Invariant Probability Measure Research Articles

Hopf bifurcation without parameters in deterministic and stochastic modeling of cancer virotherapy, part II

In part II, we analyze our stochastic model which incorporates microenvironmental noises and uncertainties related to immune responses. Outcomes of the therapy in our model are largely determined by the infectivity constant, the infection value, and stochastic relative immune clearance rates. The infection value is a universal critical value for immune-free ergodic invariant probability measures and persistence in all cases. Asymptotic behaviors of the stochastic model are similar to those of its deterministic counterpart. Our stochastic model displays an interesting dynamical behavior, stochastic Hopf bifurcation without parameters, which is a new phenomenon. We perform numerical study to demonstrate how stochastic Hopf bifurcation without parameters occurs. In addition, we give biological implications about our analytical results in stochastic setting versus deterministic setting.

Hidden spatiotemporal symmetries and intermittency in turbulence

We consider general infinite-dimensional dynamical systems with the Galilean and spatiotemporal scaling symmetry groups. Introducing the equivalence relation with respect to temporal scalings and Galilean transformations, we define a representative set containing a single element within each equivalence class. Temporal scalings and Galilean transformations do not commute with the evolution operator (flow) and, hence, the equivalence relation is not invariant. Despite of that, we prove that a normalized flow with an invariant probability measure can be introduced on the representative set, such that symmetries are preserved in the statistical sense. We focus on hidden symmetries, which are broken in the original system but restored in the normalized system. The central motivation and application of this construction is the intermittency phenomenon in turbulence. We show that hidden symmetries yield power law scaling for structure functions, and derive formulas for their exponents in terms of normalized measures. The use of Galilean transformation in the equivalence relation leads to the quasi-Lagrangian description, making the developed theory applicable to the Euler and Navier–Stokes systems.

A countable partition for singular flows, and its application on the entropy theory

In this paper, we construct a countable partition $${\cal A}$$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $${{\cal A}^\infty}$$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows, provided that the limiting measure satisfies μ(Sing(X)) = 0.

Long-Time Behavior for Subcritical Measure-Valued Branching Processes with Immigration

AbstractIn this work we study the long-time behavior for subcritical measure-valued branching processes with immigration on the space of tempered measures. Under some reasonable assumptions on the spatial motion, the branching and immigration mechanisms, we prove the existence and uniqueness of an invariant probability measure for the corresponding Markov transition semigroup. Moreover, we show that it converges with exponential rate to the unique invariant measure in the Wasserstein distance as well as in a distance defined in terms of Laplace transforms. Finally, we consider an application of our results to super-Lévy processes as well as branching particle systems on the lattice with noncompact spins.

Distal strongly ergodic actions

Let \eta be an arbitrary countable ordinal. Using results of Bourgain, Gamburd, and Sarnak on compact systems with spectral gap, we show the existence of an action of the free group on three generators F_3 on a compact metric space X , admitting an invariant probability measure \mu , such that the resulting dynamical system (X,\mu,F_3) is strongly ergodic and distal of rank \eta . In particular, this shows that there is an F_3 system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.

Specific Properties of the ODE’s Flow in Dimension Two Versus Dimension Three

This paper deals with the asymptotics of the ODE’s flow induced by a regular vector field b on the d-dimensional torus \({\mathbb {R}}^d/{\mathbb {Z}}^d\). First, we start by revisiting the Franks-Misiurewicz theorem which claims that the Herman rotation set of any two-dimensional continuous flow is a closed line segment of \({\mathbb {R}}^2\). Various general examples illustrate this result, among which a complete study of the Stepanoff flow associated with a vector field \(b=a\,\zeta \), where \(\zeta \) is a constant vector in \({\mathbb {R}}^2\). Furthermore, several extensions of the Franks-Misiurewicz theorem are obtained in the two-dimensional ODE’s context. On the one hand, we provide some interesting stability properties in the case where the Herman rotation set has a commensurable direction. On the other hand, we present new results highlighting the exceptional character of the opposite case, i.e. when the Herman rotation set is a closed line segment with \(0_{{\mathbb {R}}^2}\) at one end and with an irrational slope, if it is not reduced to a single point. Besides this, given a pair \((\mu ,\nu )\) of invariant probability measures for the flow, we establish new Fourier relations between the determinant \(\det \,(\widehat{\mu b}(j),\widehat{\nu b}(k))\) and the determinant \(\det \,(j,k)\) for any pair (j, k) of non null integer vectors, which can be regarded as an extension of the Franks-Misiurewicz theorem. Next, in contrast with dimension two, any three-dimensional closed convex polyhedron with rational vertices is shown to be the rotation set associated with a suitable vector field b. Finally, in the case of an invariant measure \(\mu \) with a regular density and a non null mass \(\mu (b)\) with respect to b, we show that the homogenization of the two-dimensional transport equation with the oscillating velocity \(b(x/\varepsilon )\) as \(\varepsilon \) tends to 0, leads us to a nonlocal limit transport equation, but with the effective constant velocity \(\mu (b)\).

