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This paper considers fractional Rayleigh–Stokes equations with a power-type nonlinearity. The linear equation can be simulated a non-Newtonian fluid for a generalized second grade fluid and display a nonlocal behavior in time. Because the coexistence of fractional and classical derivatives leads to the lack of semigroup structure of the solution operator, we need to develop a suitable tool to establish some L p − L q estimates in the framework of L p spaces and Besov spaces, respectively. Further, global existence of solutions is showed in spaces of Besov type. • The fractional Rayleigh–Stokes equations with a power-type nonlinearity is investigated. • The linear equation can be simulated a non-Newtonian fluid for a generalized second grade fluid and display a nonlocal behavior in time. • The global existence of solutions is proved in spaces of Besov type. [ABSTRACT FROM AUTHOR]

While the total variation between two probability measures cannot be bounded by Wasserstein metrics in general, it is possible if they possess smooth densities. More generally, we demonstrate that L p -distances between densities in Besov spaces can be bounded by powers of the Wasserstein metrics. [ABSTRACT FROM AUTHOR]

Abstract We study the one-dimensional stochastic heat equation in the mild form driven by a general stochastic measure μ , for μ we assume only σ -additivity in probability. The time averaging of the equation is considered, uniform a. s. convergence to the solution of the averaged equation is obtained. [ABSTRACT FROM AUTHOR]

A time-fractional wave equation with an exponential growth source function is considered. This model can be regarded as a modified version of the one studied by Nakamura and Ozawa (1999). The main feature of this model is that the exponential growth nonlinearity is essentially different from the polynomial growth one, and harder in controlling. We first prove the local existence and uniqueness of mild solutions in an Orlicz space by deriving a nonlinear estimate and L p − L q estimates the source term and solution operators, respectively. Furthermore, by additionally using specific techniques for estimating integrals, we show that small data solutions exist globally over time in such Orlicz space. The last theoretical result is about the second global-in-time existence of solutions in Besov spaces. Such a result is based on a different nonlinear estimate which is derived from a logarithmic interpolation inequality. The main ingredients of proof are the efficient application of function spaces such as Lebesgue spaces, Orlicz space,Besov spaces and computational techniques involving generalized integrals. In addition, some numerical examples are provided to illustrate theoretical results. • Time-fractional wave equation with exponential growth function. • Local existence and uniqueness of mild solutions in Orlicz space. • Global in time existence of solutions in Besov spaces. • Numerical examples are provided to illustrate theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers a lower bound estimation over L p ( R d ) ( 1 ≤ p < ∞ ) risk for d dimensional regression functions in Besov spaces based on biased data. We provide the best possible lower bound up to a ln n factor by using wavelet methods. When the weight function ω ( x , y ) ≡ 1 and d = 1 , our result reduces to Chesneau’s theorem, see Chesneau (2007). [ABSTRACT FROM AUTHOR]

We provide necessary and sufficient conditions on the characteristics of an infinitely divisible distribution under which its characteristic function φ decays polynomially. Under a mild regularity condition this polynomial decay is equivalent to 1 / φ being a Fourier multiplier on Besov spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the 3D Magneto-micropolar fluid equations in Besov spaces. We improve and extend some known regularity criterion of weak solution for the 3D Magneto-micropolar fluid equations by means of the Fourier localization technique and Bony’s para-product decomposition. [Copyright &y& Elsevier]

Abstract: This paper proposes a nonparametric method for producing smooth and positive estimates of the density of a Lévy process, which is widely used in mathematical finance. We use the method of logwavelet density estimation to estimate the Lévy density with discretely sampled observations. Since Lévy densities are not necessarily probability densities, we introduce a divergence measure similar to Kullback–Leibler information to measure the difference between two Lévy densities. Rates of convergence are established over Besov spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a simple Bayesian block wavelet shrinkage method for estimating an unknown function in the presence of Gaussian noise. A data-driven procedure which can adaptively choose the block size and the shrinkage level at each resolution level is provided. The asymptotic property of the proposed method, BBN (Bayesian BlockNorm shrinkage), is investigated in the Besov sequence space. The numerical performance and comparisons with some of existing wavelet denoising methods show that the new method can achieve good performance but with the least computational time. [Copyright &y& Elsevier]

Abstract: We derive minimax results in the functional deconvolution model under the -risk, . Lower bounds are given when the unknown response function is assumed to belong to a Besov ball and under appropriate smoothness assumptions on the blurring function, including both regular-smooth and super-smooth convolutions. Furthermore, we investigate the asymptotic minimax properties of an adaptive wavelet estimator over a wide range of Besov balls. The new findings extend recently obtained results under the -risk. As an illustration, we discuss particular examples for both continuous and discrete settings. [Copyright &y& Elsevier]

Abstract: Based on the notion of first order dyadic -variation, we give a new characterization of Besov spaces for , and . We also give results in the case where . Hence we provide simple tools that enable us to derive new regularity properties for the trajectories of various continuous time stochastic processes. [Copyright &y& Elsevier]

Abstract: Autin [2007. Maxisets for -thresholding rules. Test, to appear, see http://www.seio.es/test/Archivos/issues/Ab_TESTAutin.html ] has established the following estimation result: by considering the Gaussian white noise model and the Besov risk , the BlockShrink estimator is better in the maxiset sense than the hard thresholding estimator. In the present paper, we show that this maxiset superiority is strict and can be extended to the risk for numerous sophisticated models (regression with random uniform design, convolution model in Gaussian white noise,…). [Copyright &y& Elsevier]

Abstract: In this paper we focus on nonparametric estimation of a constrained regression function using penalized wavelet regression techniques. This results into a convex optimization problem under linear constraints. Necessary and sufficient conditions for existence of a unique solution are discussed. The estimator is easily obtained via the dual formulation of the optimization problem. In particular we investigate a penalized wavelet monotone regression estimator. We establish the rate of convergence of this estimator, and illustrate its finite sample performance via a simulation study. We also compare its performance with that of a recently proposed constrained estimator. An illustration to some real data is given. [Copyright &y& Elsevier]

Abstract: Let be a random -additive in probability set function defined on Borel subsets of . We prove that if the process , has continuous paths, then they belong a.s. to the Besov space for all . [Copyright &y& Elsevier]

Abstract: The problem of estimating a regression function based on a regression model with (known) random design is considered. By adopting the framework of wavelet analysis, we establish the asymptotic minimax rate of convergence under the risk over Besov balls. A part of this paper is devoted to the case where the design density is vanishing. [Copyright &y& Elsevier]

We study maxisets for linear procedures in the framework of the heteroscedastic white noise model. This enables us to point out the nature of the spaces naturally connected to these procedures and to compare the performance of linear and nonlinear estimates by comparing their respective maxisets. [Copyright &y& Elsevier]

We prove the existence and the Besov regularity of the density of the solution to a general parabolic SPDE which includes the stochastic Burgers equation on an unbounded domain. We use an elementary approach based on the fractional integration by parts. [ABSTRACT FROM AUTHOR]