constant smaller than the one for Gaussian measure in the classical context. [Copyright &y& Elsevier]

Abstract: Functions of bounded variation in an abstract Wiener space, i.e., an infinite-dimensional Banach space endowed with a Gaussian measure and a related differential structure, have been introduced by M. Fukushima and M. Hino using Dirichlet forms, and their properties have been studied with tools from analysis and stochastics. In this paper we reformulate, in an integral-geometric vein and with purely analytical tools, the definition and the main properties of BV functions, and investigate further properties. [Copyright &y& Elsevier]

Abstract: A reduction theorem is established, showing that any Sobolev inequality, involving arbitrary rearrangement-invariant norms with respect to the Gauss measure in , is equivalent to a one-dimensional inequality, for a suitable Hardy-type operator, involving the same norms with respect to the standard Lebesgue measure on the unit interval. This result is exploited to provide a general characterization of optimal range and domain norms in Gaussian Sobolev inequalities. Applications to special instances yield optimal Gaussian Sobolev inequalities in Orlicz and Lorentz(–Zygmund) spaces, point out new phenomena, such as the existence of self-optimal spaces, and provide further insight into classical results. [Copyright &y& Elsevier]

The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on L 2 (T) with covariance [ 1 + (− Δ) s ] − 1 under these transformations for any s > 1 2. This extends previous achievements by Nahmod, Ray-Bellet, Sheffield and Staffilani (2011) and Genovese, Lucà and Valeri (2018), who proved the result for integer values of the regularity parameter s. [ABSTRACT FROM AUTHOR]

Let L be a lattice in R n and φ the L -periodic Gaussian function on R n given by φ (x) = ∑ y ∈ L e − ‖ x − y ‖ 2 . The paper was motivated by the following observation: φ (x + y) φ (x − y) is a positive definite function of two variables x , y. We say that a positive definite function φ on an Abelian group G is cross positive definite (c.p.d.) if, for each z ∈ G , the function φ (x + y + z) φ (x − y) of two variables x , y is positive definite. We say that a finite Radon measure on a locally compact Abelian group is c.p.d. if its Fourier transform is a c.p.d. function on the dual group. We investigate properties of c.p.d. functions and measures and give some integral characterizations. It is proved that the classes of c.p.d. functions and measures are closed with respect to certain natural operations. In particular, products, convolutions and Fourier transforms of c.p.d. functions and measures are c.p.d., whenever defined. [ABSTRACT FROM AUTHOR]

Abstract: Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede–Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko–Pastur law, and relate this example to C.M. Newman''s “triangle” law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede–Kadison determinant and Voiculescu''s S-transform. [Copyright &y& Elsevier]

Abstract: Let γ be the Gaussian measure in and , , the corresponding Ornstein–Uhlenbeck semigroup, whose infinitesimal generator is . For each p with , let be the closure of the region of holomorphy of the map taking values in the space of bounded operators on . We examine the maximal operator . The known results about concern mainly the case . We prove that for this operator is of weak type but not of strong type for γ. However, if a neighbourhood of the origin is deleted from in the definition of , the resulting operator is shown to be of strong type. [Copyright &y& Elsevier]

Abstract: In the present work an analog of the quasiregular representation which is well known for locally-compact groups is constructed for the nilpotent infinite-dimensional group and a criterion for its irreducibility is presented. This construction uses the infinite tensor product of arbitrary Gaussian measures in the spaces with extending in a rather subtle way previous work of the second author for the infinite tensor product of one-dimensional Gaussian measures. [Copyright &y& Elsevier]

Abstract: Let E be an infinite-dimensional locally convex space, let be a weakly convergent sequence of probability measures on E, and let be a sequence of Dirichlet forms on E such that is defined on . General sufficient conditions for Mosco convergence of the gradient Dirichlet forms are obtained. Applications to Gibbs states on a lattice and to the Gaussian case are given. Weak convergence of the associated processes is discussed. [Copyright &y& Elsevier]

Abstract: In order to develop a differential calculus for error propagation of Bouleau [Error Calculus for Finance and Physics, the Language of Dirichlet forms, De Gruyter, Berlin, 2003], we study local Dirichlet forms on probability spaces with carré du champ —i.e. error structures—and we are looking for an object related to which is linear and with a good behaviour by images. For this we introduce a new notion called the measure-valued gradient which is a randomized square root of . The exposition begins with inspecting some natural notions candidate to solve the problem before proposing the measure-valued gradient and proving its satisfactory properties. [Copyright &y& Elsevier]