Entropy, Pressure, Ground States and Calibrated Sub-actions for Linear Dynamics

Denote by $X$ a Banach space and by $T : X \to X$ a bounded linear operator with non-trivial kernel satisfying suitable conditions. We consider the concepts of entropy - for $T$-invariant probability measures - and pressure for H\"older continuous potentials. We also prove the existence of ground states (the limit when temperature goes to zero) associated with such class of potentials when the Banach space $X$ is equipped with a Schauder basis. We produce an example concerning weighted shift operators defined on the Banach spaces $c_0(\mathbb{R})$ and $l^p(\mathbb{R})$, $1 \leq p < +\infty$, where our results do apply. In addition, we prove the existence of calibrated sub-actions when the potential satisfies certain regularity conditions using properties of the so-called Ma\~n\'e potential. We also exhibit examples of selection at zero temperature and explicit sub-actions in the class of H\"older continuous potentials.

Household epidemic models and McKean–Vlasov Poisson driven stochastic differential equations

This paper presents a new view of household epidemic models, where the interaction between the households is of mean field type. We thus obtain in the limit of infinitely many households a nonlinear Markov process solution of a McKean - Vlasov type Poisson driven SDE, and a propagation of chaos result. We also define a basic reproduction number R0, and show that if R0>1, then the nonlinear Markov process has a unique non trivial ergodic invariant probability measure, whereas if R0<=1, it converges to 0 as t tends to infinity.

An Efficient Jet Marcher for Computing the Quasipotential for 2D SDEs

We present a new algorithm, the efficient jet marching method (EJM), for computing the quasipotential and its gradient for two-dimensional SDEs. The quasipotential is a potential-like function for nongradient SDEs that gives asymptotic estimates for the invariant probability measure, expected escape times from basins of attractors, and maximum likelihood escape paths. The quasipotential is a solution to an optimal control problem with an anisotropic cost function which can be solved for numerically via Dijkstra-like label-setting methods. Previous Dijkstra-like quasipotential solvers have displayed in general 1st order accuracy in the mesh spacing. However, by utilizing higher order interpolations of the quasipotential as well as more accurate approximations of the minimum action paths, EJM achieves second-order accuracy for the quasipotential and nearly second-order for its gradient. Moreover, by using targeted search neighborhoods for the fastest characteristics following the ideas of Mirebeau, EJM also enjoys a reduction in computation time. This highly accurate solver enables us to compute the prefactor for the WKB approximation for the invariant probability measure and the Bouchet–Reygner sharp estimate for the expected escape time for the Maier–Stein SDE. Our codes are available on GitHub.

The adaptive biasing force algorithm with non-conservative forces and related topics

We propose a study of the Adaptive Biasing Force method’s robustness under generic (possibly non-conservative) forces. We first ensure the flat histogram property is satisfied in all cases. We then introduce a fixed point problem yielding the existence of a stationary state for both the Adaptive Biasing Force and Projected Adapted Biasing Force algorithms, relying on generic bounds on the invariant probability measures of homogeneous diffusions. Using classical entropy techniques, we prove the exponential convergence of both biasing force and law as time goes to infinity, for both the Adaptive Biasing Force and the Projected Adaptive Biasing Force methods.

Ergodic Optimization and Zero Temperature Limits in Negative Curvature

In this paper, we study aspects of the ergodic theory of the geodesic flow on a non-compact negatively curved manifold. In topological dynamics, it is a well-known fact that every continuous potential on a compact metric space has a maximizing measure. Unfortunately, for non-compact spaces this fact is no longer true. For the geodesic flow we provide a criterion that ensures the existence of a maximizing measure for uniformly continuous potentials. We prove that the only obstruction to the existence of a maximizing measure is the full escape of mass phenomenon. To the best of our knowledge, this is the first general result on the existence of maximizing measures for non-compact topological spaces which does not require the potential to be coercive. We study zero temperature limits of equilibrium measures for a suitable family of potentials. We prove some convergence and divergence results for the limiting behavior of such measures. Among some consequences, we obtain that the geodesic flow has the intermediate entropy property and that equilibrium states are dense in the space of invariant probability measures.

On the asymptotic behavior of the Diaconis–Freedman chain in a multi-dimensional simplex

AbstractWe give a setting of the Diaconis–Freedman chain in a multi-dimensional simplex and consider its asymptotic behavior. By using techniques from random iterated function theory and quasi-compact operator theory, we first give some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure and, in particular cases, explicit formulas for the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimension two. Some other settings of the chain are also discussed.

