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A Bayesian treatment of the proportional hazard (PH) model is revisited using Dirichlet process and gamma process priors for the baseline survival and cumulative hazard functions respectively. Such priors, due to their discrete support, conflict with the absolutely continuous nature of survival responses expressed through the hazard function structure that defines the PH model. We resolve this conflict through the use of a proposed e -grid likelihood approach which we apply to the PH model to consider marginal inference for the regression parameter β using expansions as e ↓ 0 . Using Dirichlet process priors, the e -grid likelihood approach leads to a new explicit marginal posterior density expansion for β which accommodates the most general setting with arbitrary ties and right censoring. In the previously considered case of gamma process priors, the approach extends the results of Kalbfleisch (1978) to deal with arbitrary arrangements of ties and also provides a rigorous justification for his posterior expressions. As in Kalbfleisch (1978), we show that the leading terms of both expansions approximate Cox’s partial likelihood when there are no ties and the process priors are diffuse. The e -grid likelihood approach is similar in concept to the grouped data likelihood approach of Sinha et al. (2003) but differs in the likelihood approximation. Whereas our expansions are Poincare expansions (with relative error O ( e ) as e ↓ 0 ), those of Sinha et al. (2003) are stochastic expansions and not proper Poincare expansion (as we show), so that they lack relative error O ( e ) . Using a flat improper prior for β , the two marginal posterior expressions for β are shown to be integrable for general arrangements of ties and censoring under weak conditions on the design and failures.

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $Z^d$-periodic vector field from $R^d$ in $R^d$. We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: - the everywhere asymptotics of the flow $X$, - the almost-everywhere asymptotics of the flow $X$, - the global rectification of the vector field $b$ in $Y_d$, - the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, - the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, - the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, - the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach involves reducing a variational problem to a (nondynamic) problem of constrained optimization in a normed space and then applying the results recently obtained for the latter class using generalized differentiation. In this way, we derive necessary optimality conditions for nonconvex problems of the calculus of variations with velocity constraints under the weakest metric subregularity-type constraint qualification. The developed approach leads us to a short and simple proof of the First-order necessary optimality conditions for such and related problems in broad spaces of functions, including those of class C^k.

For two independent Levy processes ξ and η and an exponentially distributed random variable τ with parameter q > 0 , independent of ξ and η , the killed exponential functional is given by V q , ξ , η ≔ ∫ 0 τ e − ξ s − d η s . Interpreting the case q = 0 as τ = ∞ , the random variable V q , ξ , η is a natural generalisation of the exponential functional ∫ 0 ∞ e − ξ s − d η s , the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein–Uhlenbeck process. In this paper we show that also the law of the killed exponential functional V q , ξ , η arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein–Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of ∫ 0 t e − ξ s − d η s for fixed t ≥ 0 , as well as for integrals of the form ∫ 0 ∞ f ( s ) d η s for deterministic functions f . Furthermore, applying the same techniques to the case q = 0 , new results on the absolute continuity of the improper integral ∫ 0 ∞ e − ξ s − d η s are derived.

In this paper, we discuss estimates on transition densities for subordinators, which are global in time. We establish the sharp two-sided estimates on the transition densities for subordinators whose L\'evy measures are absolutely continuous and decaying in mixed polynomial orders. Under a weaker assumption on L\'evy measures, we also obtain a precise asymptotic behaviors of the transition densities at infinity. Our results cover geometric stable subordinators, Gamma subordinators and much more., Comment: 47 pages. Example 3.4 is updated

We consider the Vlasov–Poisson system with repulsive interactions. For initial data a small, radial, absolutely continuous perturbation of a point charge, we show that the solution is global and disperses to infinity via a modified scattering along trajectories of the linearized flow. This is done by an exact integration of the linearized equation, followed by the analysis of the perturbed Hamiltonian equation in action-angle coordinates.

