Stationary Measure Research Articles

Stationary measures and the continuous-state branching process conditioned on extinction

Abstract We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related Lévy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

Charmenability and stiffness of arithmetic groups

We characterize charmenability among arithmetic groups and deduce dichotomy statements pertaining to normal subgroups, characters, dynamics, representations and associated operator algebras. We do this by studying the stationary dynamics on the space of characters of the amenable radical, and in particular we establish stiffness: any stationary probability measure is invariant. This generalizes a classical result of Furstenberg for dynamics on the torus. Under a higher rank assumption, we show that any action on the space of characters of a finitely generated virtually nilpotent group is stiff.

On the Dimension of Limit Sets on ℙ(ℝ3) via Stationary Measures: Variational Principles and Applications

Abstract This paper investigates the (semi)group action of $\textrm{SL}_{n}({\mathbb R})$ on ${\mathbb P}({\mathbb R}^{n})$, a primary example of non-conformal, non-linear, and non-strictly contracting action. We establish variational principles of the affinity exponent for two main examples: the Borel Anosov representations and the Rauzy gasket. In [ 32], they obtain a dimension formula for the stationary measures on ${\mathbb P}({\mathbb R}^{3})$. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in $\textrm{SL}_{3}({\mathbb R})$ and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings, generalizing the classical Patterson–Sullivan formula. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to $1.5$.

Scaling Limit of Multi-Type Invariant Measures via the Directed Landscape

Abstract This paper studies the large scale limits of multi-type invariant distributions and Busemann functions of planar stochastic growth models in the Kardar–Parisi–Zhang (KPZ) class. We identify a set of sufficient hypotheses for convergence of multi-type invariant measures of last-passage percolation (LPP) models to the stationary horizon (SH), which is the unique multi-type stationary measure of the KPZ fixed point. Our limit theorem utilizes conditions that are expected to hold broadly in the KPZ class, including convergence of the scaled last-passage process to the directed landscape. We verify these conditions for the six exactly solvable models whose scaled bulk versions converge to the directed landscape, as shown by Dauvergne and Virág. We also present a second, more general, convergence theorem with future applications to polymer models and particle systems. Our paper is the first to show convergence to the SH without relying on information about the structure of the multi-type invariant measures of the prelimit models. These results are consistent with the conjecture that the SH is the universal scaling limit of multi-type invariant measures in the KPZ class.

Stationary measures for SL2(ℝ)-actions on homogeneous bundles over flag varieties

Abstract Let 𝐺 be a real semisimple Lie group with finite centre and without compact factors, Q < G Q<G a parabolic subgroup and 𝑋 a homogeneous space of 𝐺 admitting an equivariant projection on the flag variety G / Q G/Q with fibres given by copies of lattice quotients of a semisimple factor of 𝑄. Given a probability measure 𝜇, Zariski-dense in a copy of H = SL 2 ⁡ ( R ) H=\operatorname{SL}_{2}(\mathbb{R}) in 𝐺, we give a description of 𝜇-stationary probability measures on 𝑋 and prove corresponding equidistribution results. Contrary to the results of Benoist–Quint corresponding to the case G = Q G=Q , the type of stationary measures that 𝜇 admits depends strongly on the position of 𝐻 relative to 𝑄. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin–Mirzakhani and Eskin–Lindenstrauss.

Duality for the multispecies stirring process with open boundaries

We study the stirring process with N − 1 species on a generic graph G=(V,E) with reservoirs. The multispecies stirring process generalizes the symmetric exclusion process, which is recovered in the case N = 2. We prove the existence of a dual process defined on an extended graph G~=(V~,E~) which includes additional sites in V~∖V where dual particles get absorbed in the long-time limit. We thus obtain a characterization of the non-equilibrium steady state of the boundary-driven system in terms of the absorption probabilities of dual particles. The process is integrable for the case of the one-dimensional chain with two reservoirs at the boundaries and with maximally one particle per site. We compute the absorption probabilities by relying on the underlying gl(N) symmetry and the matrix product ansatz. Thus one gets a closed-formula for (long-ranged) correlations and for the non-equilibrium stationary measure. Extensions beyond this integrable set-up are also discussed.

Transient Analysis for a Queuing System with Impatient Customers and Its Applications to the Pricing Strategy of a Video Website

In this paper, we consider a queuing system with impatient customers, which includes infinite servers and two types of customers. During the service process, Type-1 customers may leave the system or upgrade to be Type-2 customers due to their impatience. By solving the partial differential equations, we obtain the generating functions of the transient distribution of the queue length, and many stationary performance measures are further derived. Then, as an application, we formulate an expected profit function for a video website, and maximize it by determining the optimal pricing strategy. Finally, numerical examples are provided to demonstrate the impacts of parameters on the optimal website profit.

Stationary measures for integrable polymers on a strip

Stationary measures for integrable polymers on a strip

Askey–Wilson Signed Measures and Open ASEP in the Shock Region

Abstract We introduce a family of multi-dimensional Askey–Wilson signed measures. We offer an explicit description of the stationary measure of the open asymmetric simple exclusion process (ASEP) in the full phase diagram, in terms of integrations with respect to these Askey–Wilson signed measures. Using our description, we provide a rigorous derivation of the density profile and limit fluctuations of open ASEP in the entire shock region, including the high and low density phases as well as the coexistence line. This in particular confirms the existing physics postulations of the density profile.