The α-dependence of the invariant measure of stochastic real Ginzburg-Landau equation driven by α-stable Lévy processes

This paper is intended to establish the connection between stochastic dynamical systems with non-Gaussian noises and stochastic dynamical systems with Gaussian noises, by considering the convergence behavior of invariant probability measures for stochastic real Ginzburg-Landau equation driven by cylindrical α-stable Lévy process, in the limit α→2. Indeed, we prove that the invariant measure of stochastic real Ginzburg-Landau equation on torus and driven by cylindrical α-stable Lévy processes converges to the invariant measure of stochastic real Ginzburg-Landau equation forced by cylindrical Brownian motions under the Wasserstein distance, as α tends to 2. We state our strategy as below.First, by using an abstract result given by Hairer and Mattingly, we prove that a type of Wasserstein distance is contracting for the dual operator of Markov semigroup associated with the limit equation which forced by Brownian motions. Then, through server priori uniform moment estimates and a convergence result on stochastic convolutions of cylindrical subordinated Brownian motions, we establish a strong convergence result on the solution of stochastic real Ginzburg-Landau equation driven by cylindrical subordinated Brownian motions conditioned on the initial data is distributed as an invariant measure, in the limit α→2. Finally, by the Monge-Kantorovich duality, we prove that the invariant probability measure of stochastic real Ginzburg-Landau equation driven by cylindrical α-stable Lévy processes converges to the invariant measure of stochastic real Ginzburg-Landau equation forced by Brownian motions under the Wasserstein distance.

Limits of invariant measures of stochastic Burgers equations driven by two kinds of [formula omitted]-stable processes

In this work, we study the convergence behavior, as α→2, of the invariant probability measures of stochastic Burgers equations driven by two kinds of cylindrical α-stable processes (i.e., cylindrical subordinated Brownian motions and subordinated cylindrical Brownian motions) on torus. We prove that the invariant probability measures of stochastic Burgers equations driven by cylindrical subordinated Brownian motions or subordinated cylindrical Brownian motions converge to the invariant probability measure of stochastic Burgers equations forced by cylindrical Brownian motions in Wasserstein distance. This result shows a connection between stochastic dynamical systems with non-Gaussian noises and stochastic dynamical systems with Gaussian noises.

Exponential ergodicity for regime-switching diffusion processes in total variation norm

&lt;p style='text-indent:20px;'&gt;We investigate the long time behavior for a class of regime-switching diffusion processes. Based on direct evaluation of moments and exponential functionals of hitting time of the underlying process, we adopt coupling method to obtain existence and uniqueness of the invariant probability measure and establish explicit exponential bounds for the rate of convergence to the invariant probability measure in total variation norm. In addition, we provide some concrete examples to illustrate our main results which reveal impact of random switching on stochastic stability and convergence rate of the system.&lt;/p&gt;

Robust uniform persistence for structured models of delay differential equations

&lt;p style='text-indent:20px;'&gt;We consider a general class of delay differential equations systems, typically used to model the dynamics of structured biological populations, and establish necessary conditions for the part of the attractor contained in the boundary of the state space to repel the complementary dynamics contained in the interior of the state space. The conditions are formulated in terms of Lyapunov exponents and invariant probability measures and we use them to prove a robust uniform persistence result.&lt;/p&gt;

Ergodicity for stochastic equations of Navier–Stokes type

In this note we consider a simple example of a finite dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic Navier-Stokes Equations (NSEs), and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant probability measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant probability measure, there exist another one. In particular, the generator of the system is not hypoelliptic.

Statistical solution and piecewise Liouville theorem for the impulsive discrete Zakharov equations

&lt;abstract&gt;&lt;p&gt;This article studies the discrete Zakharov equations with impulsive effect. The authors first prove that the problem is global well-posed and that the process formed by the solution operators possesses a pullback attractor. Then they establish that there is a family of invariant Borel probability measures contained in the pullback attractor, and that this family of measures satisfies the Liouville type theorem piecewise and is a statistical solution of the impulsive discrete Zakharov equations.&lt;/p&gt;&lt;/abstract&gt;

Existence of invariant probability measures for functional McKean-Vlasov SDEs

We show existence of an invariant probability measure for a class of functional McKean-Vlasov SDEs by applying Kakutani’s fixed point theorem to a suitable class of probability measures on a space of continuous functions. Unlike some previous works [1, 25], we do not assume a monotonicity condition to hold. Further, our conditions are even weaker than some results in the literature on invariant probability measures for functional SDEs without dependence on the law of the solution [12].

SUFFICIENT AND NECESSARY CONDITIONS FOR PERSISTENCE AND EXTINCTION OF A STOCHASTIC TWO-PREY ONE-PREDATOR SYSTEM

This paper applies a new approach for stochastic Kolmogorov systems generalized by Hening and Nguyen to describe the dynamics of a stochastic two independent prey one predator system perturbed by white noise. Through calculating Lyapunov exponents, we thoroughly address the stability of the ergodic invariant probability measures. Sufficient and necessary conditions under which the species persist as well as conditions under which some species go extinct are established for this three dimensional models. One of the key points is that the critical cases for Lyapunov exponents being zero are considered. Finally, some numerical simulations illustrate the analytical results.