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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.

We prove an irreducible decomposition for Markov processes associated with quasi-regular symmetric Dirichlet forms or local semi-Dirichlet forms under the absolute continuity condition of transition probability with respect to the underlying measure. We do not assume the conservativeness nor the existence of invariant measures for the processes. As applications, we establish a concrete expression for Chacon–Ornstein type ratio ergodic theorem for such Markov processes and show a compactness of semi-groups under the Green-tightness of measures in the framework of symmetric resolvent strong Feller processes without irreducibility.

We consider Dirac operators with dislocation potentials on the line. The dislocation potential is a periodic potential for x 0 and the same potential but shifted by t ∈ R for x > 0 . Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each gap. Its resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface. We prove that it has only two simple poles on each open gap: eigenvalues or resonances. These poles are called states and there are no other poles. We prove: 1) states are continuous functions of t, and we obtain their local asymptotics; 2) for each t states in the gap are distinct; 3) states can be monotone or non-monotone functions of t; 4) we construct examples of operators with different types of states in gaps.

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.

In this paper, we propose three types of absolute continuity for monotone measures and present some of their basic properties. By means of these three types of absolute continuity, we establish generalized Egoroff's theorem, generalized Riesz's theorem and generalized Lebesgue's theorem in the framework involving the ordered pair of monotone measures. The Egoroff theorem, the Riesz theorem and the Lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case. These three generalized convergence theorems include as special cases several previous versions of Egoroff-like theorem, Riesz-like theorem and Lebesgue-like theorem for monotone measures, respectively.

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 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

In this article, we address the problem of stabilizing a discrete-time deterministic nonlinear control system to a target invariant measure using time-invariant stochastic feedback laws. This problem can be viewed as an extension of the problem of designing the transition probabilities of a Markov chain so that the process is exponentially stabilized to a target stationary distribution. Alternatively, it can be seen as an extension of the classical control problem of asymptotically stabilizing a discrete-time system to a single point, which corresponds to the Dirac measure in the measure stabilization framework. We assume that the target measure is supported on the entire state space of the system and is absolutely continuous with respect to the Lebesgue measure. Under the condition that the system is locally controllable at every point in the state space within one time step, we show that the associated measure stabilization problem is well-posed. Given this well-posedness result, we then frame an infinite-dimensional convex optimization problem to construct feedback control laws that stabilize the system to a target invariant measure at a maximized rate of convergence. We validate our optimization approach with numerical simulations of two-dimensional linear and nonlinear discrete-time control systems.

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 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 .

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 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.

Partially observed Markov decision processes (POMDPs) are an important class of control problems that are ubiquitous in a wide range of fields. Unfortunately, these problems are generally intractable, so, in general, we must be satisfied with suboptimal policies. But how do we evaluate the quality of these policies? This question has been addressed in recent years in the Markov decision process (MDP) literature through the use of information-relaxation-based duality, where the nonanticipativity constraints are relaxed, but a penalty is imposed for violations of these constraints. In this paper, we extend the information relaxation approach to POMDPs. It is of course well known that the belief-state formulation of a POMDP is an MDP, and therefore, the previously developed results for MDPs also apply to POMDPs. Under the belief-state formulation, we use recently developed change-of-measure arguments to solve the so-called inner problems, and we use standard filtering arguments to identify the appropriate Radon–Nikodym derivatives. We also show, however, that dual bounds can also be constructed without resorting to the belief-state formulation. In this case, change-of-measure arguments are required for the evaluation of the so-called dual feasible penalties rather than for the solution of the inner problems. We compare dual bounds for both formulations and argue that, in general, the belief-state formulation provides tighter bounds. The second main contribution of this paper is to show that several value function approximations for POMDPs are in fact supersolutions . This is of interest because it can be particularly advantageous to construct penalties from supersolutions, since absolute continuity (of the change of measure) is no longer required, and therefore, significant variance reduction can be achieved when estimating the duality gap directly. Dual bounds constructed from supersolution-based penalties are also guaranteed to provide tighter bounds than the bounds provided by the supersolutions themselves. We use applications from robotic navigation and telecommunication to demonstrate our results.

We consider scattering matrix for Schrodinger-type operators on $$\mathbb {R}^d$$ with perturbation $$V(x)=O(\langle x \rangle ^{-1})$$ as $$|x|\rightarrow \infty $$ . We show that the scattering matrix (with time-independent modifiers) is a pseudodifferential operator and analyze its spectrum. We present examples of which the spectrum of the scattering matrices has dense point spectrum, and absolutely continuous spectrum, respectively. These give a partial answer to an open question posed by Yafaev (Scattering theory: some old and new problems. Springer Lecture Notes in Mathematical, vol 1735, 2000).

