Invariant Measure Research Articles

New non degenerate invariant measures for the Benjamin–Ono equation

New non degenerate invariant measures for the Benjamin–Ono equation

Stability and Invariant Measure Asymptotics in a Model for Heavy Particles in Rough Turbulent Flows

Stability and Invariant Measure Asymptotics in a Model for Heavy Particles in Rough Turbulent Flows

Convergence and Approximation of Invariant Measures for Neural Field Lattice Models under Noise Perturbation

Convergence and Approximation of Invariant Measures for Neural Field Lattice Models under Noise Perturbation

The Statistical Theory of the Angiogenesis Equations

Angiogenesis is a multiscale process by which a primary blood vessel issues secondary vessel sprouts that reach regions lacking oxygen. Angiogenesis can be a natural process of organ growth and development or a pathological one induced by a cancerous tumor. A mean-field approximation for a stochastic model of angiogenesis consists of a partial differential equation (PDE) for the density of active vessel tips. Addition of Gaussian and jump noise terms to this equation produces a stochastic PDE that defines an infinite-dimensional Lévy process and is the basis of a statistical theory of angiogenesis. The associated functional equation has been solved and the invariant measure obtained. The results of this theory are compared to direct numerical simulations of the underlying angiogenesis model. The invariant measure and the moments are functions of a Korteweg–de Vries-like soliton, which approximates the deterministic density of active vessel tips.

Solvable Stationary Non Equilibrium States

We consider the one dimensional boundary driven harmonic model and its continuous version, both introduced in (Frassek et al. in J Stat Phys 180: 135–171, 2020). By combining duality and integrability the authors of (Frassek and Giardiná in J Math Phys 63: 103301, 2022) obtained the invariant measures in a combinatorial representation. Here we give an integral representation of the invariant measures which turns out to be a convex combination of inhomogeneous product of geometric distributions for the discrete model and a convex combination of inhomogeneous product of exponential distributions for the continuous one. The mean values of the geometric and of the exponential variables are distributed according to the order statistics of i.i.d. uniform random variables on a suitable interval fixed by the boundary sources. The result is obtained solving exactly the stationary condition written in terms of the joint generating function. The method has an interest in itself and can be generalized to study other models. We briefly discuss some applications.

Exotic B-Series and S-Series: Algebraic Structures and Order Conditions for Invariant Measure Sampling

AbstractB-Series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-Series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-Series and S-Series. Precisely, we prove the relationship between the Grossman–Larson algebras over exotic and grafted forests and the corresponding duals to the Connes–Kreimer coalgebras and use it to study the natural composition laws on exotic S-Series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge–Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.

Cultural adaptation to Bolivian Quechua and psychometric analysis of the Patient Health Questionnaire PHQ-9

ObjectiveCultural adaptation of the Patient Health Questionnaire-PHQ-9 to Bolivian Quechua and analysis of the internal structure validity, reliability, and measurement invariance by sociodemographic variables.MethodsThe PHQ-9 was translated and back-translated (English-Quechua-English) to optimise translation. For the cultural adaptation, experts, and people from the target population (e.g., in focus groups) verified the suitability of the translated PHQ-9. For the psychometric analysis, we performed a Confirmatory Factor Analysis (CFA) to evaluate internal validity, calculated α and ω indices to assess reliability, and performed a Multiple Indicator, Multiple Cause (MIMIC) model for evaluating measurement invariance by sex, age, marital status, educational level and residence. We used standard goodness-of-fit indices to interpret both CFA results.ResultsThe experts and focus groups improved the translated PHQ-9, making it clear and culturally equivalent. For the psychometric analysis, we included data from 397 participants, from which 73.3% were female, 33.0% were 18–30 years old, 56.7% reported primary school studies, 63.2% were single, and 62.0% resided in urban areas. In the CFA, the single-factor model showed adequate fit (Comparative Fit Index = 0.983; Tucker-Lewis Index = 0.977; Standardized Root Mean Squared Residual = 0.046; Root Mean Squared Error of Approximation = 0.069), while the reliability was optimal (α = 0.869—0.877; ω = 0.874—0.885). The invariance was confirmed across all sociodemographic variables (Change in Comparative Fit Index (delta) or Root Mean Square Error of Approximation (delta) < 0.01).ConclusionsThe PHQ-9 adapted to Bolivian Quechua offers a valid, reliable and invariant unidimensional measurement across groups by sex, age, marital status, educational level and residence.

