Random Manifolds Research Articles

Color confinement and random matrices. A random walk down group manifold toward Casimir scaling

We explain the microscopic origin of linear confinement potential with the Casimir scaling in generic confining gauge theories. In the low-temperature regime of confining gauge theories such as QCD, Polyakov lines are slowly varying Haar random modulo exponentially small corrections with respect to the inverse temperature, as shown by one of the authors (M. H.) and Watanabe. With exact Haar randomness, computation of the two-point correlator of Polyakov loops reduces to the problem of random walk on group manifold. Linear confinement potential with approximate Casimir scaling except at short distances follows naturally from slowly varying Haar randomness. With exponentially small corrections to Haar randomness, string breaking and loss of Casimir scaling at long distance follow. Hence we obtain the Casimir scaling which is only approximate and holds only at intermediate distance, which is precisely needed to explain the results of lattice simulations. For (1 + 1)-dimensional theories, there is a simplification that admits the Casimir scaling at short distances as well.

Random Manifold Sampling and Joint Sparse Regularization for Multi-Label Feature Selection

Random Manifold Sampling and Joint Sparse Regularization for Multi-Label Feature Selection

Approximations of center manifolds for delay stochastic differential equations with additive noise

Abstract This article deals with approximations of center manifolds for delay stochastic differential equations with additive noise. We first prove the existence and smoothness of random center manifolds for these approximation equations. Then we show that the C k {C}^{k} invariant center manifolds of the system with colored noise approximate that of the original system.

Stability and constructions of the Poincaré maps for a class of stochastic partial differential equations

This work is concerned with a class of stochastic partial differential equations with a multiplicative noise. The Poincaré map, as a main tool to study the periodic orbits, plays a crucially important role in the invariant sets of the system, which captures the dominated dynamical behavior of the trajectories. It derives out the existence and the stability of the stochastic Poincaré maps of the system, which return to a small neighborhood of a fixed point of a deterministic Poincaré map one time or even infinite times with the small noise disturbing. It further expresses the constructions of the stochastic Poincaré maps through the random invariant manifold and the moving orthonormal system.

Limiting behavior of center manifolds for stochastic evolutionary equations with delay in varying phase spaces

In this article, we consider random center manifolds for a class of stochastic evolutionary equations with delay driven by multiplicative white noise. We first prove the existence and Ck smoothness of random center manifolds for the equations with delay. Then, we show the Ck smooth convergence of the center manifolds as the phase spaces approach to their singular limit.

Effective filtering for slow-fast systems via Wong-Zakai approximation

Nonlinear filtering problem has been widely investigated in diverse fields. One important topic related to it is how to formulate an effective approximation scheme for a given observation. By using Wong-Zakai approximation and random invariant manifold theory, we will propose an effective approximation result for a class of slow-fast systems with respect to filtering. We will firstly establish the smooth reduced system via random invariant manifold theory, and then show exponential attractive property of it. At last, we will prove the filtering of the smooth reduced system can well approximate that of the original system.

Center manifolds for rough partial differential equations

We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a γ-Hölder rough path, for γ∈(1∕3,1∕2]. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical systems (RDS). We also illustrate our main theorem for reaction-diffusion equations as well as for the Swift-Hohenberg equation.

Mean-square invariant manifolds for stochastic weak-damping wave equations with nonlinear noise

We study mean-square invariant manifolds for stochastic wave equations with nonlinear infinite-dimensional noise and weak damping, where all nonlinear terms satisfy the nonhomogeneous Lipschitz conditions. First, we consider the existence of a mean-square random dynamical system generated from the mild solution. Second, we give a careful analysis for the spectrum of the wave operator and show that the wave operator satisfies the pseudo exponential dichotomy. Finally, by using the Lyapunov-Perron method and analyzing the conditional expectation solution, we show the existence of a mean-square random invariant unstable manifold as well as a mean-square random invariant stable set.

A model for random three-manifolds

We study compact three-manifolds with boundary obtained by randomly gluing together truncated tetrahedra along their faces. We prove that, asymptotically almost surely as the number of tetrahedra tends to infinity, these manifolds are connected and have a single boundary component.We prove a lawof large numbers for the genus of this boundary component, we show that the Heegaard genus of these manifolds is linear in the number of tetrahedra and we bound their first Betti number. We also show that, asymptotically almost surely as the number of tetrahedra tends to infinity, our manifolds admit a unique hyperbolic metric with totally geodesic boundary.We prove a law of large numbers for the volume of this metric, prove that the associated Laplacian has a uniform spectral gap and show that the diameter of our manifolds is logarithmic as a function of their volume. Finally, we determine the Benjamini–Schramm limit of our sequence of random manifolds.

The Poincaré maps of a slow-fast stochastic system

This work is concerned with the Poincaré maps of a slow-fast stochastic system. Applying the random manifold reduced theory and the moving orthonormal system argument, it derives the effective construction of the stochastic Poincaré map. Moreover, employing the geometric distribution of the first exit time, it builds the stability of the stochastic Poincaré maps, that is, the stochastic Poincaré maps return to a small neighborhood of a fixed point of a deterministic Poincaré map with one time or even infinite times as the noise intensity tends to zero. It further investigates the limit distribution of the first exit location on the boundary of the domain via the harmonic measure.

Stability analysis of the projectile based on random center manifold reduction

The center manifold method has been widely used in the field of stochastic dynamics as a dimensionality reduction method. This paper studied the angular motion stability of a projectile system under random disturbances. The random bifurcation of the projectile is studied using the idea of the Routh-Hurwitz stability criterion, the center manifold reduction, and the polar coordinates transformation. Then, an approximate analytical presentation for the stationary probability density function is found from the related Fokker–Planck equation. From the results, the random dynamical system of projectile generates three different dynamical behaviors with the changes of the bifurcation parameter and the noise strength, which can be a reference for projectile design.