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.

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.

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 ...

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.

We revisit and update estimating variances, fundamental quantities in a time series forecasting approach called kriging, in time series models known as FDSLRMs, whose observations can be described by a linear mixed model (LMM). As a result of applying the convex optimization, we resolved two open problems in FDSLRM research: (1) theoretical existence and equivalence between two standard estimation methods—least squares estimators, non-negative (M)DOOLSE, and maximum likelihood estimators, (RE)MLE, (2) and a practical lack of free available computational implementation for FDSLRM. As for computing (RE)MLE in the case of n observed time series values, we also discovered a new algorithm of order $${\mathcal {O}}(n)$$ , which at the default precision is $$10^7$$ times more accurate and $$n^2$$ times faster than the best current Python(or R)-based computational packages, namely CVXPY, CVXR, nlme, sommer and mixed. The LMM framework led us to the proposal of a two-stage estimation method of variance components based on the empirical (plug-in) best linear unbiased predictions of unobservable random components in FDSLRM. The method, providing non-negative invariant estimators with a simple explicit analytic form and performance comparable with (RE)MLE in the Gaussian case, can be used for any absolutely continuous probability distribution of time series data. We illustrate our results via applications and simulations on three real data sets (electricity consumption, tourism and cyber security), which are easily available, reproducible, sharable and modifiable in the form of interactive Jupyter notebooks.

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

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.

This paper investigates the problem of nonsmooth feedback stabilization for the higher order uncertain chain of integrators. For achieving the specified goal, the integral term of classical Proportional-Integral (PI) controller is replaced by an integral of the discontinuous function. Replacing this integrator, the overall control becomes absolutely continuous rather than discontinuous as in the first order sliding mode control. With this proposed scheme, the property of invariance concerning the matched Lipschitz uncertainty is still preserved. The main technical contribution of the paper is a sound and non-trivial Lyapunov analysis of the closed loop system controlled by nonsmooth PI controller. The effectiveness of the proposed controller is illustrated with the help of numerical simulation on the magnetic suspension system.

We define a class of periodic electric potentials for which the spectrum of the two-dimensional Schrodinger operator is absolutely continuous in the case of a homogeneous magnetic field B with a rational flux η = (2π)−1Bυ(K), where υ(K) is the area of an elementary cell K in the lattice of potential periods. Using properties of functions in this class, we prove that in the space of periodic electric potentials in Lloc2(ℝ2) with a given period lattice and identified with L2(K), there exists a second-category set (in the sense of Baire) such that for any electric potential in this set and any homogeneous magnetic field with a rational flow η, the spectrum of the two-dimensional Schrodinger operator is absolutely continuous.

We consider a simple example of a one-parameter family of random maps in the interval that exhibits a phase transition phenomenon in the sense of a spontaneous transition from non-existence to existence of an absolutely continuous invariant measure by changing the parameter. For this example of random maps, we estimate numerically the critical value at which the so-called transition occurs. This is done through the numerical computation of its invariant densities and the Lyapunov exponent. We further investigate the behavior of the rate of decay of correlations for the different values of the parameter, changing from power-law-like to exponential-like behavior. We also study a family of random maps with a non-expansive branch having no phase transition. For this family of random maps, we compute the invariant densities for all values of the parameter.

We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesaro–Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley–Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.

Let $$\mathcal{G}$$ be a metric, finite, noncompact, and connected graph with finitely many edges and vertices. Assume that the length of at least one of the edges is infinite. The main object of this paper is the Hamiltonian $${{{\mathbf{H}}}_{\alpha }}$$ associated in $${{L}^{2}}(\mathcal{G};{{\mathbb{C}}^{m}})$$ with a matrix Sturm–Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian $${{{\mathbf{H}}}_{\alpha }}$$ and any other self-adjoint realization of the Sturm–Liouville expression is empty. We also indicate conditions on the graph ensuring that the positive part of $${\mathbf{H}_{\alpha }}$$ is purely absolutely continuous. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of the operator $${{{\mathbf{H}}}_{\alpha }}$$ is obtained. Additionally, a formula is found for the scattering matrix of the pair $$\{ {{{\mathbf{H}}}_{\alpha }},{{{\mathbf{H}}}_{D}}\} $$, where $${{{\mathbf{H}}}_{D}}$$ is the operator of the Dirichlet problem on the graph.

