Nontrivial Probability Measure Research Articles

Almost sure global well-posedness for the energy supercritical Schrödinger equations

We consider the Schrödinger equations with arbitrary (large) power non-linearity on the three-dimensional torus. We construct non-trivial probability measures supported on Sobolev spaces and show that the equations are globally well-posed on the supports of these measures, respectively. Moreover, these measures are invariant under the flows that are constructed. Therefore, the constructed solutions are recurrent in time.Also, we show slow growth control on the time evolution of the solutions. A generalization to any dimension is given. Our proof relies on a new approach combining the fluctuation-dissipation method and some features of the Gibbs measures theory for Hamiltonian PDEs. The strategy of the paper applies to other contexts.

Orthogonal polynomials and quadrature rules on the unit circle associated with perturbations of symmetric measures

It was shown recently that given a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, we can associate a unique nontrivial probability measure μ on the unit circle, and conversely. In this paper, we consider the set Q(c) of all nontrivial probability measures for which cn=(−1)nc, with c∈R. We show that every measure μ∈Q(c) can be obtained from a perturbation of a symmetric measure η on [−1,1]. Moreover, the sequence of orthogonal polynomials associated with μ can be given in terms of a perturbation of symmetric orthogonal polynomials associated with η. We also prove that every measure in Q(c) is quasi-symmetric, that is, there exists a complex valued function q(c)(z) satisfying dμ(z)=−q(c)(z)dμ(z¯), and such that q(c)(z)→1 when c→0. Quadrature rules with quasi-symmetric weights are also considered. Finally, some examples of orthogonal polynomials on the unit circle and its associated quasi-symmetric nontrivial probability measures are obtained.

Large Deviations and One-Sided Scaling Limit of Randomized Multicolor Box-Ball System

The basic $\kappa$-color box-ball (BBS) system is an integrable cellular automaton on one dimensional lattice whose local states take $\{0,1,\cdots,\kappa \}$ with $0$ regarded as an empty box. The time evolution is defined by a combinatorial rule of quantum group theoretical origin, and the complete set of conserved quantities is given by a $\kappa$-tuple of Young diagrams. In the randomized BBS, a probability distribution on $\{0,1,\cdots,\kappa \}$ to independently fill the consecutive $n$ sites in the initial state induces a highly nontrivial probability measure on the $\kappa$-tuple of those invariant Young diagrams. In a recent work \cite{kuniba2018randomized}, their large $n$ `equilibrium shape' has been determined in terms of Schur polynomials by a Markov chain method and also by a very different approach of Thermodynamic Bethe Ansatz (TBA). In this paper, we establish a large deviations principle for the row lengths of the invariant Young diagrams. As a corollary, they are shown to converge almost surely to the equilibrium shape at an exponential rate. We also refine the TBA analysis and obtain the exact scaling form of the vacancy, the row length and the column multiplicity, which exhibit nontrivial factorization in a one-parameter specialization.

Translation-invariant probability measures on entire functions

We study non-trivial translation-invariant probability measures on the space of entire functions of one complex variable. The existence (and even an abundance) of such measures was proven by Benjamin Weiss. Answering Weiss’ question, we find a relatively sharp lower bound for the growth of entire functions in the support of such measures. The proof of this result consists of two independent parts: the proof of the lower bound and the construction, which yields its sharpness. Each of these parts combines various tools (both classical and new) from the theory of entire and subharmonic functions and from the ergodic theory. We also prove several companion results, which concern the decay of the tails of non-trivial translation-invariant probability measures on the space of entire functions and the growth of locally uniformly recurrent entire and meromorphic functions.

Periods and factors of weak model sets

There is a renewed interest in weak model sets due to their connection to $\mathcal B$-free systems, which emerged from Sarnak's program on the M\"obius disjointness conjecture. Here we continue our recent investigation [arXiv:1511.06137] of the extended hull ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, a dynamical system naturally associated to a weak model set in an abelian group $G$ with relatively compact window $W$. For windows having a nowhere dense boundary (this includes compact windows), we identify the maximal equicontinuous factor of ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ and give a sufficient condition when ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ is an almost 1:1 extension of its maximal equicontinuous factor. If the window is measurable with positive Haar measure and is almost compact, then the system ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ equipped with its Mirsky measure is isomorphic to its Kronecker factor. For general nontrivial ergodic probability measures on ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, we provide a kind of lower bound for the Kronecker factor. All relevant factor systems are natural $G$-actions on quotient subgroups of the torus underlying the weak model set. These are obtained by factoring out suitable window periods. Our results are specialised to the usual hull of the weak model set, and they are also interpreted for ${\mathcal B}$-free systems.

Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences

It was shown recently that associated with a pair of real sequences $$\{\{c_{n}\}_{n=1}^{\infty }, \{d_{n}\}_{n=1}^{\infty }\}$$ , with $$\{d_{n}\}_{n=1}^{\infty }$$ a positive chain sequence, there exists a unique nontrivial probability measure $$\mu $$ on the unit circle. The Verblunsky coefficients $$\{\alpha _{n}\}_{n=0}^{\infty }$$ associated with the orthogonal polynomials with respect to $$\mu $$ are given by the relation $$\begin{aligned} \alpha _{n-1}=\overline{\tau }_{n-1}\left[ \frac{1-2m_{n}-ic_{n}}{1-ic_{n}}\right] , \quad n \ge 1, \end{aligned}$$ where $$\tau _0 = 1$$ , $$\tau _{n}=\prod _{k=1}^{n}(1-ic_{k})/(1+ic_{k})$$ , $$n \ge 1$$ and $$\{m_{n}\}_{n=0}^{\infty }$$ is the minimal parameter sequence of $$\{d_{n}\}_{n=1}^{\infty }$$ . In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences $$\{c_{n}\}_{n=1}^{\infty }$$ and $$\{m_{n}\}_{n=1}^{\infty }$$ . When the sequence $$ \{c_{n}\}_{n=1}^{\infty }$$ is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the measure in the neighbourhood of $$z= -1$$ . Furthermore, we show that it is possible to generate periodic Verblunsky coefficients by choosing periodic sequences $$\{c_{n}\}_{n=1}^{\infty }$$ and $$\{m_{n}\}_{n=1}^{\infty }$$ with the additional restriction $$c_{2n}=-c_{2n-1}, \, n\ge 1.$$ We also give some results on periodic Verblunsky coefficients from the point of view of positive chain sequences. An example is provided to illustrate the results obtained.

Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle

It is known that given a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, we can associate a unique nontrivial probability measure μ on the unit circle. Precisely, the measure is such that the corresponding Verblunsky coefficients {αn}n=0∞ are given by the relationαn−1=ρ‾n−1[1−2mn−icn1−icn],n≥1, where ρ0=1, ρn=∏k=1n(1−ick)/(1+ick), n≥1 and {mn}n=0∞ is the minimal parameter sequence of {dn}n=1∞. In this paper we consider the space, denoted by Np, of all nontrivial probability measures such that the associated real sequences {cn}n=1∞ and {mn}n=1∞ are periodic with period p, for p∈N. By assuming an appropriate metric on the space of all nontrivial probability measures on the unit circle, we show that there exists a homeomorphism gp between the metric subspaces Np and Vp, where Vp denotes the space of nontrivial probability measures with associated p-periodic Verblunsky coefficients. Moreover, it is shown that the set Fp of fixed points of gp is exactly Vp∩Np and this set is characterized by a (p−1)-dimensional submanifold of Rp. We also prove that the study of probability measures in Np is equivalent to the study of probability measures in Vp. Furthermore, it is shown that the pure points of measures in Np are, in fact, zeros of associated para-orthogonal polynomials of degree p. We also look at the essential support of probability measures in the limit periodic case, i.e., when the sequences {cn}n=1∞ and {mn}n=1∞ are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.

Explicit formulas for OPUC and POPUC associated with measures which are simple modifications of the Lebesgue measure

We consider nontrivial probability measures, obtained as simple modifications of the Lebesgue measure, which include mass points at z=1 and z=i. The orthogonal polynomials, para-orthogonal polynomials and Toeplitz matrices associated with these measures are presented.

Zeros of para–orthogonal polynomials and linear spectral transformations on the unit circle

We study the interlacing properties of zeros of para---orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle dµ and para---orthogonal polynomials associated with a modification of dµ by the addition of a pure mass point, also called Uvarov transformation. Moreover, as a direct consequence of our approach, we present some results related with the Christoffel transformation.

A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula Rn+1(z)=[(1+icn+1)z+(1−icn+1)]Rn(z)−4dn+1zRn−1(z),n≥1, with R0(z)=1 and R1(z)=(1+ic1)z+(1−ic1), where {cn}n=1∞ is a real sequence and {dn}n=1∞ is a positive chain sequence. We establish that there exists a unique nontrivial probability measure μ on the unit circle for which {Rn(z)−2(1−mn)Rn−1(z)} gives the sequence of orthogonal polynomials. Here, {mn}n=0∞ is the minimal parameter sequence of the positive chain sequence {dn}n=1∞. The element d1 of the chain sequence, which does not affect the polynomials Rn, has an influence in the derived probability measure μ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {Mn}n=0∞ is the maximal parameter sequence of the chain sequence, then the measure μ is such that M0 is the size of its mass at z=1. An example is also provided to completely illustrate the results obtained.

