Stationary Random Environment Research Articles

Modified time domain randomization technique for multi-shaker non-stationary non-Gaussian random vibration control

Traditional random vibration environmental tests simulate stationary Gaussian random vibration environments. In this work, the generation and closed-loop control of multi-shaker non-stationary non-Gaussian random vibration signals with the modified time domain randomization technique are studied. The amplitude modulation function is introduced to control the non-stationary and non-Gaussian characteristics that the root mean square value of the generated signal changes with time. The kurtosis relationship between the one frame pseudo-random signal and consecutive true random signal is derived theoretically. It is found that the kurtosis of the consecutive non-stationary random signal is determined by the amplitude modulation function and kurtosis of the pseudo-random signal. The standard deviation of the time-varying root mean square reflects the non-stationary characteristics of the generated signal. A control strategy for multi-shaker non-stationary non-Gaussian random vibration environmental test is proposed. The non-stationary and non-Gaussian characteristics are extracted from the measured response signals so that the power spectral densities, kurtoses and non-stationary characteristics can be controlled simultaneously. The results from the simulation examples and multi-shaker vibration experiments prove the feasibility and practicability of the proposed non-stationary non-Gaussian random signal generation method and control strategy.

Limit Theorems for Additive Functionals of Continuous-Time Lattice Random Walks in a Stationary Random Environment

For a continuous-time lattice random walk $X^\Lambda=\set{X^\Lambda_t,t\ge 0}$ in a random environment $\Lambda$, we study the asymptotic behavior, as $t\rightarrow \infty$, of the normalized additive functional $c_t\int_0^{t} f(X^\Lambda_s)ds$, $t\ge 0$. We establish a limit theorem for it, which is similar to that in the non-lattice case, under less restrictive assumptions.

On the Ergodicity of Certain Markov Chains in Random Environments

We study the ergodic behaviour of a discrete-time process X which is a Markov chain in a stationary random environment. The laws of X_t are shown to converge to a limiting law in (weighted) total variation distance as trightarrow infty . Convergence speed is estimated, and an ergodic theorem is established for functionals of X. Our hypotheses on X combine the standard “drift” and “small set” conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain “maximal process” of the random environment. We are able to cover a wide range of models that have heretofore been intractable. In particular, our results are pertinent to difference equations modulated by a stationary (Gaussian) process. Such equations arise in applications such as discretized stochastic volatility models of mathematical finance.

Optimal metabolic strategies for microbial growth in stationary random environments

In order to grow in any given environment, bacteria need to collect information about the medium composition and implement suitable growth strategies by adjusting their regulatory and metabolic degrees of freedom. In the standard sense, optimal strategy selection is achieved when bacteria grow at the fastest rate possible in that medium. While this view of optimality is well suited for cells that have perfect knowledge about their surroundings (e.g. nutrient levels), things are more involved in uncertain or fluctuating conditions, especially when changes occur over timescales comparable to (or faster than) those required to organize a response. Information theory however provides recipes for how cells can choose the optimal growth strategy under uncertainty about the stress levels they will face. Here we analyse the theoretically optimal scenarios for a coarse-grained, experiment-inspired model of bacterial metabolism for growth in a medium described by the (static) probability density of a single variable (the ‘stress level’). We show that heterogeneity in growth rates consistently emerges as the optimal response when the environment is sufficiently complex and/or when perfect adjustment of metabolic degrees of freedom is not possible (e.g. due to limited resources). In addition, outcomes close to those achievable with unlimited resources are often attained effectively with a modest amount of fine tuning. In other terms, heterogeneous population structures in complex media may be rather robust with respect to the resources available to probe the environment and adjust reaction rates.

Off-Diagonal Heat Kernel Estimates for Symmetric Diffusions in a Degenerate Ergodic Environment

We study a symmetric diffusion process on mathbb {R}^{d}, d ≥ 2, in divergence form in a stationary and ergodic random environment. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also shown for a natural choice of speed measure, under an additional mixing assumption on the environment. Using these estimates, a scaling limit for the Green function is proven.

Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension

We prove homogenization for a class of nonconvex (possibly degenerate) viscous Hamilton-Jacobi equations in stationary ergodic random environments in one space dimension. The results concern Hamiltonians of the form G(p)+V(x,ω), where the nonlinearity G is the minimum of two or more convex functions with the same absolute minimum, and the potential V is a bounded stationary process satisfying an additional scaled hill and valley condition. This condition is trivially satisfied in the inviscid case, while it is equivalent to the original hill and valley condition of A. Yilmaz and O. Zeitouni [32] in the uniformly elliptic case. Our approach is based on PDE methods and does not rely on representation formulas for solutions. Using only comparison with suitably constructed super- and sub- solutions, we obtain tight upper and lower bounds for solutions with linear initial data x↦θx. Another important ingredient is a general result of P. Cardaliaguet and P. E. Souganidis [13] which guarantees the existence of sublinear correctors for all θ outside “flat parts” of effective Hamiltonians associated with the convex functions from which G is built. We derive crucial derivative estimates for these correctors which allow us to use them as correctors for G.