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.

A stochastic framework for evaluating CAR T cell therapy efficacy and variability

Based on a deterministic and stochastic process hybrid model, we use white noises to account for patient variabilities in treatment outcomes, use a hyperparameter to represent patient heterogeneity in a cohort, and construct a stochastic model in terms of Ito stochastic differential equations for testing the efficacy of three different treatment protocols in CAR T cell therapy. The stochastic model has three ergodic invariant measures which correspond to three unstable equilibrium solutions of the deterministic system, while the ergodic invariant measures are attractors under some conditions for tumor growth. As the stable dynamics of the stochastic system reflects long-term outcomes of the therapy, the transient dynamics provide chances of cure in short-term. Two stopping times, the time to cure and time to progress, allow us to conduct numerical simulations with three different protocols of CAR T cell treatment through the transient dynamics of the stochastic model. The probability distributions of the time to cure and time to progress present outcome details of different protocols, which are significant for current clinical study of CAR T cell therapy.

General criterion applicable to exclusion processes with an Ising-like invariant measure

This paper explores extensions of the totally asymmetric simple exclusion process in one dimension, assuming periodic boundary conditions. Specifically, we investigate situations where a particle’s movement rate is influenced by the positions of its two nearest neighbors. We establish the necessary and sufficient conditions for these exclusion processes to maintain an Ising-like measure as their invariant measure. Our analysis covers several renowned models, such as the cooperative exclusion process (Antal and Schütz, 2000), its dual model (Gabel and Redner, 2011), the Katz–Lebowitz–Spohn model (Katz et al., 1984), and the facilitated exclusion process (Basu and Mohanty, 2009; Gabel et al., 2010). To compare the obtained models, we will examine the average particle velocity and flux. Upon doing so, we highlight that this research contributes to the comprehension of the collective pushing phenomenon observed, for example, in biological traffic models (Epshtein and Nudler, 2003; Epshtein et al., 2003).

On the Kolmogorov constant explicit form in the theory of discrete-time stochastic branching systems

AbstractWe consider a discrete-time population growth system called the Bienaymé–Galton–Watson stochastic branching system. We deal with a noncritical case, in which the per capita offspring mean $m\neq1$ . The famous Kolmogorov theorem asserts that the expectation of the population size in the subcritical case $m&lt;1$ on positive trajectories of the system asymptotically stabilizes and approaches ${1}/\mathcal{K}$ , where $\mathcal{K}$ is called the Kolmogorov constant. The paper is devoted to the search for an explicit expression of this constant depending on the structural parameters of the system. Our argumentation is essentially based on the basic lemma describing the asymptotic expansion of the probability-generating function of the number of individuals. We state this lemma for the noncritical case. Subsequently, we find an extended analogue of the Kolmogorov constant in the noncritical case. An important role in our discussion is also played by the asymptotic properties of transition probabilities of the Q-process and their convergence to invariant measures. Obtaining the explicit form of the extended Kolmogorov constant, we refine several limit theorems of the theory of noncritical branching systems, showing explicit leading terms in the asymptotic expansions.

A large deviation principle for the empirical measures of Metropolis–Hastings chains

To sample from a given target distribution, Markov chain Monte Carlo (MCMC) sampling relies on constructing an ergodic Markov chain with the target distribution as its invariant measure. For any MCMC method, an important question is how to evaluate its efficiency. One approach is to consider the associated empirical measure and how fast it converges to the stationary distribution of the underlying Markov process. Recently, this question has been considered from the perspective of large deviation theory, for different types of MCMC methods, including, e.g., non-reversible Metropolis–Hastings on a finite state space, non-reversible Langevin samplers, the zig-zag sampler, and parallel tempering. This approach, based on large deviations, has proven successful in analysing existing methods and designing new, efficient ones. However, for the Metropolis–Hastings algorithm on more general state spaces, the workhorse of MCMC sampling, the same techniques have not been available for analysing performance, as the underlying Markov chain dynamics violate the conditions used to prove existing large deviation results for empirical measures of a Markov chain. This also extends to methods built on the same idea as Metropolis–Hastings, such as the Metropolis-Adjusted Langevin Method or ABC-MCMC. In this paper, we take the first steps towards such a large-deviations based analysis of Metropolis–Hastings-like methods, by proving a large deviation principle for the empirical measures of Metropolis–Hastings chains. In addition, we also characterize the rate function and its properties in terms of the acceptance- and rejection-part of the Metropolis–Hastings dynamics.

