Invariant Distribution Research Articles

Comparison of Homologies and Automatic Extensions of Invariant Distributions

Comparison of Homologies and Automatic Extensions of Invariant Distributions

Infectious diseases and social distancing under state-dependent probabilities.

We analyze the implications of infectious diseases and social distancing in an extended SIS framework to allow for the presence of stochastic shocks with state dependent probabilities. Random shocks give rise to the diffusion of a new strain of the disease which affects both the number of infectives and the average biological characteristics of the pathogen causing the disease. The probability of such shock realizations changes with the level of disease prevalence and we analyze how the properties of the state-dependent probability function affect the long run epidemiological outcome which is characterized by an invariant probability distribution supported on a range of positive prevalence levels. We show that social distancing reduces the size of the support of the steady state distribution decreasing thus the variability of disease prevalence, but in so doing it also shifts the support rightward allowing eventually for more infectives than in an uncontrolled framework. Nevertheless, social distancing is an effective control measure since it concentrates most of the mass of the distribution toward the lower extreme of its support.

Error analysis of time-discrete random batch method for interacting particle systems and associated mean-field limits

Abstract The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially in terms of approximating the invariant distribution. The triangle inequality framework employed in this paper is a convenient approach to estimate the long-time sampling error of the numerical methods. Using the triangle inequality framework, we show that the long-time error of the discrete random batch method is $O(\sqrt {\tau } + e^{-\lambda t})$, where $\tau $ is the time step and $\lambda $ is the convergence rate, which does not depend on the time step $\tau $ or the number of particles $N$. Our results also apply to the McKean–Vlasov process, which is the mean-field limit of the interacting particle system as the number of particles $N\rightarrow \infty $.

Stabilization in distribution of hybrid stochastic systems by intermittent feedback controls

In this paper, stabilization in distribution of hybrid stochastic differential equations (HSDEs) via periodically intermittent feedback controls is investigated. It is revealed that probability distributions of the solution will converge to a stationary distribution is revealed. Firstly, by using the theory of M-matrix and intermittent control strategy, we establish sufficient conditions for stabilization in distribution of HSDEs. Then, we prove the existence and uniqueness of invariant probability distribution, the lower bound of the intermittent parameter δ∗ is obtained. Three numerical examples are discussed to support our results of theoretical analysis.

Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains

We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and π-canonical kernels, we show that we can recover the convergence rate of Arcones and Giné who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels hi,j with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein’s type inequality. Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms.

Collisionless dissipation at the boundary sheath of magnetized low temperature plasmas

In partially magnetized technological plasmas like magnetron discharges or Hall-effect thrusters, the interaction of gyrating electrons with the plasma boundary sheath plays an important role. This study shows that the interaction can be described as a dissipative process. It is assumed that the Debye length is much smaller than the electron gyroradius , and in turn much smaller than the mean free path λ and the field gradient length l. Focusing on the scale of the gyroradius, the sheath is assumed as thin (), collisions are neglected (), the magnetic field is taken as homogeneous, and electric fields (= potential gradients) in the bulk are neglected (). The interaction of an electron with the electric field of the plasma boundary sheath is represented by specular reflection, , where v is the electron velocity and n is the sheath normal. A previous study showed that this set-up leads to a scattering process where gyrophase invariant incoming velocity distributions are scattered into outgoing distributions with fractal dependence on the gyrophase (Krüger and Brinkmann 2017 Plasma Sources Sci. Technol. 26 115009). The present research focuses on the subsequent evolution. It is shown that the gyromotion results in fast phase mixing with respect to the gyrophase which quickly restores the gyrophase invariance. In combination, the two phases of the sheath interaction constitute a mapping within the space of gyrophase invariant distributions. This combined process is dissipative and leads to isotropization. As an explicit comparison shows, it acts similarly to elastic electron-neutral collisions and can even dominate the latter in low pressure plasmas. A simplified version of the operator that applies for weakly anisotropic distributions is suited for practical simulation work.

Increasing the synchronization stability in complex networks.