Asymptotic behaviour of Sobolev orthogonal polynomials on the unit circle

In this contribution, we deal with analytic properties of sequences of polynomials orthogonal with respect to a Sobolev-type inner product where μ is a non-trivial probability measure supported on the unit circle. We focus our attention on the outer relative asymptotics of these polynomials in terms of those associated with the measure μ. The behaviour of their zeros in terms of the parameter λ is studied in some illustrative examples.

Quasi-stationary distributions and Yaglom limits of self-similar Markov processes

We discuss the existence and characterization of quasi-stationary distributions and Yaglom limits of self-similar Markov processes that reach 0 in finite time. By Yaglom limit, we mean the existence of a deterministic function g and a non-trivial probability measure ν such that the process rescaled by g and conditioned on non-extinction converges in distribution towards ν. We will see that a Yaglom limit exists if and only if the extinction time at 0 of the process is in the domain of attraction of an extreme law and we will then treat separately three cases, according to whether the extinction time is in the domain of attraction of a Gumbel, Weibull or Fréchet law. In each of these cases, necessary and sufficient conditions on the parameters of the underlying Lévy process are given for the extinction time to be in the required domain of attraction. The limit of the process conditioned to be positive is then characterized by a multiplicative equation which is connected to a factorization of the exponential distribution in the Gumbel case, a factorization of a Beta distribution in the Weibull case and a factorization of a Pareto distribution in the Fréchet case.This approach relies partly on results on the tail distribution of the extinction time, which is known to be distributed as the exponential functional of a Lévy process. In that aim, new results on such tail distributions are given, which may be of independent interest.

Zeros of Sobolev orthogonal polynomials on the unit circle

In this contribution, we study the sequences of orthogonal polynomials with respect to the Sobolev inner product $$ \langle{f,g}\rangle_{S}:=\int_{\mathbb{T}} f(z) \overline{g(z)} d\mu(z) + \lambda f^{(j\,)}(\alpha)\overline{g^{(j\,)}(\alpha)}, $$ where μ is a nontrivial probability measure supported on the unit circle, ?????, $\lambda \in {\mathbb{R}}_+\backslash\{0\}$ , and j????. In particular, we analyze the behavior of their zeros when n and ? tend to infinity, respectively. We also provide some numerical examples to illustrate the behavior of these zeros with respect to ?.

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{\delta,\beta}^{(n)} \prod_{1\leq k -1/2$. If $e$ is a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well parameterized by its Verblunsky coefficients $(\alpha_0, ..., \alpha_{n-1})$. We introduce here a deformation $(\gamma_0, >..., \gamma_{n-1})$ of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product $r(\gamma_0)... r(\gamma_{n-1})$ of elementary reflections parameterized by these coefficients. If $\gamma_0, ..., \gamma_{n-1}$ are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above. These deformed Verblunsky coefficients also allow to prove that, in the regime $\delta = \delta(n)$ with $\delta(n)/n \to \dd$, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distribution. Formula with Formula . If e is a cyclic vector for a unitary n x n matrix U, the spectral measure of the pair (U, e) is well parameterized by its Verblunsky coefficients ({alpha}0, ..., {alpha}n–1). We introduce here a deformation ({gamma}0, ..., {gamma}n–1) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r({gamma}0)··· r({gamma}n–1) of elementary reflections parameterized by these coefficients. If {gamma}0, ..., {gamma}n–1 are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above. These deformed Verblunsky coefficients also allow us to prove that, in the regime {delta} = {delta} (n) with {delta} (n)/ n -> β d/2, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distribution.

Szegő transformations and Nth order associated polynomials on the unit circle

In this paper we analyze the Stieltjes functions defined by the Szegő inverse transformation of a nontrivial probability measure supported on the unit circle such that the corresponding sequence of orthogonal polynomials is defined by either backward or forward shifts in their Verblunsky parameters. Such polynomials are called anti-associated (respectively associated) orthogonal polynomials. Thus, rational spectral transformations appear in a natural way.