Spatial Characterization of Electromagnetic Random Channels

The majority of stochastic channel models rely on the electromagnetic far-field assumption, which allows to decompose the channel in terms of plane waves. The far-field assumption breaks down in future applications that push towards the electromagnetic near-field region, such as those where the use of electromagnetically large antenna arrays is envisioned. Motivated by this consideration, we show how physical principles can be used to derive a plane-wave scalar channel model that is also valid in the reactive near-field region. Precisely, we show that narrowband wave propagation through a three-dimensional scattered medium can be generally modeled as a linear and space-variant system. We first review the physics principles that lead to a closed-form deterministic plane-wave representation of the channel impulse response. This serves as a basis for deriving a stochastic representation of the channel in terms of statistically independent Gaussian random coefficients for spatially stationary random propagation environments. The very desirable property of spatial stationarity can always be retained in the radiative near-field region by excluding reactive propagation mechanisms confined in close proximity to the source. Remarkably, the provided stochastic representation is directly connected to the Fourier spectral representation of a general stationary spatial random field.

Stochastic homogenization of deterministic control problems

In this paper we study homogenization of a class of control problems in a stationary and ergodic random environment. This problem has been mostly studied in the calculus of variations setting in connection to the homogenization of the Hamilton–Jacobi equation. We extend the result to control problems with more general state dynamics and macroscopically inhomogeneous Lagrangians. Moreover, our approach proves homogenization under weaker growth assumptions on the Lagrangian, even in the well-studied calculus of variations setting.

On the Behavior of One Stochastic Automaton in a Random Environment

A construction (behavior algorithm) of a finite stochastic automaton functioning in a stationary random environment with three classes of reactions (encouragement, punishment, indifference) is proposed. Using the methods of the theory of random walks, formulas are obtained for the generating function of the probability of changing the action and for calculating the probability characteristics of the behavior of an automaton. The convergence of sequences of finite automata (when the memory of the automaton n → ∞) to the corresponding infinite automaton (with a countable number of states) of a similar structure is established and given a classification of his possible behavior in this stationary random environment.

ПРИМЕНЕНИЕ ВЕРОЯТНОСТНЫХ И ВРЕМЕННЫХ АВТОМАТОВ В ПРОГРАММАХ УПРАВЛЕНИЯ МНОГОАГЕНТНЫХ СИСТЕМ

Classical models of the time and probabilistic automaton constructed using the theory of Markov chains - the system "automaton-random environment" described by the Markov chain are considered. The uncertainty of the probabilistic automaton can be determined by various reasons, including the nondeterminism of the rules of state change of the simulated system, which is one of the most important obstacles in the modeling of multi-agent systems. The developed model is intended for the control program of multi-agent systems, takes into account the stochastic behavior of the environment when used in multi-agent systems. The key differences between the classical stationary random environment and multi-agent systems are presented, including the rationality of agent behavior and the need for cooperation or counteraction. The model does not provide trainable algorithms, their application requires a large sample for training, not all multi-agent systems have enough time to set the necessary statistics and behavior adaptation. Approaches to building the architecture of the agent control program in a multi-agent environment based on the theories of time and probabilistic automata are described, a format for storing models in file storage is proposed. The proposed architecture takes into account the possibility of changing the state of time automata based on the state of probabilistic automata and vice versa. The generalized algorithm of the control program functioning is described. The proposed algorithm makes it possible to implement control programs for multi-agent systems in the case of a finite number of agent states and uses flexible control mechanisms taken from the time machine model. The main results and proposals for further research, including the construction of hierarchical models are presented.

Absolute continuity of the martingale limit in branching processes in random environment

We consider a supercritical branching process $Z_{n}$ in a stationary and ergodic random environment $\xi =(\xi _{n})_{n\ge 0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_{n}=Z_{n}/(\mathbb{E} [Z_{n}|\xi ])$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $\xi $ the law of $W$ conditioned on the environment $\xi $ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of [10], and of course it covers the well-known case of the martingale limit of a Galton-Watson process. Our proof combines analytical arguments with the recursive description of $W$.

