Best approximation characterization of Sobolev spaces with common smoothness in the probabilistic setting and Sq norm

In this article, we mainly studied the Gel’fand widths of Sobolev space in the probabilistic and average settings. And, we estimated the sharp bounds of the probabilistic Gel’fand (N,δ)-widths of multivariate Sobolev space MW2r(Td) with mixed derivative equipped with the Gaussian measure in the Sq-norm by discretization methods. Later, we estimated the sharp bounds of the p-average Gel’fand N-widths of univariate Sobolev space W2r(T) and multivariate Sobolev space MW2r(Td) with mixed derivative equipped with the Gaussian measure in the Sq-norm.

In this article, we give the sharp bounds of probabilistic Kolmogorov N,δ-widths and probabilistic linear N,δ-widths of the multivariate Sobolev space W2A with common smoothness on a Sq norm equipped with the Gaussian measure μ, where A⊂Rd is a finite set. And we obtain the sharp bounds of average width from the results of the probabilistic widths. These results develop the theory of approximation of functions and play important roles in the research of related approximation algorithms for Sobolev spaces.

In this paper, we discuss the best approximation of functions on the sphere by spherical polynomials and the approximation by the Fourier partial summation operators and the Vallée-Poussin operators, on a Sobolev space with a Gaussian measure in the probabilistic case setting, and get the probabilistic error estimation. We show that in the probabilistic case setting, the Fourier partial summation operators and the Vallée-Poussin operators are the order optimal linear operators in the Lq space for 1 ≤ q ≤ ∞, but the spherical polynomial spaces are not order optimal in the Lq space for 2 < q ≤ ∞. This is completely different from the situation in the average case setting, which the spherical polynomial spaces are order optimal in the Lq space for 1 ≤ q < ∞. Also, in the Lq space for 1 ≤ q ≤ ∞, worst-case order optimal subspaces are also order optimal in the probabilistic case setting.

We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the two-dimensional defocusing cubic nonlinear wave equation (NLW). Under some regularity condition, we prove quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces for the NLW dynamics. We achieve this by introducing a simultaneous renormalization of the energy functional and its time derivative and establishing a renormalized energy estimate in the probabilistic setting.

Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure

In this article, we first put forward a new definition of probabilistic Gel’fand-(N,delta )-width which is the classical Gel’fand-N-width in a probabilistic setting. Then we estimate the sharp order of the probabilistic Gel’fand-(N,delta )-width of finite-dimensional space. Furthermore, we obtain the exact order of probabilistic Gel’fand-(N, delta )-width of univariate Sobolev space by the discretization method according to the result of finite-dimensional space.

We define a Poisson structure on the Nualart-Pardoux test algebra associated to the path space of a finite dimensional Lie algebra.

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Vallee-Poussin operators, Ces`aro operators, Abel operators, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the Lq space for 1 ⩽ q < ∞, and the Fourier partial summation operators and the Vallee-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the Lq space for 1 ⩽ q < ∞.

We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we establish an optimal regularity result for quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main new ingredient is an improved energy estimate established by performing an infinite iteration of normal form reductions on the energy functional. Furthermore, we show that the dispersion is essential for such a quasi-invariance result by proving non quasi-invariance of the Gaussian measures under the dynamics of the dispersionless model.

Complexity of approximation with relative error criterion in worst, average, and probabilistic settings

In this paper, we determine the asymptotic values of the probabilistic adaptive widths of the space of multivariate functions with bounded mixed derivative (MW2r(Td),μ) relative to the manifold (YN,ν) in the Lq(Td)-norm, 1 < q ≤ 2, where μ and ν are two given Gaussian measures.

In this paper, we define the entropy number in probabilistic setting and determine the exact order of entropy number of finite-dimensional space in probabilistic setting. Moreover, we also estimate the sharp order of entropy number of univariate Sobolev space in probabilistic setting by discretization method.

In this paper, we study the probabilistic Gel’fand N,δ-width of multivariate Sobolev spaces MW2rTd with mixed derivative that are equipped with Gaussian measure μ in LqTd. The sharp asymptotic estimates are determined by employing the discretization method.

We extend the theory of tent spaces from Euclidean spaces to various types of metric measure spaces. For doubling spaces we show that the usual ‘global’ theory remains valid, and for ‘non-uniformly locally doubling’ spaces (including R with the Gaussian measure) we establish a satisfactory local theory. In the doubling context we show that Hardy–Littlewood–Sobolev-type embeddings hold in the scale of weighted tent spaces, and in the special case of unbounded ADregular metric measure spaces we identify the real interpolants (the ‘Z-spaces’) of weighted tent spaces. Weighted tent spaces and Z-spaces on R are used to construct Hardy–Sobolev and Besov spaces adapted to perturbed Dirac operators. These spaces play a key role in the classification of solutions to first-order Cauchy–Riemann systems (or equivalently, the classification of conormal gradients of solutions to second-order elliptic systems) within weighted tent spaces and Z-spaces. We establish this classification, and as a corollary we obtain a useful characterisation of well-posedness of Regularity and Neumann problems for second-order complex-coefficient elliptic systems with boundary data in Hardy–Sobolev and Besov spaces of order s ∈ (−1, 0).

We prove quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the three-dimensional defocusing cubic nonlinear wave equation. As in the previous work on the two-dimensional case, we employ a simultaneous renormalization on the energy functional and its time derivative. Two new ingredients in the three-dimensional case are (i) the construction of the weighted Gaussian measures, based on a variational formula for the partition function inspired by Barashkov and Gubinelli (2018), and (ii) an improved argument in controlling the growth of the truncated weighted Gaussian measures, where we combine a deterministic growth bound of solutions with stochastic estimates on random distributions.