States on Bold Algebras: Categorical Aspects

We study bold algebras and states on bold algebras in the context of transition from classical probability theory to fuzzy probability theory. Our aim is to point out the role of bold algebras and states on bold algebras in a categorical approach to probability theory. In particular, we formulate several fundamental questions related to basic probability notions and constructions and provide possible answers in terms of bold algebras and states on bold algebras. We show that the category ID of D-posets of fuzzy sets and sequentially continuous difference homomorphisms can serve as a base category in which both classical and fuzzy probability theory can be developed and generalized. Classical and fuzzy random events such as fields of sets and measurable real-valued functions into the interval [0,1], considered as bold algebras, become special objects. Observables, considered as morphisms between objects, become dual to generalized random variables. States become morphisms into [0,1], considered as an object of ID. Properties of objects of ID follow from classical theorems of analysis such as the Lebesgue Dominated Convergence Theorem (states are sequentially continuous) and categorical constructions such as the product (the structure of a probability domain is completely determined by the states as the initial structure). We prove that each generated Łukasiewicz tribe is the epireflection of its underlying Butnariu–Klement σ-field of sets. This helps to understand the transition from classical crisp random events to fuzzy random events. Indeed, the corresponding fuzzification is necessary to cover generalized random variables having a quantum character, i.e. fuzzy random variables in the Gudder–Bugajski sense sending a classical elementary event (point measure) to a non-trivial probability measure.

Anomalous Dissipation in a Stochastically Forced Infinite-Dimensional System of Coupled Oscillators

We study a system of stochastically forced infinite-dimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has a nontrivial invariant probability measure. This phenomenon, which has no finite dimensional equivalent, is due to the appearance of some anomalous dissipation mechanism which transports energy to infinity. This prevents the energy from building up locally and allows the system to converge to the invariant measure. The invariant measure is constructed explicitly and some of its properties are analyzed.

Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems

Let \(\mu\) be a nontrivial probability measure on the unit circle \(\partial\mbox{\bf D}, w\) the density of its absolutely continuous part, \(\alpha_n\) its Verblunsky coefficients, and \(\Phi_n\) its monic orthogonal polynomials. In this paper we compute the coefficients of \(\Phi_n\) in terms of the \(\alpha_n\). If the function \(\log w\) is in \(L^1(d\theta)\), we do the same for its Fourier coefficients. As an application we prove that if \(\alpha_n\in\ell^4\) and if \(Q(z) \equiv\sum_{m=0}^N q_m z^m\) is a polynomial, then with \(\bar Q(z) \equiv\sum_{m=0}^N \bar q_m z^m\) and S the left-shift operator on sequences we have $$| Q(e^{i\theta}) |^2\log w(\theta) \in L^1(d\theta) \quad \Leftrightarrow \quad \{\bar Q(S)\alpha\}_n\in\ell^2.$$ We also study relative ratio asymptotics of the reversed polynomials \(\Phi_{n+1}^*(\mu)/\Phi_n^*(\mu)-\Phi_{n+1}^*(\nu)/\Phi_n^*(\nu)\) and provide a necessary and sufficient condition in terms of the Verblunsky coefficients of the measures \(\mu\) and \(\nu\) for this difference to converge to zero uniformly on compact subsets of \(\mbox{\bf D}\).

Stochastic Processes for Bounded Noise

The scalar differential inclusion x˙ ∈ f(x) + g(x) u, u ∈ [-1,1], x(0) = x 0 (0.1) is considered as a model of the dynamical system x˙ = f(x) perturbed by the bounded noise g(x)u, u ∈ [-1,1], and the problem of constructing a nontrivial probability measure on the set {\cal S} of solutions to (0.1) is studied. In particular, it is shown that: (i) every Markov process whose probability measure is supported on {\cal S} is degenerate, in a sense to be specified (see Theorem 3.1); (ii) given a flow of probability measures μt on the reachable sets R t of (0.1), satisfying a certain compatibility condition, a Markov process X t is constructed such that its marginals are exactly μt and (0.1) is satisfied “from one side” (see Theorem 4.1); its finite-dimensional distributions are computed and the regularity of its sample paths is investigated (see Section 5.2); (iii) given a process of a type previously considered, another process Y t is constructed through its finite-dimensional distributions, and its distribution is shown to be supported exactly on {\cal S}. Finally, a model example is considered (see Section 7).

Spectral properties for a class of continuous state branching processes with immigration

We explicitly construct nontrivial invariant probability measures for a class of continuous state branching processes with immigration. The class of these measures include random Gamma measures and path space measures of Lévy subordinators as particular examples. Using the explicit construction we study long-time behaviour and hypercontractivity of the transition semigroups in corresponding L 2-spaces.