Unique Invariant Measure Research Articles

Self-interacting CBO: Existence, uniqueness, and long-time convergence

A self-interacting dynamics that mimics the standard Consensus-Based Optimization (CBO) model is introduced. This single-particle dynamics is shown to converge to a unique invariant measure that approximates the global minimum of a given function. As an application, its connection to CBO with Personal Best introduced by C. Totzeck and M.-T. Wolfram (Math. Biosci. Eng., 2020) has been established.

Stability analysis of a stochastic port-Hamiltonian car-following model

Port-Hamiltonian systems are pertinent representations of many nonlinear physical systems. In this study, we formulate and analyse a general class of stochastic car-following models with a systematic port-Hamiltonian structure. The model class is a generalisation of classical car-following approaches, including the optimal velocity model of Bando et al (1995 Phys. Rev. E 51 1035), the full velocity difference model of Jiang et al (2001 Phys. Rev. E 64 017101), and recent stochastic following models based on the Ornstein–Uhlenbeck process. In contrast to traditional models where the interaction is totally asymmetric (i.e. depending only on the speed and distance to the predecessor), the port-Hamiltonian car-following model also depends on the distance to the follower. We determine the exact stability condition of the finite system with N vehicles and periodic boundaries. The stable system is ergodic with a unique Gaussian invariant measure. Other properties of the model are studied using numerical simulation. It turns out that the Hamiltonian component improves the flow stability and reduces the total energy in the system. Furthermore, it prevents the problematic formation of stop-and-go waves with oscillatory dynamics, even in the presence of stochastic perturbations.

Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori–Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard–Jones and Coulomb functions.

Lattice regularization of reduced Kähler-Dirac fermions and connections to chiral fermions

We show how a path integral for reduced Kähler-Dirac fermions suffers from a phase ambiguity associated with the fermion measure that is an analog of the measure problem seen for chiral fermions. However, unlike the case of chiral symmetry, a doubler free lattice action exists which is invariant under the corresponding onsite symmetry. This allows for a clear diagnosis and solution to the problem using mirror fermions resulting in a unique gauge invariant measure. By introducing an appropriate set of Yukawa interactions which are consistent with ’t Hooft anomaly cancellation we conjecture the mirrors can be decoupled from low energy physics. Moreover, the minimal such Kähler-Dirac mirror model yields a light sector which corresponds, in the flat space continuum limit, to the Pati-Salam GUT model.

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

AbstractWe show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

Stability and dynamical bifurcation of a stochastic regime-switching predator–prey model

This paper considers a stochastic regime-switching predator–prey system with general functional response. We show that the system exactly possesses two dynamical bifurcation points (DBPs), and the expressions of these DBPs are given explicitly. In addition, we provide the sharp sufficient criteria under which the system admits a unique ergodic invariant measure (UEIM), and show that the transition function of the solution of the system exponentially converges to the UEIM under the total variation norm. The findings show that the stability and dynamical bifurcation of the system have close relationships with the random perturbations. Additionally, the main findings are applied to several special cases, and a number of recent reports are improved.

Invariant measures for stochastic FitzHugh-Nagumo delay lattice systems with long-range interactions in weighted space

<abstract><p>The focus of this paper lies in exploring the limiting dynamics of stochastic FitzHugh-Nagumo delay lattice systems with long-range interactions and nonlinear noise in weighted space. To begin, we established the well-posedness of solutions to these stochastic delay lattice systems and subsequently proved the existence and uniqueness of invariant measures.</p></abstract>

Well-posedness and invariant measures for 2D stochastic Oldroyd model of order one with pure jumps

In this paper, we consider the stochastic two-dimensional viscoelastic fluid flow equations driven by a multiplicative Lévy noise, arising from the Oldroyd model for the non-Newtonian fluid flows. The system carries the natural Lipschitz condition on balls and linear growth assumptions on the jump coefficient. We first prove the existence and uniqueness of a global strong solution by using stopping time technique and Banach fixed point theorem. Meanwhile, we overcome the difficulties posed by the jump and memory term, the latter makes some convergence findings more natural. By demonstrating the exponential stability of strong solutions, we establish the existence of a unique invariant measure for the stochastic system. Numerical results provide further support for these theoretical findings, the dissipative velocity of solution is more rapid and rigorous when compared with the stochastic Navier-Stokes equation.

Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O(λN−γ) in space and O(τγ) in time, where γ∈(0,1] from the assumption ∥Aγ−12Q12∥L2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings.

Stability and moment estimates for the stochastic singular Φ-Laplace equation

We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular stochastic p-Laplace equations and, more generally, singular stochastic Φ-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by Es-Sarhir et al. (2012) [22], and improve the results by Liu and Tölle (2011) [41].

Partially hyperbolic random dynamics on Grassmannians

A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions, it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure, showing that with high probability, the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, in a regime where the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained.

Invariant measures for stochastic conservation laws on the line

We consider a stochastic conservation law on the line with solution-dependent diffusivity, a super-linear, sub-quadratic Hamiltonian, and smooth, spatially-homogeneous kick-type random forcing. We show that this Markov process admits a unique ergodic spatially-homogeneous invariant measure for each mean in a non-explicit unbounded set. This generalises previous work on the stochastic Burgers equation.

Limit theorems for quantum trajectories

Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Under purification and irreducibility assumptions, these Markov processes admit a unique invariant measure — see Benoist et al. (2019). In this article we prove finer limit theorems such as Law of Large Numbers (LLN), Functional Central Limit Theorem, Law of Iterated Logarithm and Moderate Deviation Principle. The proof of the LLN is based on Birkhoff’s ergodic theorem and an analysis of harmonic functions. The other theorems are proved using martingale approximation of empirical sums.