We aim to increase the ability of coupled phase oscillators to maintain synchronization when the system is affected by stochastic disturbances. We model the disturbances by Gaussian noise and use the mean first hitting time when the state hits the boundary of a secure domain, that is a subset of the basin of attraction, to measure synchronization stability. Based on the invariant probability distribution of a system of phase oscillators subject to Gaussian disturbances, we propose an optimization method to increase the mean first hitting time and, thus, increase synchronization stability. In this method, a new metric for synchronization stability is defined as the probability of the state being absent from the secure domain, which reflects the impact of all the system parameters and the strength of disturbances. Furthermore, by this new metric, one may identify those edges that may lead to desynchronization with a high risk. A case study shows that the mean first hitting time is dramatically increased after solving corresponding optimization problems, and vulnerable edges are effectively identified. It is also found that optimizing synchronization by maximizing the order parameter or the phase cohesiveness may dramatically increase the value of the metric and decrease the mean first hitting time, thus decrease synchronization stability.

Optimal subsampling design for polynomial regression in one covariate

Improvements in technology lead to increasing availability of large data sets which makes the need for data reduction and informative subsamples ever more important. In this paper we construct D-optimal subsampling designs for polynomial regression in one covariate for invariant distributions of the covariate. We study quadratic regression more closely for specific distributions. In particular we make statements on the shape of the resulting optimal subsampling designs and the effect of the subsample size on the design. To illustrate the advantage of the optimal subsampling designs we examine the efficiency of uniform random subsampling.

A Micro-Macro Markov Chain Monte Carlo Method for Molecular Dynamics using Reaction Coordinate Proposals

We introduce a new micro-macro Markov chain Monte Carlo method to sample invariant distributions of molecular dynamics systems that exhibit a time-scale separation between the microscopic (fast) dynamics, and the macroscopic (slow) dynamics of some low-dimensional set of reaction coordinates. The algorithm enhances exploration of the state space in the presence of metastability by allowing larger proposed moves at the macroscopic level, on which a conditional accept-reject procedure is applied. Only when the macroscopic proposal is accepted, is the full microscopic state reconstructed from the newly sampled reaction coordinate value and is subjected to a second accept/reject procedure. The computational gain stems from the fact that most proposals are rejected at the macroscopic level, at low computational cost, while microscopic states, once reconstructed, are almost always accepted. We analytically show convergence and discuss the rate of convergence of the proposed algorithm, and numerically illustrate its efficiency on two standard molecular test cases.

Synchronization of power systems under stochastic disturbances

The synchronization of power generators is an important condition for the proper functioning of a power system, in which the fluctuations in frequency and the phase angle differences between the generators are sufficiently small when subjected to stochastic disturbances. Serious fluctuations can prompt desynchronization, which may lead to widespread power outages. Here, we model the stochastic disturbance by a Brownian motion process in the linearized system of the non-linear power systems and characterize the fluctuations by the variances of the frequency and the phase angle differences in the invariant probability distribution. We propose a method to calculate the variances of the frequency and the phase angle differences. For the system with uniform disturbance-damping ratio, we derive explicit formulas for the variance matrices of the frequency and the phase angle differences. It is shown that the fluctuation of the frequency at a node depends on the disturbance-damping ratio and the inertia at this node only, and the fluctuations of the phase angle differences in the lines are independent of the inertia. In particular, the synchronization stability is related to the cycle space of the network. We reveal the influences of constructing new lines and increasing capacities of lines on the fluctuations in the phase angle differences in the existing lines. The results are illustrated for the transmission system of Shandong Province of China. For the system with non-uniform disturbance-damping ratio, we further obtain bounds of the variance matrices.

Designing robust scale-free networks under targeted link attack using local information

We study the effects of deliberate attacks to important links in scale-free networks and propose simple strategies to mitigate the damage. While strategies that require global network information and heavy computation have been proposed, here we focus on heuristics that make use of local information only and are much easier to compute. Using model scale-free networks and under the constraint of an invariant degree distribution, we show by numerical simulation that our approach in the average allows to enhance the network robustness notably, as measured by the integrity of the largest connected component. Because it is fast, the approach is also applicable to real-world networks of which we give two typical examples. Overall, the results are equivalent to those obtained in more complex settings that require global network knowledge and are much more computer-intensive, an important point for large networks.