We analyze the one dimensional Cucker–Smale (in short CS) model with a weak singular communication weight ψ ( x ) = | x | − β with β ∈ ( 0 , 1 ) . We first establish a global-in-time existence of measure-valued solutions to the kinetic CS equation. For this, we use a proper change of variable to reformulate the particle CS model as a first-order particle system and provide the uniform-in-time stability for that particle system. We then extend this stability estimate for the singular CS particle system. By using that stability estimate, we construct the measure-valued solutions to the kinetic CS equation globally in time. Moreover, as a direct application of the uniform-in-time stability estimate, we show the quantitative uniform-in-time mean-field limit from the particle system to that kinetic CS equation in p-Wasserstein distance with p ∈ [ 1 , ∞ ] . Our result gives the uniqueness of measure-valued solution in the sense of mean-field limits, i.e., the measure-valued solutions, approximated by the empirical measures associated to the particle system, uniquely exist. Similar results for the first-order model also follow as a by-product. We also reformulate the continuity-type equation, which is derived from the first-order model, as an integro-differential equation by employing the pseudo-inverse of the accumulative particle distribution. By making use of a modified p-Wasserstein distance, we provide the contractivity estimate for absolutely continuous solutions of the continuum equation.

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.

We consider a weighted initial value problem for first-order functional differential equations with monotone non-linearities which may have a non-integrable singularity with respect to the independent variable. Conditions for the existence of solutions in a suitable weighted space of locally absolutely continuous functions are given.

We consider the action of Mandelbrot multiplicative cascades on probability measures supported on a symbolic space. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on the lower and upper Hausdorff dimensions of the measure and the entropy of the random weights. We also obtain sharp bounds for the lower Hausdorff and upper packing dimensions of the limiting measure. When the original measure is a Gibbs measure associated with a potential of certain modulus of continuity (weaker than H\"older), all our results are sharp. This improves results previously obtained by Kahane and Peyri\`ere, Ben Nasr, and Fan. We exploit our results to derive dimension estimates and absolute continuity for some random fractal measures., Comment: 27 pages

Focusing first on the inner $\alpha$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $\mu^A$ the inner $\alpha$-Riesz balayage of a Radon measure $\mu$ to $A\subset\mathbb R^n$ arbitrary), we describe its Euclidean support, provide a formula for evaluation of its total mass, establish the vague continuity of the map $y\mapsto\varepsilon_y^A$ outside the inner $\alpha$-irregular points for $A$, and obtain necessary and sufficient conditions for $\varepsilon_y^A$ to be of finite energy (more generally, for $\varepsilon_y^A$ to be absolutely continuous with respect to inner capacity) as well as for $\varepsilon_y^A(\mathbb R^n)\equiv1$ to hold. Those criteria are given in terms of the newly defined concepts of $\alpha$-thinness and $\alpha$-ultrathinness at infinity that generalize the concepts of thinness at infinity by Doob and Brelot, respectively. Further, we extend some of these results to $\mu^A$ general by verifying the formula $\mu^A=\int\varepsilon_y^A\,d\mu(y)$. We also show that there is a $K_\sigma$-set $A_0\subset A$ such that $\mu^A=\mu^{A_0}$ for all $\mu$, and give various applications of this theorem. In particular, we prove the vague and strong continuity of the inner swept, resp. equilibrium, measure under the approximation of $A$ arbitrary, thereby strengthening Fuglede's result established for $A$ Borel (Acta Math., 1960). Being new even for $\alpha=2$, the results obtained also present a further development of the theory of inner Newtonian capacities and of inner Newtonian balayage, originated by Cartan., Comment: 23 pages, 1 figure

A convex sweeping process is considered in a separable Hilbert space. The majority of works on sweeping processes use the Hausdorff distance to describe the movement of the convex set generating the process. However, for unbounded sets the use of the Hausdorff distance does not always guarantee the fulfilment of conditions under which a solution exists. In the present work, instead of the Hausdorff distance we use the $$\rho $$ -excesses of sets. These excesses are subjected to positive Radon measures depending on $$\rho $$ . We prove the existence of right continuous BV solutions and establish their dependence on single-valued perturbations depending only on time. The results we obtain are applied to prove the theorems on existence and relaxation of extremal right continuous BV solutions of a sweeping process with multivalued perturbations. Some results on absolutely continuous solutions are derived as corollaries.

We consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. Under minimal assumptions on the type of disorder, we prove the following dichotomy: Either every realization of the random operator has purely absolutely continuous spectrum or spectral and exponential dynamical localization hold. In particular, we establish Anderson localization for Schrodinger operators with Bernoulli-type random singular potential and singular density.

In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument. In this paper, we construct and study the Gibbs measure. Our approach is based on earlier work of Barashkov and Gubinelli for the $\Phi^4_3$-model. Most importantly, our truncated Gibbs measures are tailored towards the dynamical aspects in the second part of the series. In addition, we develop new tools dealing with the non-locality of the Hartree interaction. We also determine the exact threshold between singularity and absolute continuity of the Gibbs measure depending on the regularity of the interaction potential., Comment: Added uniqueness statement and corrected typos

In this paper, we prove that any projection in the convex hull of three surjective linear isometries on $$\mathrm {AC}(X)$$ is a generalized bi-circular projection, where $$\mathrm {AC}(X)$$ denotes the Banach space of all absolutely continuous functions on a compact subset of $$\mathbb {R}$$ with at least two points. We also show that the trivial projections are the only projections on $$\mathrm {AC}(X)$$ which can be represented as the average of three surjective linear isometries.

In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space L loc 1 . The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system. The method used is Lagrangian representation, the essence of which is characteristic analysis. The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables. We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.

Let $$f = P[F]$$ denote the Poisson integral of F in the unit disk $${\mathbb {D}}$$ with F being absolutely continuous in the unit circle $${\mathbb {T}}$$ and $${\dot{F}}\in L^{p}({\mathbb {T}})$$ , where $${\dot{F}}(e^{it})=\frac{d}{dt} F(e^{it})$$ and $$p\ge 1$$ . Recently, the author in Zhu (J Geom Anal, 2020) proved that (1) if f is a harmonic mapping and $$1\le p< 2$$ , then $$f_{z}$$ and $$\overline{f_{{\overline{z}}}}\in \mathcal {B}^{p}({\mathbb {D}}),$$ the classical Bergman spaces of $${\mathbb {D}}$$ [12, Theorem 1.2]; (2) if f is a harmonic quasiregular mapping and $$1\le p\le \infty $$ , then $$f_{z},$$ $$\overline{f_{{\overline{z}}}}\in \mathcal {H}^{p}({\mathbb {D}}),$$ the classical Hardy spaces of $${\mathbb {D}}$$ [12, Theorem 1.3]. These are the main results in Zhu (J Geom Anal, 2020). The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, [12, Theorem 1.2] is true when $$1\le p< \infty $$ . Also, we show that [12, Theorem 1.2] is not true when $$p=\infty $$ . Second, we demonstrate that [12, Theorem 1.3] still holds true when the assumption f being a harmonic quasiregular mapping is replaced by the weaker one f being a harmonic elliptic mapping.

International audience; We study the semilinear elliptic equation −∆u + g(u)σ = µ with Dirichlet boundary conditions in a smooth bounded domain where σ is a nonnegative Radon measure, µ a Radon measure and g is an absorbing nonlinearity. We show that the problem is well posed if we assume that σ belongs to some Morrey class. Under this condition we give a general existence result for any bounded measure provided g satisfies a subcritical integral assumption. We study also the supercritical case when g(r) = |r| q−1 r, with q > 1 and µ satisfies an absolute continuity condition expressed in terms of some capacities involving σ. 2010 Mathematics Subject Classification. 35 J 61; 31 B 15; 28 C 05 .

For a class of piecewise convex maps T with an indifferent fixed point and critical points on the interval [0,1], we show that T has a unique absolutely continuous invariant probability measure μ, ...

A new class of statistics obtained by ordering the absolute values of the observations arising from absolutely continuous distributions which are symmetrically distributed about zero is introduced ...

Given i.i.d. ℝd-valued stochastic processes X1(t), . . . ,Xp(t), p ≥ 2, with stationary increments, a minimal condition is provided for the occupation measure $$ {\mu}_t(B)=\underset{\left[0,t\right]}{\int }1B\left({X}_1\left({s}_1\right)-{X}_2\left({s}_2\right),\dots, {X}_{p-1}\left({s}_{p-1}\right)-{X}_p\left({s}_p\right)\right){ds}_1\dots {ds}_p, $$ B ⊂ ℝd(p−1), to be absolutely continuous with respect to the Lebesgue measure on ℝd(p−1). An isometry identity related to the resulting density (known as intersection local time) is also established.

