Invariant Distribution Research Articles

On the differential equations satisfied by weighted orbital integrals

Weighted orbital integrals are distributions on reductive groups over local fields appearing both in the local and global trace formulas. There are associated invariant distributions, which play the same role in the invariant trace formulas. In the case of real groups, the Fourier transforms of these distributions satisfy a system of differential equations. As a step towards determining those Fourier transforms, we show that this system is holonomic and has a simple singularity at infinity. We deduce that any solution has a series expansion and is a linear combination of certain canonical solutions. For some groups of small rank, we solve the recursion formula for the coefficients explicitly.

Triangulated Laman graphs, local stochastic matrices, and limits of their products

We derive conditions on the products of stochastic matrices guaranteeing the existence of a unique limit invariant distribution. Belying our approach is the hereby defined notion of restricted triangulated Laman graphs. The main idea is the following: to each triangle in the graph, we assign a stochastic matrix. Two matrices can be adjacent in a product only if their corresponding triangles share an edge in the graph. We provide an explicit formula for the limit invariant distribution of the product in terms of the individual stochastic matrices.

Dynamics of a Fleming–Viot type particle system on the cycle graph

We study the Fleming–Viot particle process formed by N interacting continuous-time asymmetric random walks on the cycle graph, with uniform killing. We show that this model has a remarkable exact solvability, despite the fact that it is non-reversible with non-explicit invariant distribution. Our main results include quantitative propagation of chaos and exponential ergodicity with explicit constants, as well as formulas for covariances at equilibrium in terms of the Chebyshev polynomials. We also obtain a bound uniform in time for the convergence of the proportion of particles in each state when the number of particles goes to infinity.

Camera-Based Batch Normalization: An Effective Distribution Alignment Method for Person Re-Identification

Person re-identification (ReID) aims at matching identities across disjoint cameras. Its fundamental difficulty lies in associating images across individual cameras, where a key clue, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , identity appearance, is prone to the environmental factors of cameras and, consequently, subject to distinct image distributions due to the environmental differences between cameras. To associate images from training cameras, ReID methods strongly demand expensive inter-camera annotations for learning the relations between the distribution of these cameras, yet trained models are still not guaranteed to transfer well to unseen cameras. This problem significantly limits the application of ReID. This paper rethinks the working mechanism of conventional ReID approaches and puts forward a new solution. With an effective operator named Camera-based Batch Normalization (CBN), we guarantee an invariant input distribution independent of all cameras. Thus, the training and testing procedures are always conducted under the same input distribution. This alignment brings three benefits. First, ReID models enjoy better abilities to generalize across testing scenarios with unseen cameras and transfer across multiple training sets. Second, it makes better use of intra-camera annotations, which have been undervalued before due to the lack of cross-camera information. Ideally, the cost of inter-camera annotations can be largely reduced. Third, cross-modality tasks can be better defined through aligning visible/infrared cameras’ distributions. Experiments on a wide range of ReID tasks demonstrate the effectiveness of our approach.

Large Deviation Properties of the Empirical Measure of a Metastable Small Noise Diffusion

The aim of this paper is to develop tractable large deviation approximations for the empirical measure of a small noise diffusion. The starting point is the Freidlin–Wentzell theory, which shows how to approximate via a large deviation principle the invariant distribution of such a diffusion. The rate function of the invariant measure is formulated in terms of quasipotentials, quantities that measure the difficulty of a transition from the neighborhood of one metastable set to another. The theory provides an intuitive and useful approximation for the invariant measure, and along the way many useful related results (e.g., transition rates between metastable states) are also developed. With the specific goal of design of Monte Carlo schemes in mind, we prove large deviation limits for integrals with respect to the empirical measure, where the process is considered over a time interval whose length grows as the noise decreases to zero. In particular, we show how the first and second moments of these integrals can be expressed in terms of quasipotentials. When the dynamics of the process depend on parameters, these approximations can be used for algorithm design, and applications of this sort will appear elsewhere. The use of a small noise limit is well motivated, since in this limit good sampling of the state space becomes most challenging. The proof exploits a regenerative structure, and a number of new techniques are needed to turn large deviation estimates over a regenerative cycle into estimates for the empirical measure and its moments.