Distributed-Observer-Based Distributed Control Law for Affine Nonlinear Systems and Its Application on Interconnected Cruise Control of Intelligent Vehicles

Distributed-observer-based distributed control law for interconnected systems has become the research focus in recent years. This paper extends this study from linear systems to a class of affine nonlinear systems. Before this, distributed observer for nonlinear systems with control inputs has not been studied, as well as the distributed-observer-based distributed control law. To fill this gap, we overcome the difficulty of proving joint observability for distributed nonlinear observer by an original analysis method based on maximum invariant distribution. In addition, the influence of model mismatch on distributed observer and distributed controller is also investigated and analyzed in detail. By analyzing the complex coupling relationship between observer dynamics, control input, and model mismatch, we successfully deal with the dissatisfaction of separation principle and prove the stability of observer error systems and closed-loop systems. The results show both the error dynamics of distributed observer and the system states of closed-loop system formed by the observer-based distributed control law can converge to an arbitrarily small invariant set around the origin. In addition to the original work in distributed-observer-based distributed control law, this paper also contributes to the design of distributed observer. Firstly, the proposed design method is also applicable to distributed observer of autonomous systems. Compared to the existing results, our method admits weaker assumptions. Specifically, one could find an approximate system to meet the assumptions required by designing distributed observer when the underlying system fails to meet the requirements. At last, the automated highway system is employed to verify the results.

Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

In this paper, we focus on nonasymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (nonconstant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step (usually called unadjusted Langevin algorithm in the Monte Carlo literature) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under uniform ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for total variation and L1-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using stochastic analysis adapted to decreasing step setting.

Ena/VASP clustering at microspike tips involves lamellipodin but not I-BAR proteins, and absolutely requires unconventional myosin-X

Sheet-like membrane protrusions at the leading edge, termed lamellipodia, drive 2D-cell migration using active actin polymerization. Microspikes comprise actin-filament bundles embedded within lamellipodia, but the molecular mechanisms driving their formation and their potential functional relevance have remained elusive. Microspike formation requires the specific activity of clustered Ena/VASP proteins at their tips to enable processive actin assembly in the presence of capping protein, but the factors and mechanisms mediating Ena/VASP clustering are poorly understood. Systematic analyses of B16-F1 melanoma mutants lacking potential candidate proteins revealed that neither inverse BAR-domain proteins, nor lamellipodin or Abi is essential for clustering, although they differentially contribute to lamellipodial VASP accumulation. In contrast, unconventional myosin-X (MyoX) identified here as proximal to VASP was obligatory for Ena/VASP clustering and microspike formation. Interestingly, and despite the invariable distribution of other relevant marker proteins, the width of lamellipodia in MyoX-KO mutants was significantly reduced as compared with B16-F1 control, suggesting that microspikes contribute to lamellipodium stability. Consistently, MyoX removal caused marked defects in protrusion and random 2D-cell migration. Strikingly, Ena/VASP-deficiency also uncoupled MyoX cluster dynamics from actin assembly in lamellipodia, establishing their tight functional association in microspike formation.

Approximation of the invariant distribution for a class of ergodic SDEs with one-sided Lipschitz continuous drift coefficient using an explicit tamed Euler scheme

We study the behavior in a large time regime of an explicit tamed Euler-Maruyama scheme applied to a class of ergodic Ito stochastic differential equations with one-sided Lipschitz continuous drift coefficient and bounded globally Lipschitz diffusion coefficient. Our first main contribution is to prove moments for the numerical scheme, which, on the one hand, are uniform with respect to the time-step size, and which, on the other hand, may not be uniform but have at most polynomial growth with respect to time. Our second main contribution is to apply this result to obtain weak error estimates to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process, as a function of the time-step size and of the time horizon. The explicit tamed Euler scheme is shown to be computationally effective for the approximation of the invariant distribution: even if the moment bounds and error estimates are not proved to be uniform with respect to time, the obtained polynomial growth results in a marginal increase in the upper bound of the computational cost. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for stochastic differential equations with non-globally Lipschitz coefficients using an explicit tamed Euler-Maruyama scheme.