In this paper, we study norm almost periodic measures on locally compact Abelian groups. First, we show that the norm almost periodicity of $\mu$ is equivalent to the equi-Bohr almost periodicity of $\mu*g$ for all $g$ in a fixed family of functions. Then, we show that, for absolutely continuous measures, norm almost periodicity is equivalent to the Stepanov almost periodicity of the Radon--Nikodym density., Comment: 23 pages, added many examples

In this paper, we consider a location-scale family arising out of Absolutely Continuous Trivariate Exponential (ACTVE) distribution with equal marginal due to Weier and Basu. The distribution of th...

We give the cumulative distribution function of $M_n$, the maximum of a sequence of $n$ observations from an ARMA(1, 1) process. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. The distribution of $M_n$ is then given as a weighted sum of the $n$th powers of the eigenvalues of a non-symmetric Fredholm kernel. The weights are given in terms of the left and right eigenfunctions of the kernel. These results are large deviations expansions for estimates, since the maximum need not be standardized to have a limit. In fact, such a limit need not exist.

We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure., Comment: 8 pages

We are concerned with the weakly asymmetric normalized binary contact path process. The symmetric version was introduced in Griffeath (1983). The model describes the spread of an infectious disease on the lattice Z d . The configuration at each site x ∈ Z d takes value in [ 0 , ∞ ) . Site x is recovered at rate 1, and is infected by its neighbor x ± e i at rate λ ∓ λ 1 ∕ N , where { e i } 1 ≤ i ≤ d is the canonical basis of the d -dimensional lattice, λ , λ 1 are non-negative constants and N is the scaling parameter. When the infection occurs, the seriousness of the disease at site x is added with that of y . When there is neither recovery nor infection occurring during some time interval, the value at site x evolves according to some ODE. We prove that for d ≥ 3 and large value λ , the empirical measure of the process, under diffusive scaling, converges in probability to a deterministic measure whose density is the unique weak solution to a linear parabolic equation, while for small λ and in all dimensions, the limit is zero. We also conjecture that the fluctuations field in the former case is driven by a generalized Ornstein–Uhlenbeck process, while a rigorous proof is absent. The main difficulty in proving the hydrodynamics is to prove the absolute continuity of the limiting path, where the theory of linear systems introduced in Liggett (1985) is utilized.

In a Hilbert space H with an unconditional basis of reproducing kernels, we study the existence of a weighted integral norm with respect to an absolutely continuous measure, which is equivalent to the original H-norm. If the space H is defined via weighted integrals, the problem can be interpreted as restoring the original structure.

Let µ be a positive measure on the unit circle that is regular in the sense of Stahl, Totik, and Ullmann. Assume that in some subarc J, µ is absolutely continuous, while µ′ is positive and continuous. Let {φn} be the orthonormal polynomials for µ. We show that for appropriate ζn ∈ J, $${{\rm{\{ }}{{{\varphi _n}({\zeta _n}(1 + {\textstyle{z \over n}}))} \over {{\varphi _n}({\zeta _n})}}{\rm{\}}}_{n \ge 1}}$$ is a normal family in compact subsets of ℂ. Using universality limits, we show that limits of subsequences have the form ez + C(ez − 1) for some constant C. Under additional conditions, we can set C = 0.

In this paper, we introduce the Bezier variant of two new families of generalized Bernstein type operators. We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity. By means of construction of suitable functions and the method of Bojanic and Cheng, we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.

Let H 0 be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and V be a compact perturbation. We relate the regularity properties of V to various spectral properties of the perturbed operator H 0 + V . The structures of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended. For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.