We consider de Finetti’s problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

The main results of the Aronszajn–Donoghue–Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations $$\tilde {A}$$ of an operator $${{A}_{0}} = A_{0}^{ * }$$ defined on a Hilbert space $$\mathfrak{H}$$ . By applying the technique of boundary triplets, the singular continuous and point spectra of extensions AB of a symmetric operator A are described in terms of the Weyl function $$M( \cdot )$$ of the pair {A, A0} and an n-dimensional boundary operator B = B*. Assuming that the multiplicity of the singular spectrum of A0 is maximal, we establish the orthogonality of the singular parts $$E_{{{{A}_{B}}}}^{s}$$ and $$E_{{{{A}_{0}}}}^{s}$$ of the spectral measure $${{E}_{{{{A}_{B}}}}}$$ and $${{E}_{{{{A}_{0}}}}}$$ of the operators AB and A0, respectively. The multiplicity of the singular spectrum of special extensions of direct sums $$A = {{A}^{{(1)}}} \oplus {{A}^{{(2)}}}$$ is investigated. In particular, it is shown that this multiplicity cannot be maximal, as distinguished from the multiplicity of the absolutely continuous spectrum. This result generalizes and refines the Kac theorem on the multiplicity of the singular spectrum of the Schrodinger operator on the line.

In this paper, we establish, in the setting of infinite dimensional Hilbert space, a new existence result for nonconvex sweeping process with right uniformly lower semicontinuous sets. This class of sets is more general than the classical assumption of absolutely continuous sets, and contains hyperplanes and half-spaces. Further, we apply on the problem an unbounded set-valued perturbation and we state the existence of solution.

The present paper offers a new approach to the construction of the coordinate asymptotics of the kernel of the resolvent of the Schrodinger operator in the scattering problem of three onedimensional quantum particles with short-range pair potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrodinger operator can be constructed. In the paper, the possibility of a generalization of the suggested approach to the case of the scattering problem of N particles with arbitrary masses is discussed.

We consider a family of convergent positive normed series with real terms defined by the conditions where (an) is a nondecreasing sequence of real numbers. The structural properties of these series are investigated. For a special case, namely, $$ \left({a}_n\right)={2}^{n-1},{c}_n=\left(n+1\right){\tilde{r}}_n,n\in \mathbb{N} $$ , we study the geometry of the series (i.e., the properties of cylindrical sets, metric relations generated by them, and topological and metric properties of the set of all incomplete sums of the series). For the infinite Bernoulli convolution determined by this series, we describe its Lebesgue structure (discrete, absolutely continuous, and singular components) and spectral properties, as well as the behavior of the absolute value of the characteristic function at infinity. We also study finite self-convolutions of the distributions of this kind.

We consider Anosov diffeomorphisms on $\mathbb{T}^3$ such that the tangent bundle splits into three subbundles $E^s_f \oplus E^{wu}_f \oplus E^{su}_f.$ We show that if $f$ is $C^r, r \geq 2,$ volume preserving, then $f$ is $C^1$ conjugated with its linear part $A$ if and only if the center foliation $\mathcal{F}^{wu}_f$ is absolutely continuous and the equality $\lambda^{wu}_f(x) = \lambda^{wu}_A,$ between center Lyapunov exponents of $f$ and $A,$ holds for $m$ a.e. $x \in \mathbb{T}^3.$ We also conclude rigidity of derived from Anosov diffeomorphism, assuming an strong absolute continuity property (Uniform bounded density property) of strong stable and strong unstable foliations.

We consider the equation Δu = Vu in the half-space ${\open R}_ + ^d $, d ⩾ 2 where V has certain periodicity properties. In particular, we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation Δu = Vu is studied as part of a broader class of elliptic evolution equations.

In recent collaborative work we studied existence and uniqueness of a Lagrangian description for absolutely continuous curves in spaces of Borel probabilities on the real line with finite moments of given order. Of course, a measurable velocity driving the evolution in Eulerian coordinates is necessary to define the Eulerian and Lagrangian descriptions of fluid flow; here we prove that in this case it is also sufficient for a Lagrangian description. More precisely, we argue that the existence of the integrable velocity along an absolutely continuous curve in the set of Borel probabilities on the line is enough to produce a canonical Lagrangian description for the curve; this is given by the family of optimal maps between the uniform distribution on the unit interval and the measures on the curve. Moreover, we identify a necessary and sufficient condition on said family of optimal maps which ensures that the measurable velocity along the curve exists.