Stochastic parametrization with VARX processes

In this study we investigate a data-driven stochastic methodology to parameterize small-scale features in a prototype multiscale dynamical system, the Lorenz '96 (L96) model. We propose to model the small-scale features using a vector autoregressive process with exogenous variable (VARX), estimated from given sample data. To reduce the number of parameters of the VARX we impose a diagonal structure on its coefficient matrices. We apply the VARX to two different configurations of the 2-layer L96 model, one with common parameter choices giving unimodal invariant probability distributions for the L96 model variables, and one with non-standard parameters giving trimodal distributions. We show through various statistical criteria that the proposed VARX performs very well for the unimodal configuration, while keeping the number of parameters linear in the number of model variables. We also show that the parameterization performs accurately for the very challenging trimodal L96 configuration by allowing for a dense (non-diagonal) VARX covariance matrix.

Charged current semi-inclusive deeply inelastic scattering at the Electron-Ion Collider

We present a systematic calculation of the charged current semi-inclusive deeply inelastic scattering process at leading order twist-3 level in the parton model. We consider the general form of the negatively charged beam scattering off the polarized target which has spin-1. The calculations are carried out by applying the collinear expansion where multiple gluon scattering is taken into account and gauge links are obtained automatically. We first present the general form of the differential cross section in terms of the structure functions by kinematic analysis and then present the structure functions in terms of the the gauge invariant parton distribution functions up to twist-3 level. Considering the angle modulations and polarizations of the cross section, we calculate the complete azimuthal asymmetries in the charged current semi-inclusive deeply inelastic scattering process. The charge asymmetries are also considered in this paper with the introduction of the definitions of the $plus$ and $minus$ cross sections.

Parameter estimation for diffusion process from perturbed discrete observations

We study the parameter estimation for ergodic diffusion process Xt from perturbed observations where are observation times and the noise is a strongly mixing stationary noisy process with the density function g. We construct an estimator of the diffusion parameters based on the minimum Hellinger distance between the density of the invariant distribution of diffusion process Xt and the nonparametric deconvolution kernel estimator of this density. This article focuses on the ordinary smooth noise density class with the assumption (where and is characteristic function of noisy random variable). This assumption is more general than the condition (where C is a constant) which is used in a lot of articles. We also discuss the asymptotic normality for both the estimator of deconvolution kernel density and the estimator of diffusion parameters. Finally, we illustrate the properties of the estimator by two examples of diffusion processes.

The entropy based goodness of fit tests for generalized von Mises-Fisher distributions and beyond

We introduce some new classes of unimodal rotational invariant directional distributions, which generalize von Mises-Fisher distribution. We propose three types of distributions, one of which represents axial data. For each new type we provide formulae and short computational study of parameter estimators by the method of moments and the method of maximum likelihood. The main goal of the paper is to develop the goodness of fit test to detect that sample entries follow one of the introduced generalized von Mises--Fisher distribution based on the maximum entropy principle. We use $k$th nearest neighbour distances estimator of Shannon entropy and prove its $L^2$-consistency. We examine the behaviour of the test statistics, find critical values and compute power of the test on simulated samples. We apply the goodness of fit test to local fiber directions in a glass fibre reinforced composite material and detect the samples which follow axial generalized von Mises--Fisher distribution.

Elliptic Optimal Control and Symmetric Sub-Riemannian Spaces

In this paper we study the integrability properties of Lie-Poisson adjoint systems associated to four-dimensional sub-Riemannian Lie groups with invariant quasi-contact distributions. We show an equivalence between the facts that this system admits three quadratic functionally independent first integrals, the optimal controls are elliptic functions, and the sub-Riemannian space is symmetric.

Survey and Review

The use of probabilistic ideas by applied mathematicians has seen a continued increase in recent decades. Probability now appears frequently at the modeling stage. There is widespread interest in investigating the effects of noise and uncertainty. Probabilistic algorithms are routinely applied with much success to the solution of deterministic problems (think of Monte Carlo quadrature or stochastic gradient descent). Markov chains give a simple, widely used tool to describe systems that evolve randomly. They are also a very popular topic in applied mathematics courses. At times \(0\), \(1\), \(2\), łdots, a Markov chain jumps from its current state \(x\) to a randomly chosen new state \(y\); the set of possible states is discrete. The requirement that the jumping times be uniformly spaced is a clear limitation in many situations, and this shortcoming is avoided by considering continuous-time Markov chains. In the continuous-time setting, if the chain has reached state \(x\) at time \(t_i\), it will jump to the randomly chosen \(y\) at time \(t_i+1=t_i+\tau_i\), where the waiting time \(\tau_i>0\) is itself random. Markov chains in continuous time are featured in many applications, including chemistry, ecology, and epidemiology. In a chemical system containing \(n\) species \(S_1\), łdots, \(S_n\), each state corresponds to vector \((c_1\), łdots, \( c_n)\) where \(c_j\) is the number of molecules of species \(S_j\). The species may take part in a number of chemical reactions, let us say \(S_1+2S_2 \rightarrow S_3\), \(2S_1+3S_3 \rightarrow S_4+2S_6\), and so on. At random times, one or another of the \(m\) possible reactions will take place and the state will change. Such a stochastic, molecular approach typically provides a more accurate description of the system than deterministic treatments where the concentrations of the different species are regarded as continuous variables that evolve according to a set of differential equations. This is particularly true in systems where, for some species, the number \(c_j\) is low, as is the case in many biological reactions. The following Survey and Review paper, “Stationary Distributions of Continuous-Time Markov Chains: A Review of Theory and Truncation-Based Approximations,” has been written by Juan Kuntz, Philipp Thomas, Guy-Bart Stan, and Mauricio Barahona. Section 2 provides a reader-friendly introduction to continuous-time Markov chains and their stationary distributions; these are important because they determine the long-time behavior of the chain. A salient future is that the authors present, in an accessible way, results that do not assume the chain to be irreducible (roughly speaking, irreducibility means that the chain is not the juxtaposition of two or more smaller chains that do not talk to each other; irreducibility simplifies the math, but is not a reasonable hypothesis in some applications). After that, the authors concentrate on how to compute invariant distributions. The paper contains numerical results, an extensive bibliography, and a detailed discussion of open problems.