We show that for each $\gamma \in (0,2)$, there is a unique metric (i.e., distance function) associated with $\gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) $h$, there is a unique random metric $D_h$ associated with the Riemannian metric tensor "$e^{\gamma h} (dx^2 + dy^2)$" on $\mathbb C$ which is characterized by a certain list of axioms: it is locally determined by $h$ and it transforms appropriately when either adding a continuous function to $h$ or applying a conformal automorphism of $\mathbb C$ (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The $\gamma$-LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding, Dub\'edat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when $\gamma = \sqrt{8/3}$, our metric coincides with the $\sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general $\gamma \in (0,2)$, we conjecture that our metric is the Gromov-Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems., Comment: 90 pages, 12 figures; to appear in Inventiones Mathematicae

Optimal transport maps and plans between two absolutely continuous measures $\mu$ and $\nu$ can be approximated by solving semidiscrete or fully discrete optimal transport problems. These two problems ensue from approximating $\mu$ or both $\mu$ and $\nu$ by Dirac measures. Extending an idea from Gigli (2011, On Hölder continuity-in-time of the optimal transport map towards measures along a curve. Proc. Edinb. Math. Soc. (2), 54, 401–409), we characterize how transport plans change under the perturbation of both $\mu$ and $\nu$. We apply this insight to prove error estimates for semidiscrete and fully discrete algorithms in terms of errors solely arising from approximating measures. We obtain weighted $L^2$ error estimates for both types of algorithms with a convergence rate $O(h^{1/2})$. This coincides with the rate in Theorem 5.4 in Berman (2018, Convergence rates for discretized Monge–Ampère equations and quantitative stability of optimal transport. Preprint available at arXiv:1803.00785) for semidiscrete methods, but the error notion is different.

We describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound Poisson distributed. We establish a relation between the limiting return times distribution and the probability of the cluster sizes, where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity. In the special case of periodic points we recover the known Polya-Aeppli distribution which is associated with geometrically distributed cluster sizes. We apply this method to several examples the most important of which is synchronisation of coupled map lattices. For the invariant absolutely continuous measure we establish that the returns to the diagonal is compound Poisson distributed where the coefficients are given by certain integrals along the diagonal.

In this article we are interested in the regularity properties of the probability measure induced by the solution process by a Levy process or a fractional Brownian motion driven Navier–Stokes equations on the two-dimensional torus T . We mainly investigate under which conditions on the characteristic measure of the Levy process or the Hurst parameter of the fractal Brownian motion the law of the projection of u ( t ) onto any finite dimensional F ⊂ L 2 ( T ) is absolutely continuous with respect to the Lebesgue measure on F .

This paper deals with approximation of smooth convex functions f on an interval by convex algebraic polynomials which interpolate f and its derivatives at the endpoints of this interval. We call such estimates “interpolatory”. One important corollary of our main theorem is the following result on approximation of $$f \in \Delta ^{(2)}$$ , the set of convex functions, from $$W^r$$ , the space of functions on $$[-1, 1]$$ for which $$f^{(r-1)}$$ is absolutely continuous and $$\Vert f^{(r)}\Vert_\infty := \mathrm{ess~sup}_{x\in [-1,1]}|f^{(r)}(x)| < \infty $$ : For any $$f \in W^{r} \cap \Delta ^{(2)}, r \in \mathbb {N}$$ , there exists a number $$ \mathcal {N} =\mathcal {N}(f, r)$$ , such that for every $$n \ge \mathcal {N}$$ , there is an algebraic polynomial of degree $$ \le n$$ which is in $$\Delta ^{(2)}$$ and such that $$\begin{aligned}{\left\| \frac{f-P_n}{\varphi ^{r}}\right\| }_{\infty } \le \frac{c(r)}{n^r}\Vert f^{(r)}\Vert _\infty ,\end{aligned}$$ where $$\varphi (x) := \sqrt{1 - x^2}$$ . For $$r = 1$$ and $$r = 2$$ , the above result holds with $$\mathcal {N} = 1$$ and is well known. For $$ r\ge 3$$ , it is not true, in general, with $$\mathcal {N}$$ independent of f.

Let f be the characteristic function of a probability measure μf on ℝn, and let σ > 0. We study the following extrapolation problem: under what conditions on the neighborhood of infinity Vσ = {x ∈ ℝn : |xk| > σ, k = 1, …n} in ℝndoes there exist a characteristic function g on ℝn, such that g = f on Vσ but g ≢ f? Let μf have a nonzero absolutely continuous part with continuous density 𝜑. In this paper, we give certain sufficient conditions on 𝜑 and Vσ under which the latter question has an affirmative answer. We also address the optimality of these conditions. Our results indicate that not only does the size of both Vσ and the support supp 𝜑 matter, but also certain arithmetic properties of supp 𝜑.