The curvature effect in Gaussian random fields

Random field models are mathematical structures used in the study of stochastic complex systems. In this paper, we compute the shape operator of Gaussian random field manifolds using the first and second fundamental forms (Fisher information matrices). Using Markov chain Monte Carlo techniques, we simulate the dynamics of these random fields and compute the Gaussian, mean and principal curvatures of the parametric space, analyzing how these quantities change along dynamics exhibiting phase transitions. During the simulations, we have observed an unexpected phenomenon that we called the curvature effect, which indicates that a highly asymmetric geometric deformation happens in the underlying parametric space when there are significant increase/decrease in the system’s entropy. When the system undergoes a phase transition from randomness to clustered behavior the curvature is smaller than that observed in the reverse phase transition. This asymmetric pattern relates to the emergence of hysteresis phenomenon, leading to an intrinsic arrow of time along the random field dynamics.

Random Perturbation of Invariant Manifolds for Non-Autonomous Dynamical Systems

Random invariant manifolds are geometric objects useful for understanding dynamics near the random fixed point under stochastic influences. Under the framework of a dynamical system, we compared perturbed random non-autonomous partial differential equations with original stochastic non-autonomous partial differential equations. Mainly, we derived some pathwise approximation results of random invariant manifolds when the Gaussian white noise was replaced by colored noise, which is a type of Wong–Zakai approximation.

Mean-square invariant manifolds for ill-posed stochastic evolution equations driven by nonlinear noise

This paper discerns the invariant manifold of a class of ill-posed stochastic evolution equations driven by a nonlinear multiplicative noise. To be more precise, we establish the existence of mean-square random unstable invariant manifold and only mean-square stable invariant set. Due to the lack of the Hille-Yosida condition, we construct a modified variation of constants formula by the resolvent operator. With the price of imposing an unusual condition involving a non-decreasing map, we set up the Lyapunov-Perron method and derive the required estimates. We also emphasize that the Lyapunov-Perron map in the forward time loses the invariant due to the adaptedness, we alternatively establish the existence of mean-square random stable sets.

Random invariant manifolds and foliations for slow-fast PDEs with strong multiplicative noise

This article is devoted to the dynamical behaviors of a class of slow-fast PDEs perturbed by strong multiplicative noise. We will accomplish the existence of random invariant manifolds and foliations, and show exponential tracking property of them. Moreover, the asymptotic approximation for both objects will be presented.

Random invariant manifolds of stochastic evolution equations driven by Gaussian and non-Gaussian noises

The goal of this work is to compare the invariant manifold of the stochastic evolution equation driven by an α-stable process with the invariant manifold of the stochastic evolution equation forced by Brownian motion. First, we show that the solution of the Marcus stochastic evolution equation driven by a type of α-stable process converges to the solution of the related Stratonovich stochastic evolution equation forced by Brownian motion. Then, we study the invariant stable manifold of the stochastic evolution equation driven by an α-stable process. Finally, we prove that the invariant stable manifold of the Marcus stochastic evolution equation driven by an α-stable process converges in probability to the invariant stable manifold of the Stratonovich stochastic evolution equation forced by Brownian motion. The connection between the random dynamical system driven by non-Gaussian noise and the random dynamical system driven by Gaussian noise is established.

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

Effective Filtering for Multiscale Stochastic Dynamical Systems Driven by Lévy Processes*

The work is about multiscale stochastic dynamical systems driven by Lévy processes. We prove that such a system can be approximated by a low-dimensional system on a random invariant manifold, and the original filter can be also approximated by the reduced low-dimensional filter. Finally, we investigate the reduction for $$\varepsilon =0$$ and obtain that these reduced systems does not approximate these multiscale stochastic dynamical systems.

Inferring latent temporal progression and regulatory networks from cross-sectional transcriptomic data of cancer samples.

Unraveling molecular regulatory networks underlying disease progression is critically important for understanding disease mechanisms and identifying drug targets. The existing methods for inferring gene regulatory networks (GRNs) rely mainly on time-course gene expression data. However, most available omics data from cross-sectional studies of cancer patients often lack sufficient temporal information, leading to a key challenge for GRN inference. Through quantifying the latent progression using random walks-based manifold distance, we propose a latent-temporal progression-based Bayesian method, PROB, for inferring GRNs from the cross-sectional transcriptomic data of tumor samples. The robustness of PROB to the measurement variabilities in the data is mathematically proved and numerically verified. Performance evaluation on real data indicates that PROB outperforms other methods in both pseudotime inference and GRN inference. Applications to bladder cancer and breast cancer demonstrate that our method is effective to identify key regulators of cancer progression or drug targets. The identified ACSS1 is experimentally validated to promote epithelial-to-mesenchymal transition of bladder cancer cells, and the predicted FOXM1-targets interactions are verified and are predictive of relapse in breast cancer. Our study suggests new effective ways to clinical transcriptomic data modeling for characterizing cancer progression and facilitates the translation of regulatory network-based approaches into precision medicine.

Stochastic bifurcation for two-time-scale dynamical system with α-stable Lévy noise

This work focuses on stochastic bifurcation for a slow–fast dynamical system driven by non-Gaussian α-stable Lévy noise. We prove the main result for the stochastic equilibrium states for the original system and the reduced system based on the random slow manifold. Then, it is verified that the slow reduced system bears the stochastic bifurcation phenomenon inherited from the original system. Furthermore, we investigate the number and stability type of stochastic equilibrium states for dynamical systems through numerical simulations, and it is illustrated that the slow reduced system captures the stochastic bifurcation of the original system.