Abstract: We produce two sets of results arising in the analysis of the degree of smoothness of a function that is known to be smooth along the leaves of one or more foliations. These foliations might arise from Anosov systems, and while each leaf is smooth, the leaves might vary in a nonsmooth fashion. One set of results gives microlocal regularity of such a function away from the conormal bundle of a foliation. The other set of results gives local regularity of solutions to a class of elliptic systems with fairly rough coefficients. Such a regularity theory is motivated by one attack on the foliation regularity problem. [Copyright &y& Elsevier]

Let γ be the Gauss measure on Rd and L the Ornstein–Uhlenbeck operator. For every p in [1,∞)&z.drule;{2}, set φp&ast;=arcsin|2/p−1|, and consider the sector Sφp&ast;={z∈C : |arg z|<φp&ast;}. The main results of this paper are the following. If p is in (1,∞)&z.drule;{2}, and supt>0 ⫼M(tL)⫼L p(γ)<∞, i.e., if M is an L p(γ) uniform spectral multiplier of L in our terminology, and M is continuous on R+, then M extends to a bounded holomorphic function on the sector Sφp&ast;. Furthermore, if p=1 a spectral multiplier M, continuous on R+, satisfies the condition supt>0 ⫼M(tL)⫼L1(γ)<∞ if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i·) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on Rd belonging to a wide class, which contains L. From these results we deduce that operators in this class do not admit an H∞ functional calculus in sectors smaller than Sφp&ast;. [Copyright &y& Elsevier]

We prove that if (P x) x ∈ X is a family of probability measures which satisfy the log-Sobolev inequality and whose pairwise chi-squared divergences are uniformly bounded, and μ is any mixing distribution on X , then the mixture ∫ P x d μ (x) satisfies a log-Sobolev inequality. In various settings of interest, the resulting log-Sobolev constant is dimension-free. In particular, our result implies a conjecture of Zimmermann and Bardet et al. that Gaussian convolutions of measures with bounded support enjoy dimension-free log-Sobolev inequalities. [ABSTRACT FROM AUTHOR]

We continue the study on the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation. By considering the renormalized equation, we extend the quasi-invariance results in [29,26] to Sobolev spaces of negative regularity. Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia [34] with the normal form approach in [29,26]. [ABSTRACT FROM AUTHOR]

We consider the Nonlinear Schrödinger (NLS) equation with third-order dispersion and prove that the Gaussian measure with covariance (1 − ∂ x 2) − α on L 2 (T) is quasi-invariant for the associated flow for α > 1 / 2. This is sharp and improves a previous result obtained in [20] where the values α > 3 / 4 were obtained. Also, our method is completely different and simpler, it is based on an explicit formula for the Radon-Nikodym derivative. We obtain an explicit formula for this latter in the same spirit as in [4] and [5]. The arguments are general and can be used to other Hamiltonian equations. [ABSTRACT FROM AUTHOR]

Let γ n be the standard Gaussian measure on R n. We prove that for every symmetric convex sets K , L in R n and every λ ∈ (0 , 1) , γ n (λ K + (1 − λ) L) 1 n ⩾ λ γ n (K) 1 n + (1 − λ) γ n (L) 1 n , thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn–Minkowski inequality for the Lebesgue measure. We also show that, for a fixed λ ∈ (0 , 1) , equality is attained if and only if K = L. [ABSTRACT FROM AUTHOR]

We establish new approximation results in the sense of Lusin for Sobolev functions f with | ∇ f | ∈ L log ⁡ L on infinite-dimensional spaces equipped with Gaussian measures. The proof relies on some new pointwise estimate for the approximations based on the corresponding semigroup which can be of independent interest. [ABSTRACT FROM AUTHOR]

We prove nontangential and radial maximal function characterizations for Hardy spaces associated to a non-negative self-adjoint operator satisfying Gaussian estimates on a space of homogeneous type with finite measure. This not only addresses an open point in the literature, but also gives a complete answer to the question posed by Coifman and Weiss in the case of finite measure. We then apply our results to give maximal function characterizations for Hardy spaces associated to second–order elliptic operators with Neumann and Dirichlet boundary conditions, Schrödinger operators with Dirichlet boundary conditions, and Fourier–Bessel operators. [ABSTRACT FROM AUTHOR]