We prove a completely new integral criterion for the existence and completeness of the wave operators $W_{\pm}(-\Delta_h,-\Delta_g, I_{g,h})$ corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators $-\Delta_j$, $j=1,2$, that are induced by two quasi-isometric complete Riemannian metrics $g$ and $h$ on an open manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra of $-\Delta_g$ and $-\Delta_h$ to coincide. Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup. Unlike all previous results, our integral criterion only requires some lower control on the Ricci curvatures and some upper control on the heat kernels, but no control at all on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow.

In this work, sharp Wirtinger type inequalities for double integrals are established. As applications, two sharp \v{C}eby\v{s}ev type inequalities for absolutely continuous functions whose second partial derivatives belong to $L^2$ space are proved.

Hunt’s hypothesis (H) and the related Getoor’s conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt’s hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes (Xt) and (Yt), if (Xt) satisfies (H) and (Yt) is locally absolutely continuous with respect to (Xt), then (Yt) satisfies (H). Our second theorem shows that a standard process (Xt) satisfies (H) if and only if $(X_{\tau _{t}})$ satisfies (H) for some (and hence any) subordinator (τt) which is independent of (Xt) and has a positive drift coefficient. Applications of the two theorems are given.

The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal transport is timelike on a set of full measure. In the second part the persistence of absolute continuity along an optimal transportation under obvious assumptions is proven and a solution to the relativistic Monge problem is provided., Comments are welcome

We consider the Dirac operator with a periodic potential on the half-line with the Dirichlet boundary condition at zero. Its spectrum consists of an absolutely continuous part plus at most ...

The main objective of the present paper is to prove the existence of absolutely continuous solutions for an evolution problem in a Hilbert space H, of the form −(du/dt)(t)∈A(t,u(t))u(t) where, for ...

An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vladimirov. This distance as a function of time has bounded variation with an upper bound given by a nonatomic positive Radon measure. By a solution of the inclusion one means a continuous function of bounded variation whose differential measure (Stieltjes measure) is absolutely continuous with respect to the positive Radon measure above, and the values of the density of this differential measure with respect to the Radon measure belong to the right-hand side of the inclusion almost everywhere. Under the traditional assumptions on the perturbation (measurability, Lipschitzianity in the phase variable in the Hausdorff metric, linear growth condition), the existence of solutions is proven and some properties of the solution set are established.

This paper is an extension of known results of Pesin’s entropy formula and SRB measures for random compositions of infinite-dimensional mappings to the continuous-time setting of stochastic flows. Consider a stochastic flow ϕ on a separable infinite dimensional Hilbert space preserving a probability measure μ, which is supported on a random compact set K. We show that if ϕ is C2 (on K) and satisfies some mild integrable conditions of the differentials, then Pesin’s entropy formula holds if μ has absolutely continuous conditional measures along the unstable manifolds. The converse is also true under an additional condition on K when the system has no zero Lyapunov exponent.

A pair $(\Gamma,\Lambda)$, where $\Gamma\subset\mathbb{R}^2$ is a locally rectifiable curve and $\Lambda\subset\mathbb{R}^2$ is a {\em Heisenberg uniqueness pair} if an absolutely continuous (with respect to arc length) finite complex-valued Borel measure supported on $\Gamma$ whose Fourier transform vanishes on $\Lambda$ necessarily is the zero measure. Recently, it was shown by Hedenmalm and Montes that if $\Gamma$ is the hyperbola $x_1x_2=M^2/(4\pi^2)$, where $M>0$ is the mass, and $\Lambda$ is the lattice-cross $(\alpha\mathbb{Z}\times\{0\}) \cup (\{0\}\times\beta\mathbb{Z})$, where $\alpha,\beta$ are positive reals, then $(\Gamma,\Lambda)$ is a Heisenberg uniqueness pair if and only if $\alpha\beta M^2\le4\pi^2$. The Fourier transform of a measure supported on a hyperbola solves the one-dimensional Klein-Gordon equation, so the theorem supplies very thin uniqueness sets for a class of solutions to this equation. The case of the semi-axis $\mathbb{R}_+$ as well as the holomorphic counterpart remained open. In this work, we completely solve these two problems. As for the semi-axis, we show that the restriction to $\mathbb{R}_+$ of the above exponential system spans a weak-star dense subspace of $L^\infty(\mathbb{R}_+)$ if and only if $0, Comment: 41 pages