The Analytic Theory of a Monetary Shock

We propose an analytical method to analyze the propagation of a once-and-for-all shock in a broad class of sticky price models. The method is based on the eigenvalue- eigenfunction representation of the dynamics of the cross-sectional distribution of firms’ desired adjustments. A key novelty is that, under assumptions that are appropriate for low-inflation economies, we can approximate the whole profile of the impulse response for any moment of interest in response to an aggregate shock (any displacement of the invariant distribution). We present several applications and discuss extensions. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

The Analytic Theory of a Monetary Shock

We propose an analytical method to analyze the propagation of a once-and-for-all shock in a broad class of sticky price models. The method is based on the eigenvalue- eigenfunction representation of the dynamics of the cross-sectional distribution of firms’ desired adjustments. A key novelty is that, under assumptions that are appropriate for low-inflation economies, we can approximate the whole profile of the impulse response for any moment of interest in response to an aggregate shock (any displacement of the invariant distribution). We present several applications and discuss extensions.

Resonant spaces for volume-preserving Anosov flows

We consider Anosov flows on closed 3-manifolds preserving a volume form $\Omega$. Following Dyatlov and Zworski (2017) we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class $[\iota_{X}\Omega]$ (where $X$ is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the Pollicott-Ruelle resonance multiplicities under the assumption of $\textit{semisimplicity}$. We prove various results regarding semisimplicity on 1-forms, including an example showing that it may fail for time changes of hyperbolic geodesic flows. We also study non null-homologous deformations of contact Anosov flows and we show that there is always a splitting Pollicott-Ruelle resonance on 1-forms and that semisimplicity persists in this instance. These results have consequences for the order of vanishing at zero of the Ruelle zeta function. Finally our analysis also incorporates a flat unitary twist in both, the resonant spaces and the Ruelle zeta function.

Aftershock Rate Changes at Different Ocean Tide Heights

The differential probability gain approach is used to estimate quantitatively the change in aftershock rate at various levels of ocean tides relative to the average rate model. An aftershock sequences are analyzed from two regions with high ocean tides, Kamchatka and New Zealand. The Omori-Utsu law is used to model the decay over time, hypothesizing an invariable spatial distribution. Ocean tide heights are considered rather than phases. A total of 16 sequences of M ≥6 aftershocks off Kamchatka and 15 sequences of M ≥6 aftershocks off New Zealand are examined. The heights of the ocean tides at various locations were modeled using FES 2004. Vertical stress changes due to ocean tides are here about 10–20 kPa, that is, at least several times greater than the effect due to Earth tides. An increase in aftershock rate is observed by more than two times at high water after main M ≥6 shocks in Kamchatka, with slightly less pronounced effect for the earthquakes of M = 7.8, December 15, 1971 and M = 7.8, December 5, 1997. For those two earthquakes, the maximum of the differential probability gain function is also observed at low water. For New Zealand, we also observed an increase in aftershock rate at high water after thrust type main shocks with M ≥6. After normal-faulting main shocks there was the tendency of the rate increasing at low water. For the aftershocks of the strike-slip main shocks we observed a less evident impact of the ocean tides on their rate. This suggests two main mechanisms of the impact of ocean tides on seismicity rate, an increase in pore pressure at high water, or a decrease in normal stress at low water, both resulting in a decrease of the effective friction in the fault zone.

