Invariant Probability Density Research Articles

Short Remark on (p1,p2,p3)-Complex Numbers

Movements on surfaces of centered Euclidean spheres and changes between those with different radii mean complex multiplication in R3. Here, the Euclidean norm, which generates the spheres, is replaced with an inhomogeneous functional and a product is introduced in a geometric analogy. Because a change in the radius now leads to a change in the shape of the sphere, a three-dimensional dynamic complex structure is created. Statements about invariant probability densities, generalized uniform distributions on generalized spheres, geometric measure representations, and dynamic ball numbers associated with this structure are also presented.

Stochastic generalized Kolmogorov systems with small diffusion: I. Explicit approximations for invariant probability density function

This paper presents Part I of a two-part series on studying the long-term coexistence states of stochastic generalized Kolmogorov systems with small diffusion. Part I establishes a mathematical framework for approximating the invariant probability measures (IPMs) and density functions (IPDFs) of these systems, while Part II will focus on analyzing their non-autonomous periodic counterparts. Compared with the existing approximation methods available only for systems with non-degenerate linear diffusion, this paper introduces two new and easily implementable approximation methods, the log-normal approximation (LNA) and updated normal approximation (uNA), which can be used for systems with not only non-degenerate but also degenerate diffusion. Moreover, we utilize the Kolmogorov-Fokker-Planck (KFP) operator and matrix algebra to develop algorithms for calculating the associated covariance matrix and verifying its positive definiteness. Our new approximation methods exhibit good accuracy in approximating the IPM and IPDF at both local and global levels, and significantly relaxes the minimal criteria for positive definiteness of the solution of the continuous-type Lyapunov equation. We demonstrate the utility of our methods in several application examples from biology and ecology.

Lévy processes in bounded domains: path-wise reflection scenarios and signatures of confinement

We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric α-stable Lévy processes, whose permanent residence in a finite interval on a line is secured by a two-sided reflection. Depending on the specific reflection ‘mechanism’, the inferred jump-type processes differ in their spectral and statistical characteristics, like e.g. relaxation properties, and functional shapes of invariant (equilibrium, or asymptotic near-equilibrium) probability density functions in the interval. The analysis is carried out in conjunction with attempts to give meaning to the notion of a reflecting Lévy process, in terms of the domain of its motion generator, to which an invariant pdf (actually an eigenfunction) does belong.

On the asymptotic behavior of the Diaconis–Freedman chain in a multi-dimensional simplex

AbstractWe give a setting of the Diaconis–Freedman chain in a multi-dimensional simplex and consider its asymptotic behavior. By using techniques from random iterated function theory and quasi-compact operator theory, we first give some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure and, in particular cases, explicit formulas for the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimension two. Some other settings of the chain are also discussed.

Range Sidelobe Suppression Approach for SAR Images Using Chaotic FM Signals

Range sidelobe is very common in synthetic aperture radar (SAR) images, particularly when imaging scene includes strongly scattering targets such as ships or complex buildings. As a kind of interference, it may reduce the image quality and hinder the image interpretation. Hence, range sidelobe suppression is an important mission for SAR images. The main task of mitigating the sidelobe is how to achieve the most effective suppression with the minimal resolution loss and signal-to-noise ratio (SNR) loss. However, the widely recognized classic method, spatially variant apodization (SVA), still has a lot of residual sidelobe energy and other problems. This article proposes a novel suppression approach based on time-variant transmission of chaotic frequency modulation (CFM) signals. The key is to build an appropriate transmitted signal set, where the signals are generated by various chaotic initial states and the same special map with low mixing rate and uniform invariant probability density (IPD). Due to their beneficial autocorrelation properties, the proposed approach achieves superior performance in range sidelobe suppression and resolution preservation. More importantly, it maintains the energy of the signals and overcomes the SNR loss that occurs in some classic methods, such as spectral weighting (SW) and SVA. In addition, it is suitable for both vertical and squint side-looking mode and can well reconstruct the weakly scattering targets which are severely disturbed by range sidelobe. All of them are validated by comparative experiments.