Persistence in Perturbed Contact Models in Continuum

Can a local disaster lead to extinction? We answer this question in this work. In the paper [19] we considered contact processes on locally compact metric spaces with state dependent birth and death rates and formulated suf- ficient conditions on the rates that ensure the existence of invariant measures. One of the crucial conditions in [19] was the critical regime condition, which meant the existence of a balance between birth and death rates in average. In the present work, we reject the criticality condition and suppose that the bal- ance condition is violated. This implies that the evolution of the correlation functions of the contact model under consideration is determined by a nonlocal convolution type operator perturbed by a (negative) potential. We show that local peaks in mortality do not typically lead to extinction. We prove that a family of invariant measures exists even without the criticality condition and these measures can be described using the Feynman-Kac formula.

Closing lemmas for interval translation maps

A interval translation mapping (or a circle translation mapping) is studied. Such maps can be regarded as interval exchange maps with overlaps. It is known that for any mapping of that type admits a Borel probability invariant non-atomic measure. This measure can be constructed as a weak limit of invariant measures of maps with periodic parameters. Those measures, are just normalized Lebesgue ones on a family of sub-sectors. For such limit measures, in the case of a shift of the arcs of the circle, it is shown that any point of their supports can be made periodic by arbitrarily small change of the parameters of the system without changing the number of segments. For any invariant measure, it is deduced from Poincar´es Recurrence Theorem shows every point can be made periodic by a small change in the parameters of the system, with the number of intervals increasing by two at most.

Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on Rn.

This paper is concerned with invariant measures of fractional stochastic delay Ginzburg-Landau equations on the entire space $ \mathbb{R}^n $. We first derive the uniform estimates and the mean-square uniform smallness of the tails of solutions in corresponding space. Then we deduce the weak compactness of a set of probability distributions of the solutions applying the Ascoli-Arzel$ \grave{a} $. We finally prove the existence of invariant measures by applying Krylov-Bogolyubov's method.

Invariant measures and statistical solutions of the globally modified magnetohydrodynamics equations

Invariant measures and statistical solutions of the globally modified magnetohydrodynamics equations

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

Limiting behavior of invariant or periodic measure of Hopfield neural models driven by locally Lipschitz Lévy noise

Limiting behavior of invariant or periodic measure of Hopfield neural models driven by locally Lipschitz Lévy noise

Dynamics of a stochastic epidemic model with information intervention and vertical transmission

&lt;abstract&gt;&lt;p&gt;The dynamic behavior of a stochastic epidemic model with information intervention and vertical transmission was the concern of this paper. The threshold to judge the extinction and persistence of the disease was obtained. Specifically, when $ \Delta &amp;lt; 0 $ ($ \Delta $ appears in Section 3), the three classes $ I_t $, $ M_t $, and $ R_t $ appearing in the model go extinct at an exponential rate, and the susceptible class $ S_t $ almost surely converges to the solution of the boundary equation exponentially. When $ \Delta &amp;gt; 0 $, the result that the disease in the model is persistent in the mean and the existence of invariant probability measure are proved by constructing a new form of Lyapunov functions, which results in getting sufficient and nearly necessary conditions for different properties. Moreover, one of the main characteristics of this article was the study of the critical case of $ \Delta = 0 $ under some conditions. Some examples were listed to confirm the obtained results.&lt;/p&gt;&lt;/abstract&gt;

Regularity of pullback random attractors and invariant sample measures for nonautonomous stochastic &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ p $&lt;/tex-math&gt;&lt;/inline-formula&gt;-Laplacian lattice systems

Regularity of pullback random attractors and invariant sample measures for nonautonomous stochastic &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ p $&lt;/tex-math&gt;&lt;/inline-formula&gt;-Laplacian lattice systems