Ergodicity and long-time behavior of the Random Batch Method for interacting particle systems

We study the geometric ergodicity and the long-time behavior of the Random Batch Method for interacting particle systems, which exhibits superior numerical performance in recent large-scale scientific computing experiments. We show that for both the interacting particle system (IPS) and the random batch interacting particle system (RB–IPS), the distribution laws converge to their respective invariant distributions exponentially, and the convergence rate does not depend on the number of particles [Formula: see text], the time step [Formula: see text] for batch divisions or the batch size [Formula: see text]. Moreover, the Wasserstein-1 distance between the invariant distributions of the IPS and the RB–IPS is bounded by [Formula: see text], showing that the RB–IPS can be used to sample the invariant distribution of the IPS accurately with greatly reduced computational cost.

Locally unipotent invariant measures and limit distribution of a sequence of polynomial trajectories on homogeneous spaces

Let $ G $ be a Lie group, and let $ \Gamma $ be a lattice in $ G $. We introduce the notion of locally unipotent invariant measures on $ G/\Gamma $. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on $ G/\Gamma $ is locally unipotent invariant, thus give a partial answer to an equidistribution problem for higher dimensional polynomial trajectories on homogeneous spaces, which was raised by Shah in [15].The proof relies on Ratner's measure classification theorem, a linearization technique for polynomial trajectories near singular sets, and a twisting technique of Shah.

Rate of convergence of Nummelin-type representation of the invariant distribution of a Markov chain via the residual kernel

Rate of convergence of Nummelin-type representation of the invariant distribution of a Markov chain via the residual kernel

Radionuclide 239+240Pu for Dating and Sedimentation Rate in Chinese Lakes

Radionuclide 239+240Pu dating and sedimentation rate is a crucial parameter for studying lake sedimentary processes. To comprehensively investigate 239+240Pu vertical distributions and relevant processes in Chinese lakes, this study collects and summarizes 54 sediment core samples from 40 lakes and reservoirs globally. First, the results suggest that two typical vertical distributions of 239+240Pu are summarized in Chinese lakes. The location of the 239+240Pu peak corresponds to the age of global atmospheric nuclear test deposition in 1963 ± 1. Second, 239+240Pu based on 1963 peak is far more suitable for lake dating compared to 137Cs, and a significant linear correlation (R2 = 0.924) between the depth of 239+240Pu maximum value (cm) and sedimentation rate (cm a–1) in Chinese lakes was established. For the first time, this study has found that this significant linear relationship can be generalized globally (R2 = 0.909), regardless of where the lake sediment cores were sampled. Third, we identify three typical vertical distributions for the 240Pu/239Pu atom ratio in Chinese lake sediment cores: invariant distribution, mutation distribution, and gradient distribution. This result implies that 239+240Pu in Chinese lakes mainly originate from the global fallout; yet this does not rule out the potential contribution from local sources of 239+240Pu, such as the Lop Nor nuclear tests, the atomic city nuclear activities, and the Chernobyl nuclear accident. However, the trend in the average 240Pu/239Pu ratio varies independent of latitude. Moreover, a significant positive linear correlation is detected between average 239+240Pu inventory in the soil and lake cores, with a coefficient of 0.538. In particular, the high 239+240Pu inventory background value in Northwest China closely correlates with the deposition of 239+240Pu in some lakes (e.g., Lake Sugan). This manuscript provides insights into 239+240Pu vertical patterns and processes in Chinese lakes and sheds light on future research using 239+240Pu for dating and sedimentation rate in lakes outside China.

Random Walks, Directed Cycles and Markov Chains

A Markov chain is a random process which iteratively travels around in its state space with each transition only depending on the current position and not on the past. When the state space is discrete, we can think of a Markov chain as a special type of random walk on a directed graph. Although a Markov chain normally never settles down but keeps moving around, it does usually have a well-defined limiting behavior in a statistical sense. A given finite directed graph can potentially support many different random walks or Markov chains and each one could have one or more invariant (stationary) distributions. In this paper we explore the question of characterizing the set of all possible invariant distributions. The answer turns out to be quite simple and very natural and involves the cycles on the graph.