If $\mu $ is a smooth measure supported on a real-analytic submanifold of ${\mathbb {R}}^{2n}$ which is not contained in any affine hyperplane, then the Weyl transform of $\mu $ is a compact operator.

We prove existence and uniqueness for the solution of a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized Itô rule valid for functions with an absolutely continuous derivative and applicable to solutions of mixed fractional stochastic differential equations with Lipschitz coefficients, which plays a key role in our proof of existence and uniqueness. The proof of such a formula is new and relies on showing the existence of a density of the law under mild assumptions on the diffusion coefficient.

We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact.

Relying on results due to Shmerkin and Solomyak, we show that outside a zero-dimensional set of parameters, for every planar homogeneous self-similar measure $\unicode[STIX]{x1D708}$, with strong separation, dense rotations and dimension greater than $1$, there exists $q>1$ such that $\{P_{z}\unicode[STIX]{x1D708}\}_{z\in S}\subset L^{q}(\mathbb{R})$. Here $S$ is the unit circle and $P_{z}w=\langle z,w\rangle$ for $w\in \mathbb{R}^{2}$. We then study such measures. For instance, we show that $\unicode[STIX]{x1D708}$ is dimension conserving in each direction and that the map $z\rightarrow P_{z}\unicode[STIX]{x1D708}$ is continuous with respect to the weak topology of $L^{q}(\mathbb{R})$.

There are two main aims of the paper. The first is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second is to extend the criterion for the precompactness of sets in the Lebesgue spaces Lp (ℝn ), 1 ⩽ p < ∞, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces , into Lorentz-type spaces.

We prove that all convolution products of pairs of continuous orbital measures in rank one, compact symmetric spaces are absolutely continuous and determine which convolution products are in $L^{2}$ (meaning that their density function is in $L^{2}$). We characterise the pairs whose convolution product is either absolutely continuous or in $L^{2}$ in terms of the dimensions of the corresponding double cosets. In particular, we prove that if $G/K$ is not $\text{SU}(2)/\text{SO}(2)$, then the convolution of any two regular orbital measures is in $L^{2}$, while in $\text{SU}(2)/\text{SO}(2)$ there are no pairs of orbital measures whose convolution product is in $L^{2}$.

Sharp bounds for the deviation of a real-valued function f defined on a compact interval [a,b] to the chord generated by its end points (a,f(a)) and (b,f(b)) under various assumptions for f and f′, including absolute continuity, convexity, bounded variation, and monotonicity, are given. Some applications for weighted means and f-divergence measures in information theory are also provided.

Using variational measures, we characterize additive interval functions which are indefinite Henstock–Kurzweil integrals in the Euclidean space, answering a problem posed by W. B. Jurkat and R. W. Knizia and independently by Lee Peng-Yee and Chew Tuan-Seng. This also extends a result of Bongiorno, Di Piazza and Skvortsov.

Distributional fixed points of a Poisson shot noise transform (for nonnegative and nonincreasing response functions bounded by 1) are characterized. The tail behavior of fixed points is described. Typically they have either exponential moments or their tails are proportional to a power function, with exponent greater than −1. The uniqueness of fixed points is also discussed. Finally, it is proved that in most cases fixed points are absolutely continuous, apart from the possible atom at zero.

We study second order elliptic operators with periodic coefficients in two-dimensional simply connected periodic waveguides with the Dirichlet or Neumann boundary conditions. It is proved that under some mild smoothness restrictions on the coefficients, such operators have purely absolutely continuous spectra. The proof follows a method suggested previously by A. Morame to tackle periodic operators with variable coefficients in dimension 2.

2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.

Under the assumptions of the neutral infinite alleles model, K (the total number of alleles present in a sample) is sufficient for estimating θ (the mutation rate). This is a direct result of the Ewens sampling formula, which gives a consistent, asymptotically normal estimator for θ based on K. It is shown that the same estimator used to estimate θ under neutrality is consistent and asymptotically normal, even when the assumption of selective neutrality is violated.

In this paper weak Perron integrals are characterized as n-dimensional interval functions F which are additive, differentiable almost everywhere in the weak sense and which satisfy a new continuity condition concerning the singular set. Before, only one-dimensional Perron integrals were characterized by the theorem of Hake- Alexendrov-Looman, and analogous results for strong Perron integrals (which are best analyzed, but more restrictive) are not available in higher dimensions yet. In order to formulate our continuity condition we introduce an outer measure μ by means of a new weak variation of F which is required to vanish on all null sets. The same condition is also necessary and sufficient for the integral of the weak derivative to yield the original interval function. This “fundamental theorem” is split into two fundamental inequalities of very general nature which contain additional singular terms involving our variation. These inequalities are very useful also for Lebesgue integrals.

There are some martingales associated with the branching random walk which are natural generalizations of the classical martingale occurring in the Galton–Watson process. Some continuity properties of the distributions of their limits are discussed.

We establish the absolute continuity of the limit random variables of two supercritical Galton-Watson branching processes, one allowing unrestricted immigration and the other having a state dependent immigration component.