We introduce two classes of Metropolis-Hastings algorithms for sampling target measures that are absolutely continuous with respect to non-Gaussian prior measures on infinite-dimensional Hilbert spaces. In particular, we focus on certain classes of prior measures for which prior-reversible proposal kernels of the autoregressive type can be designed. We then use these proposal kernels to design algorithms that satisfy detailed balance with respect to the target measures. Afterwards, we introduce a new class of prior measures, called the Bessel-K priors, as a generalization of the gamma distribution to measures in infinite dimensions. The Bessel-K priors interpolate between well-known priors such as the gamma distribution and Besov priors and can model sparse or compressible parameters. We present concrete instances of our algorithms for the Bessel-K priors in the context of numerical examples in density estimation, finite-dimensional denoising and deconvolution on the circle., Minor revisions

Let \begin{document}$ g $\end{document} be a smooth expanding map of degree \begin{document}$ D $\end{document} which maps a circle to itself, where \begin{document}$ D $\end{document} is a natural number greater than \begin{document}$ 1 $\end{document} . It is known that the Lyapunov exponent of \begin{document}$ g $\end{document} with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to \begin{document}$ \log D $\end{document} which, in addition, is less than or equal to the Lyapunov exponent of \begin{document}$ g $\end{document} with respect to the measure of maximal entropy. Moreover, the equalities can only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree \begin{document}$ D $\end{document} on a circle.

We are considering partially hyperbolic diffeomorphims of the torus, with \begin{document}${\rm{dim}}(E^c) > 1.$\end{document} We prove, under some conditions, that if the all center Lyapunov exponents of the linearization \begin{document}$A,$\end{document} of a DA-diffeomorphism \begin{document}$f,$\end{document} are positive and the center foliation of \begin{document}$f$\end{document} is absolutely continuous, then the sum of the center Lyapunov exponents of \begin{document}$f$\end{document} is bounded by the sum of the center Lyapunov exponents of \begin{document}$A.$\end{document} After, we construct a \begin{document}$C^1-$\end{document} open class of volume preserving DA-diffeomorphisms, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each \begin{document}$f$\end{document} in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of \begin{document}$f$\end{document} is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of \begin{document}$f$\end{document} is non absolutely continuous. We still build an example of a DA-diffeomorphism, such that the disintegration of volume along the two dimensional, non compact center foliation is neither Lebesgue nor atomic.

We study the quantum limits of Eisenstein series off the critical line for $\mathrm{PSL}_{2}(\mathcal{O}_{K})\backslash\mathbb{H}^{3}$, where $K$ is an imaginary quadratic field of class number one. This generalises the results of Petridis, Raulf and Risager on $\mathrm{PSL}_{2}(\mathbb{Z})\backslash\mathbb{H}^{2}$. We observe that the measures $\lvert E(p,\sigma_{t}+it)\rvert^{2}d\mu(p)$ become equidistributed only if $\sigma_{t}\rightarrow 1$ as $t\rightarrow\infty$. We use these computations to study measures defined in terms of the scattering states, which are shown to converge to the absolutely continuous measure $E(p,3)d\mu(p)$ under the GRH.

Using an explicit catching-up algorithm, we prove the existence of absolutely continuous as well as bounded variation continuous solutions to a second-order perturbed Moreau’s sweeping process with the normal cone of a subsmooth moving set, which depends both on the time and on the state.

We consider a general piecewise deterministic Markov process (PDMP) $X=\{X_t\}_{t\geqslant~0}$ with a measure-valued generator $\mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continuous. A general form of the exponential martingales that are associated with $X$ is given by By considering this exponential martingale to be a likelihood-ratio process, we define a new probability measure and show that the process $X$ is still a general PDMP under the new probability measure. We additionally find the new measure-valued generator and its domain. To illustrate our results, we investigate the continuous-time compound binomial model.

We use Lebesgue decomposition of two probability measures on a measurable space to obtain conditions for their equivalence and singularity in terms of the density of the absolutely continuous part of one probability measure with respect to the other. This allows us to obtain simple proofs of Kakutani’s theorem on product measures (Kakutani, 1948) and an extension of the result of Shepp (1966). In addition, using the density form of two finite-dimensional Gaussian measures, we derive analogues of major results on equivalence and singularity (Parzen, 1963; Kallianpur and Oodaira, 1963; Rozanov, 1968) for Gaussian random fields. These can be used to study the interpolation of the spatial data (Stein, 1999).