Random walk on the Poincaré disk

We consider three interpretations of the standard random walk of [Formula: see text] on a hyperbolic space. We will consider the hyperbolic disk model, and we study properties such as transient behavior, synchronicity and properties of the stationary measure.

Absolutely continuous invariant measures for random dynamical systems of beta-transformations

We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval Tβ(x)=βx (mod 1), x∈[0,1] , β > 0, which are the so-called beta-transformations. For such a random dynamical system, including the case that it is generated by uncountably many maps, we give an explicit formula for the density function of a unique stationary measure under the assumption that the random dynamics is expanding in mean. As an application, in the case that the random dynamics is generated by finitely many maps and the maps are chosen according to a Bernoulli measure, we show that the density function is analytic as a function of parameter in the Bernoulli measure and give its derivative explicitly. Furthermore, for a non-i.i.d. random dynamical system of beta-transformations, we also give an explicit formula for the random densities of a unique absolutely continuous invariant measure under a certain strong expanding condition or under the assumption that the maps randomly chosen are close to the beta-transformation for a non-simple number in the sense of parameter β.

Random walks on tori and normal numbers in self-similar sets

abstract: We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K\subset\mathbb{R}^d$ is an attractor of a finite iterated function system of $n\geq 2$ maps of the form $x\mapsto D^{-1}x+t_i$ ($i=1,\dotsc,n$), where $D$ is an expanding $d\times d$ integer matrix, and is the same for all the maps, under an irrationality condition on the translation parts $t_i$, almost every point in $K$ (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $x\mapsto Dx$ (multiplication mod $\mathbb{Z}^d$). In the one-dimensional case, this conclusion amounts to normality to base $D$. Thus for example, almost every point in an irrational dilation of the middle-thirds Cantor set is normal to base $3$.

Stationary measures for stochastic differential equations with degenerate damping

Stationary measures for stochastic differential equations with degenerate damping

The steady state of the boundary-driven multiparticle asymmetric diffusion model

We consider the multiparticle asymmetric diffusion model (MADM) introduced by Sasamoto and Wadati with integrability preserving reservoirs at the boundaries. In contrast to the open asymmetric simple exclusion process the number of particles allowed per site is unbounded in the MADM. Taking inspiration from the stationary measure in the symmetric case, i.e. the rational limit, we first obtain the length 1 solution and then show that the steady state can be expressed as an iterated product of Jackson q-integrals. In the proof of the stationarity condition, we observe a cancellation mechanism that closely resembles the one of the matrix product ansatz. To our knowledge, the occupation probabilities in the steady state of the boundary-driven MADM were not available before.

Stable laws for random dynamical systems

Abstract In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $ . We consider a class of non-square-integrable observables $\phi $ , mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$ , where $x_0$ is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index $\alpha \in (0,2)$ . The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $\alpha $ , we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $\nu $ . This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

STRONG MEASURE ZERO SETS ON FOR INACCESSIBLE

Abstract We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^\kappa $ for $\kappa $ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of $$\begin{align*}|2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \textrm{ is strong measure zero if and only if } |X| \leq \kappa^+. \end{align*}$$ Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence of the two notions is undecidable in ZFC.

Stationary measure for six-vertex model on a strip

Stationary measure for six-vertex model on a strip

Fleming–Viot couples live forever

We prove a non-extinction result for Fleming–Viot-type systems of two particles with dynamics described by an arbitrary symmetric Hunt process under the assumption that the reference measure is finite. Additionally, we describe an invariant measure for the system, we discuss its ergodicity, and we prove that the reference measure is a stationary measure for the embedded Markov chain of positions of the surviving particle at successive branching times.

SUBGEOMETRICALLY ERGODIC AUTOREGRESSIONS WITH AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY

In this paper, we consider subgeometric (specifically, polynomial) ergodicity of univariate nonlinear autoregressions with autoregressive conditional heteroskedasticity (ARCH). The notion of subgeometric ergodicity was introduced in the Markov chain literature in the 1980s, and it means that the transition probability measures converge to the stationary measure at a rate slower than geometric; this rate is also closely related to the convergence rate of $\beta $ -mixing coefficients. While the existing literature on subgeometrically ergodic autoregressions assumes a homoskedastic error term, this paper provides an extension to the case of conditionally heteroskedastic ARCH-type errors, considerably widening the scope of potential applications. Specifically, we consider suitably defined higher-order nonlinear autoregressions with possibly nonlinear ARCH errors and show that they are, under appropriate conditions, subgeometrically ergodic at a polynomial rate. An empirical example using energy sector volatility index data illustrates the use of subgeometrically ergodic AR–ARCH models.

Contracting on average iterated function systems by metric change

We study contraction conditions for an iterated function system (IFS) of continuous maps on a metric space which are chosen randomly, identically and independently. We investigate metric changes, preserving the topological structure of the space, which turn the IFS into one which is contracting on average. For the particular case of a system of C 1-diffeomorphisms of the circle which is proximal and does not have a probability measure simultaneously invariant by every map, we derive an equivalent metric which contracts on average and conclude that it carries a unique stationary probability measure. Here we strongly use local contraction properties established by Malicet in 2017, which then imply that the IFS has a negative random Lyapunov exponent and generalizes a result by Baxendale.