Thin-shell theory for rotationally invariant random simplices

For fixed functions G,H:[0,∞)→[0,∞), consider the rotationally invariant probability density on Rn of the form μn(ds)=1 ZnG(‖s‖2)e−nH(‖s‖2)ds. We show that when n is large, the Euclidean norm ‖Yn‖2 of a random vector Yn distributed according to μn satisfies a thin-shell property, in that its distribution is highly likely to concentrate around a value s0 minimizing a certain variational problem. Moreover, we show that the fluctuations of this modulus away from s0 have the order 1∕n and are approximately Gaussian when n is large. We apply these observations to rotationally invariant random simplices: the simplex whose vertices consist of the origin as well as independent random vectors Y1n,…,Ypn distributed according to μn, ultimately showing that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior. Our class of measures includes the Gaussian distribution, the beta distribution and the beta prime distribution on Rn, provided a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thäle [Limit theorems for random simplices in high dimensions, ALEA 16, 141–177 (2019)]. Finally, the volumes of random simplices may be related to the determinants of random matrices, and we use our methods with this correspondence to show that if An is an n×n random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants c0,c1∈(0,∞) and an absolute constant C∈(0,∞) such that sups∈RPlogdet(An)−log(n−1)!−c0 1 2logn+c1<s−∫−∞se−u2∕2du 2π<C log3∕2n, sharpening the 1∕log1∕3+o(1)n bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42(1) (2014), 146–167].

The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation

&lt;abstract&gt;&lt;p&gt;The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.&lt;/p&gt;&lt;/abstract&gt;

Singular invariant probability densities in dissipative dynamical systems

Dynamical systems are considered dissipative when they asymptotically produce a net contraction of volume in phase space, so that V(t) → 0 as , where V(t) is the volume at time t into which the dynamics transforms a nonzero initial volume V0. The trajectories are thereby forced to asymptotically converge toward time-invariant subsets of measure zero, which are typically strange attractors with fractal structure. Since a probability density is a probability per unit volume in phase space, any initially smooth probability density must become singular as , so invariant probability densities do not exist as proper mathematical functions in dissipative systems. For this reason, the statistical behaviour of such systems is generally analysed in terms of invariant probability measures, which remain well defined even when densities do not. The purpose of this paper is to point out that invariant probability densities in dissipative systems can nevertheless be rigorously defined as singular generalised functions which (a) are natural generalisations of the Dirac delta function, (b) are easier to work with than invariant measures, (c) satisfy the same stationary Liouville equation as continuous invariant densities, and (d) can therefore be formally manipulated in the same way. These simplifying features facilitate the analysis of the statistical properties of dissipative dynamical systems, thereby making abstract measure-theoretical theories of such systems more accessible to students and nonspecialists.

Kemeny’s constant for one-dimensional diffusions

Let $X(\cdot )$ be a non-degenerate, positive recurrent one-dimensional diffusion process on $\mathbb{R} $ with invariant probability density $\mu (x)$, and let $\tau _{y}=\inf \{t\ge 0: X(t)=y\}$ denote the first hitting time of $y$. Let $\mathcal{X} $ be a random variable independent of the diffusion process $X(\cdot )$ and distributed according to the process’s invariant probability measure $\mu (x)dx$. Denote by $\mathcal{E} ^{\mu }$ the expectation with respect to $\mathcal{X} $. Consider the expression \[ \mathcal{E} ^{\mu }E_{x}\tau _{\mathcal{X} }=\int _{-\infty }^{\infty }(E_{x}\tau _{y})\mu (y)dy, x\in \mathbb{R} . \] In words, this expression is the expected hitting time of the diffusion starting from $x$ of a point chosen randomly according to the diffusion’s invariant distribution. We show that this expression is constant in $x$, and that it is finite if and only if $\pm \infty $ are entrance boundaries for the diffusion. This result generalizes to diffusion processes the corresponding result in the setting of finite Markov chains, where the constant value is known as Kemeny’s constant.

On the stochastic parametrization of short‐scale processes

Stochastic parametrization of short‐scale processes is revisited in an idealized setting in which the short scales are manifested as small‐amplitude deterministic forcings acting on the evolution laws of the resolved variables. Conditions under which the invariant probability density of the original deterministic system tends to that induced by the action of appropriately parametrized Gaussian white noise are identified. On the other hand, substantial differences are revealed between the deterministic description and its stochastic substitute when time‐dependent properties are considered such as short‐time behaviour of the variance, recurrence and first passage time statistics. Illustrations and explicit evaluations are provided on low‐order model systems.