Inhomogeneous functionals and approximations of invariant distributions of ergodic diffusions: Central limit theorem and moderate deviation asymptotics

The paper studies asymptotics of inhomogeneous integral functionals of an ergodic diffusion process under the effect of discretization. Convergence to the corresponding functionals of the invariant distribution is shown for suitably chosen discretization steps, and the fluctuations are analyzed through central limit theorem and moderate deviation principle. The results will be particularly useful for understanding accuracy of an Euler discretization based numerical scheme for approximating functionals of invariant distribution of an ergodic diffusion. This is an infinite-time horizon problem, and the accuracy of numerical schemes in this context are comparatively much less studied than the ones used for generating approximate trajectories of diffusions over finite time intervals. The potential applications of these results also extend to other areas including mathematical physics, parameter inference of ergodic diffusions and analysis of multiscale dynamical systems with averaging.

HOMOGENEOUS RIEMANNIAN MANIFOLDS WITH NON-TRIVIAL NULLITY

We develop a general theory for irreducible homogeneous spaces M = G/H, in relation to the nullity distribution ν of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e., where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity k = 3, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that H is trivial and G is solvable if k = 3. Another of our main results is that the leaves, i.e., the integral manifolds, of the nullity are closed (we used a rather delicate argument). This implies that M is a Euclidean affine bundle over the quotient by the leaves of ν. Moreover, we prove that ν⊥ defines a metric connection on this bundle with transitive holonomy or, equivalently, ν⊥ is completely non-integrable (this is not in general true for an arbitrary autoparallel and at invariant distribution). We also found some general obstruction for the existence of non-trivial nullity: e.g., if G is reductive (in particular, if M is compact), or if G is two-step nilpotent.

Stochastic Variability of Tropical Cyclone Intensity at the Maximum Potential Intensity Equilibrium

Abstract This study examines the variability of tropical cyclone (TC) intensity associated with stochastic forcings at the maximum potential intensity (PI) equilibrium. By representing TC intensity as an Itô diffusion process in the framework of TC-scale dynamics, we show from both theoretical and numerical analyses that there exists an invariant intensity distribution whose variance is proportional to the variances of stochastic forcings. This result provides further evidence that TC dynamics possess an intrinsic variability that prevents the TC absolute intensity errors in numerical models from being reduced below an arbitrarily small threshold. Analysis of the invariant intensity distribution at the PI limit reveals also that the stochastic forcing component associated with tangential wind and the warm-core anomaly in the TC central region have the largest contribution to TC intensity variability. These results suggest that future development of stochastic representation in TC models should focus on the tangential wind and thermodynamic structure to capture proper TC intensity random fluctuations.

The Aldous chain on cladograms in the diffusion limit

In (Combin. Probab. Comput. 9 (2000) 191–204), Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in (Random Structures Algorithms 20 (2002) 59–70). In the present paper, we encode cladograms as binary, algebraic measure trees and show that this Markov chain on cladograms with a fixed number of leaves converges in distribution as the number of leaves tends to infinity. We give a rigorous construction of the limit as the solution of a well-posed martingale problem. The existence of a continuum limit diffusion was conjectured by Aldous, and we therefore refer to it as Aldous diffusion. We show that the Aldous diffusion is a Feller process with continuous paths, and the algebraic measure Brownian CRT is its unique invariant distribution. Furthermore, we consider the vector of the masses of the three subtrees connected to a sampled branch point. In the Brownian CRT, its annealed law is known to be the Dirichlet distribution. Here, we give an explicit expression for the infinitesimal evolution of its quenched law under the Aldous diffusion.

Unitary transformation for Poincar\xe9 beams on different parts of Poincar\xe9 sphere

We construct an experimental setup, consisting of conical refraction transformation in two biaxial cascade crystals and 4f-system, to realize Unitary transformation of light beam and the manipulation of Poincaré beams on the different parts of Poincaré sphere. The spatial structure of the polarization can be controlled by changing the polarization of the incident beam or rotating the angle between these two crystals. The beams with different SoPs covering the full-Poincaré sphere, part-Poincaré sphere and one point on the sphere are generated for the different angles between crystals. The Unitary transformation of light beam is proposed in the experiment with the invariant intensity distribution. Subsequently, the spin angular momentum is derived from the distribution of polarization measured in our experiment. Moreover, the conversion between orbital angular momentum and spin angular momentum of light beam is obtained by changing the angle between crystals. And the conversion progress can also be influenced by the polarization of incident beam. We realized the continuous control of the spatial structure of the angular momentum density, which has potential in the manipulation of optical trapping systems and polarization-multiplexed free-space optical communication.