We consider the problem of finding an optimal transport plan between an absolutely continuous measure and a finitely supported measure of the same total mass when the transport cost is the unsquared Euclidean distance. We may think of this problem as closest distance allocation of some resource continuously distributed over Euclidean space to a finite number of processing sites with capacity constraints. This article gives a detailed discussion of the problem, including a comparison with the much better studied case of squared Euclidean cost. We present an algorithm for computing the optimal transport plan, which is similar to the approach for the squared Euclidean cost by Aurenhammer et al. (Algorithmica 20(1):61–76, 1998) and Mérigot (Comput Graph Forum 30(5):1583–1592, 2011). We show the necessary results to make the approach work for the Euclidean cost, evaluate its performance on a set of test cases, and give a number of applications. The later include goodness-of-fit partitions, a novel visual tool for assessing whether a finite sample is consistent with a posited probability density.

In this paper, we consider r-continued fraction expansions, which are a type of generalized continued fraction expansions. We show that every real number has a valid r-continued fraction expansion. In 1960, Parry proved that a sequence is admissible as the β-expansion of a real number if and only if all of the shifts of the sequence are dominated lexicographically by the sequence obtained from the β-expansion of 1. We prove a similar result for the r-continued fraction expansions. We prove that a sequence is admissible, that is, it comes from the r-continued fraction map, if and only if all of its shifts are alternating-lexicographically less than the sequence obtained from the r-continued fraction expansion of 1. Finally, we show that the r-continued fraction map admits an ergodic absolutely continuous invariant measure that is equivalent to Lebesgue measure.

In this paper, we study the existence of solutions for evolution problems of the form \begin{document}$ -\frac{du}{dr}(t) \in A(t)u(t) + F(t, u(t))+f(t, u(t)) $\end{document} , where, for each \begin{document}$ t $\end{document} , \begin{document}$ A(t) : D(A(t)) \to 2 ^H $\end{document} is a maximal monotone operator in a Hilbert space \begin{document}$ H $\end{document} with continuous, Lipschitz or absolutely continuous variation in time. The perturbation \begin{document}$ f $\end{document} is separately integrable on \begin{document}$ [0, T] $\end{document} and separately Lipschitz on \begin{document}$ H $\end{document} , while \begin{document}$ F $\end{document} is scalarly measurable and separately scalarly upper semicontinuous on \begin{document}$ H $\end{document} , with convex and weakly compact values. Several new applications are provided.

We establish equivalent conditions for the existence of an integrable or locally integrable fixed point for a Markov operator with the maximal support. Maximal support means that almost all initial points will concentrate on the support of the invariant density under the iteration of the process. One of the equivalent conditions for the existence of a locally integrable fixed point is weak almost periodicity of the jump operator with respect to some sweep-out set. This result includes the case of the existence of an absolutely continuous \begin{document}$ \sigma $\end{document} -finite invariant measure when we consider a nonsingular transformation on a probability space. Weak almost periodicity implies the Jacobs-Deleeuw-Glicksberg splitting and we show that constrictive Markov operators which guarantee the spectral decomposition are typical weakly almost periodic operators.

We have analyzed some conditions which are essentially involved in deciding whether or not a probability distribution is unique (moment-determinate) or non-unique (moment-indeterminate) by its moments. We suggest new conditions concerning both absolutely continuous and discrete distributions. By using the new conditions, which are easily checkable, we either establish new results, or extend previous ones in both Hamburger case (distributions on the whole real line) and Stieltjes case (distributions on the positive half-line). Specific examples illustrate both the results and the relationship between the new conditions and previously available conditions., 15 pages