Impact of metallicity and star formation rate on the time-dependent, galaxy-wide stellar initial mass function

The stellar initial mass function (IMF) is commonly assumed to be an invariant probability density distribution function of initial stellar masses. These initial stellar masses are generally represented by the canonical IMF, which is defined as the result of one star formation event in an embedded cluster. As a consequence, the galaxy-wide IMF (gwIMF) should also be invariant and of the same form as the canonical IMF; gwIMF is defined as the sum of the IMFs of all star-forming regions in which embedded clusters form and spawn the galactic field population of the galaxy. Recent observational and theoretical results challenge the hypothesis that the gwIMF is invariant. In order to study the possible reasons for this variation, it is useful to relate the observed IMF to the gwIMF. Starting with the IMF determined in resolved star clusters, we apply the IGIMF-theory to calculate a comprehensive grid of gwIMF models for metallicities, [Fe/H] ∈ (−3, 1), and galaxy-wide star formation rates (SFRs), SFR ∈ (10−5, 105) M⊙ yr−1. For a galaxy with metallicity [Fe/H] &lt; 0 and SFR &gt; 1 M⊙ yr−1, which is a common condition in the early Universe, we find that the gwIMF is both bottom light (relatively fewer low-mass stars) and top heavy (more massive stars), when compared to the canonical IMF. For a SFR &lt; 1 M⊙ yr−1 the gwIMF becomes top light regardless of the metallicity. For metallicities [Fe/H] &gt; 0 the gwIMF can become bottom heavy regardless of the SFR. The IGIMF models predict that massive elliptical galaxies should have formed with a gwIMF that is top heavy within the first few hundred Myr of the life of the galaxy and that it evolves into a bottom heavy gwIMF in the metal-enriched galactic centre. Using the gwIMF grids, we study the SFR−Hα relation and its dependency on metallicity and the SFR. We also study the correction factors to the Kennicutt SFRK − Hα relation and provide new fitting functions. Late-type dwarf galaxies show significantly higher SFRs with respect to Kennicutt SFRs, while star-forming massive galaxies have significantly lower SFRs than hitherto thought. This has implications for gas-consumption timescales and for the main sequence of galaxies. We explicitly discuss Leo P and ultra-faint dwarf galaxies.

Random Evolutionary Dynamics Driven by Fitness and House-of-Cards Mutations: Sampling Formulae

We first revisit the multi-allelic mutation-fitness balance problem, especially when mutations obey a house of cards condition, where the discrete-time deterministic evolutionary dynamics of the allelic frequencies derives from a Shahshahani potential. We then consider multi-allelic Wright-Fisher stochastic models whose deviation to neutrality is from the Shahsha-hani mutation/selection potential. We next focus on the weak selection, weak mutation cases and, making use of a Gamma calculus, we compute the normalizing partition functions of the invariant probability densities appearing in their Wright-Fisher diffusive approximations. Using these results, Generalized Ewens sampling formulae (ESF) from the equilibrium distributions are derived. We start treating the ESF in the mixed mutation/selection potential case and then we restrict ourselves to the ESF in the simpler house-of-cards mutations only situation. We also address some issues concerning sampling problems from infinitely-many alleles weak limits.

Chaotic time hopping based multiple access in BPSK-UWB system

In this paper we propose a binary phase shift keying-ultra wide band (BPSK-UWB) system in which multiple access is defined by chaotic time hopping. We showed that the performance of the proposed system depends on the invariant probability density of the used chaotic transformation; more over when the width of the pulse is little enough with respect to the period of one bit then the performance of the proposed system approaches the performance of the one user BPSK system. This result is confirmed by simulating the system and computing the binary error rate (BER).

Transformations Which Preserve Cauchy Distributions and Their Ergodic Properties

This paper is concerned with invariant densities for transformations on $\mathbb{R}$ which are the boundary restrictions of inner functions of the upper half plane. G. Letac [9] proved that if the corresponding inner function has a fixed point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ or a periodic point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ with period 2, then a Cauchy distribution $(1/\pi)\mathrm{Im}\left(1/(x-z_{0}) \right)$ is an invariant probability density for the transformation. Using Cauchy's integral formula, we give an easier proof of Letac's result. An easy sufficient condition for such transformations to be isomorphic to piecewise expanding transformations on an finite interval is given by the explicit form of the density. Transformations of the forms $\alpha x + \beta - \sum ^{n }_{k=1}b_{k}/(x-a_{k})$, \:$\alpha x-\sum ^{\infty }_{k=1}\left\{ b_{k}/(x-a_{k})+b_{k}/(x+a_{k}) \right\}$ and $\alpha x +\beta\tan x$ are studied as examples.

Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation

We present an experimental method for directly observing the amplification of microscopic intrinsic noise in a high-dimensional chaotic laser system, a laser with delayed feedback. In the experiment, the chaotic laser system is repeatedly switched from a stable lasing state to a chaotic state, and the time evolution of an ensemble of chaotic states starting from the same initial state is measured. It is experimentally demonstrated that intrinsic noises amplified by the chaotic dynamics are transformed into macroscopic fluctuating signals, and the probability density of the output light intensity actually converges to a natural invariant probability density in a strongly chaotic regime. Moreover, with the experimental method, we discuss the application of the chaotic laser systems to physical random bit generators. It is experimentally shown that the convergence to the invariant density plays an important role in nondeterministic random bit generation, which could be desirable for future ultimate secure communication systems.

Harmonic Averages and New Explicit Constants for Invariant Densities of Piecewise Expanding Maps of the Interval

The statistical behavior of families of maps is important in studying the stability properties of chaotic maps. For a piecewise expanding map τ whose slope >2 in magnitude, much is known about the stability of the associated invariant density. However, when the map has slope magnitude ≤2 many different behaviors can occur as shown in (Keller in Monatsh. Math. 94(4): 313–333, 1982) for W maps. The main results of this note use a harmonic average of slopes condition to obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Since these constants are determined explicitly the results can be extended to families of approximating maps.

Phenomenological and ratio bifurcations of a class of discrete time stochastic processes

Zeeman proposed a classification of stochastic dynamical systems based on the Morse classification of their invariant probability densities; the associated bifurcations are the ‘phenomenological bifurcations’ of L. Arnold. The classification is however not invariant under diffeomorphisms of the state space. In a recent paper we proposed an alternative classification, based on an invariant that is a ratio of joint and marginal probability density functions, which does not suffer from this defect. This classification entails the concept of what we call ‘ratio bifurcations’. In this note it is shown that for a large class of dynamical systems, ratio bifurcations and phenomenological bifurcations actually coincide. Moreover, we link the ratio invariant to the transformation invariant function that Wagenmakers et al. obtained for stochastic differential equations. The results are illustrated with numerical applications to stochastic dynamical systems.

On the Efetov–Wegner terms by diagonalizing a Hermitian supermatrix

The diagonalization of Hermitian supermatrices is studied. Such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory. However, it still poses serious problems since until now the calculation of all Rothstein contributions, known as Efetov–Wegner terms in physics, was quite cumbersome. We derive the supermatrix Bessel function with all Efetov–Wegner terms for an arbitrary rotation invariant probability density function. As applications we consider representations of generating functions for Hermitian random matrices with and without an external field as integrals over eigenvalues of Hermitian supermatrices. All results are obtained with all Efetov–Wegner terms which were previously unknown in such an explicit and compact representation.

Effective phase description of noise-perturbed and noise-induced oscillations

An effective description of a general class of stochastic phase oscillators is presented. For this, the effective phase velocity is defined either by invariant probability density or via first passage times. While the first approach exhibits correct frequency and distribution density, the second one yields proper phase resetting curves. Their discrepancy is most pronounced for noise-induced oscillations and is related to non-monotonicity of the phase fluctuations.

Derivation of determinantal structures for random matrix ensembles in a new way

There are several methods to treat ensembles of random matrices in symmetric spaces, circular matrices, chiral matrices and others. Orthogonal polynomials and the supersymmetry method are particular powerful techniques. Here, we present a new approach to calculate averages over ratios of characteristic polynomials. At first sight paradoxically, one can call our approach ‘supersymmetry without supersymmetry’ because we use structures from supersymmetry without actually mapping onto superspaces. We address two kinds of integrals which cover a wide range of applications for random matrix ensembles. For probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way. As a new application we derive an expression for the k-point correlation function of an arbitrary rotation invariant probability density over the Hermitian matrices in the presence of an external field.