On the abundance of k-fold semi-monotone minimal sets in bimodal circle maps

AbstractInspired by a twist map theorem of Mather. we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map g to the k-fold cover. For each irrational in the rotation set’s interior, the collection of the k-fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$ -dimensional ball with the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool used is a generalization of a construction of Hedlund and Morse that generates symbolic analogs of these k-fold well-ordered invariant sets.

The small mass limit for long time statistics of a stochastic nonlinear damped wave equation

We study the long time statistics of a class of semi–linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase space are stochastically forced, the system is exponentially attractive toward its unique invariant measure with a convergent rate that is uniform with respect to the mass. Then, in the small mass limit, we prove the convergence of the first marginal of the invariant measures in a suitable Wasserstein distance toward the unique invariant measure of a stochastic reaction–diffusion equation. Together with the uniform geometric ergodicity, we obtain the validity of the small mass limit for the solutions on the infinite time horizon [0,∞). This extends previously known results established for the damped wave equations under Lipschitz nonlinearities.

DYNAMICAL BEHAVIORS OF A CLASS OF STOCHASTIC TUMOR–IMMUNE SYSTEMS

In this paper, we consider a class of tumor–immune systems perturbed by the environmental noise and focus on the longtime behaviors. The existence and uniqueness of the globally positive solution to the tumor–immune system are proved using stochastic Lyapunov analysis and Itô’s formula. We study the boundedness of moments for tumor cells and effector cells. By considering the dynamics on the boundary, applying the comparison theorem and the strong ergodic theorem, we obtain a threshold [Formula: see text] which is used to characterize the stochastic permanence in the sense that there is a unique invariant measure and extinction of the stochastic tumor–immune system. We also give biological interpretations about our analytical results of stochastic system. In addition, we present numerical examples and discussions to illustrate our analysis results. We find that the small noises preserve Hopf bifurcation of the deterministic system in stochastic setting and study numerically how the stochastic Hopf bifurcation with parameters occurs.

Well-posedness and stationary solutions of McKean-Vlasov (S)PDEs

This paper is composed of two parts. In the first part we consider McKean-Vlasov Partial Differential Equations (PDEs), obtained as thermodynamic limits of interacting particle systems (i.e. in the limit N→∞, where N is the number of particles). It is well-known that, even when the particle system has a unique invariant measure (stationary solution), the limiting PDE very often displays a phase transition: for certain choices of (coefficients and) parameter values, the PDE has a unique stationary solution, but as the value of the parameter varies multiple stationary states appear. In the first part of this paper, we add to this stream of literature and consider a specific instance of a McKean-Vlasov type equation, namely the Kuramoto model on the torus perturbed by a symmetric double-well potential, and show that this PDE undergoes the type of phase transition just described, as the diffusion coefficient is varied. In the second part of the paper, we consider a rather general class of McKean-Vlasov PDEs on the torus (which includes both the original Kuramoto model and the Kuramoto model in double well potential of part one) perturbed by (strong enough) infinite-dimensional additive noise. To the best of our knowledge, the resulting Stochastic PDE, which we refer to as the Stochastic McKean-Vlasov equation, has not been studied before, so we first study its well-posedness. We then show that the addition of noise to the PDE has the effect of restoring uniqueness of the stationary state in the sense that, irrespective of the choice of coefficients and parameter values in the McKean-Vlasov PDE, the Stochastic McKean-Vlasov PDE always admits at most one invariant measure.

Analysis of a Class of Multilevel Markov Chain Monte Carlo Algorithms Based on Independent Metropolis–Hastings

In this work, we present, analyze, and implement a class of multilevel Markov chain Monte Carlo (ML-MCMC) algorithms based on independent Metropolis–Hastings proposals for Bayesian inverse problems. In this context, the likelihood function involves solving a complex differential model, which is then approximated on a sequence of increasingly accurate discretizations. The key point of this algorithm is to construct highly coupled Markov chains together with the standard multilevel Monte Carlo argument to obtain a better cost-tolerance complexity than a single-level MCMC algorithm. Our method extends the ideas of Dodwell et al., [SIAM/ASA J. Uncertain. Quantif., 3 (2015), pp. 1075–1108] to a wider range of proposal distributions. We present a thorough convergence analysis of the ML-MCMC method proposed, and show, in particular, that (i) under some mild conditions on the (independent) proposals and the family of posteriors, there exists a unique invariant probability measure for the coupled chains generated by our method, and (ii) that such coupled chains are uniformly ergodic. We also generalize the cost-tolerance theorem of Dodwell et al. to our wider class of ML-MCMC algorithms. Finally, we propose a self-tuning continuation-type ML-MCMC algorithm. The presented method is tested on an array of academic examples, where some of our theoretical results are numerically verified. These numerical experiments evidence how our extended ML-MCMC method is robust when targeting some pathological posteriors, for which some of the previously proposed ML-MCMC algorithms fail.

Degenerate Kolmogorov equations and ergodicity for the stochastic Allen–Cahn equation with logarithmic potential

Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen–Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at the respective potential barriers, making thus the corresponding Kolmogorov equation not uniformly elliptic in space. First, existence and uniqueness of invariant measures and ergodicity are discussed. Then, classical solutions to some regularised Kolmogorov equations are explicitly constructed. Eventually, a sharp analysis of the blow-up rates of the regularised solutions and a passage to the limit with a specific scaling yield existence à la Friedrichs for the original Kolmogorov equation.

Ergodicity for Stochastic Conservation Laws with Multiplicative Noise

We proved that there exists a unique invariant measure for solutions of stochastic conservation laws with Dirichlet boundary condition driven by multiplicative noise. Moreover, a polynomial mixing property is established. This is done in the setting of kinetic solutions taking values in an $$L^1$$ -weighted space.