A loss function based parameterless learning automaton scheme

Learning automaton (LA) is a powerful tool for reinforcement learning in the field of Artificial Intelligence. In the evaluation of LA, it has always been a key issue how to trade off between “accuracy” and “speed”, which substantially involves in parameter tuning. Owing to the environmental randomness and complexity, it universally takes millions of interactions during the process of parameter tuning, bringing about a tremendous expense. To avoid this fatal flaw, in this paper, a novel parameterless learning automaton named LFPLA is proposed. It shows an intriguing property of not relying on manually configured parameters and possesses an ϵ-optimality property in any stationary random environment. A distinctive innovation lies in a newly defined loss function, which replaces the probability vector maintaining in conventional LA. Furthermore, a series of sampling strategies are designed for action selection, and in terms of iteration termination conditions, a sufficiently small threshold is employed. In addition to proving its advantageous performance theoretically by detailed mathematical proofs, we also carried out extensive experiments to illustrate the effectiveness in two-action as well as multi-action benchmark environments, by means of Monte Carlo simulations. The proposed LFPLA converges faster with a higher accuracy than the only parameter-free LA presently: GBLA. Moreover, it is superior to the state of the arts in multi-action LA, especially in complex and confuse environments. Most of all, it embodies a unique and extraordinary benefit with special regard to either tuning cost or interaction cost.

On the concept of subcriticality and criticality and a ratio theorem for a branching process in a random environment

We consider a branching process (Zn) in a stationary and ergodic random environment ξ=(ξn). Athreya and Karlin (1971) proved the basic result about the concept of subcriticality and criticality, by showing that under the quenched law Pξ, the conditional distribution of Zn given the non-extinction at time n converges in law to a proper distribution on N+={1,2,⋯} in the subcritical case, and to the null distribution in the critical case, under the condition that the environment sequence is exchangeable. In this paper we first improve this basic result by removing the exchangeability condition on the environment, and by establishing a more general result about the conditional law of Zn given the non-extinction at time n+k for each fixed k≥0. As a by-product of the proof we also remove the exchangeability condition in another result of Athreya and Karlin (1971) for the subcritical case about the decay rate of the survival probability given the environment. We then establish a convergence theorem about the ratio Pξ(Zn=j)/Pξ(Zn=1), which can be applicable in each of the subcritical, critical, and supercritical cases.

Stochastic homogenization of viscous superquadratic Hamilton\u2013Jacobi equations in dynamic random environment

We study the qualitative homogenization of second-order Hamilton–Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient), we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends previous work that required uniform ellipticity and space-time homogeneity for the diffusion.

Communication: Fully coherent quantum state hopping.

In this paper, we describe a new and fully coherent stochastic surface hopping method for simulating mixed quantum-classical systems. We illustrate the approach on the simple but unforgiving problem of quantum evolution of a two-state quantum system in the limit of unperturbed pure state dynamics and for dissipative evolution in the presence of both stationary and nonstationary random environments. We formulate our approach in the Liouville representation and describe the density matrix elements by ensembles of trajectories. Population dynamics are represented by stochastic surface hops for trajectories representing diagonal density matrix elements. These are combined with an unconventional coherent stochastic hopping algorithm for trajectories representing off-diagonal quantum coherences. The latter generalizes the binary (0,1) "probability" of a trajectory to be associated with a given state to allow integers that can be negative or greater than unity in magnitude. Unlike existing surface hopping methods, the dynamics of the ensembles are fully entangled, correctly capturing the coherent and nonlocal structure of quantum mechanics.

A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (P ω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (Pω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

Stochastic Homogenization for Some Nonlinear Integro-Differential Equations

In this note we prove the stochastic homogenization for a large class of fully nonlinear elliptic integro-differential equations in stationary ergodic random environments. Such equations include, but are not limited to, Bellman equations and the Isaacs equations for the control and differential games of some pure jump processes in a random, rapidly varying environment. The translation invariant and non-random effective equation is identified, and the almost everywhere in ω, uniform in x convergence of the family solutions of the original equations is obtained. Even in the linear case of the equations contained herein the results appear to be new.

Asymptotic shape for the contact process in random environment

The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if $H_{t}$ denotes the set of already occupied sites at time $t$, we show that for almost every environment, when the contact process survives, the set $H_{t}/t$ almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem.

A Symmetry Property for a Class of Random Walks in Stationary Random Environments on Z

A correspondence formula between the laws of dual Markov chains on Z with two transition jumps is established. This formula contributes to the study of random walks in stationary random environments. Counterexamples with more than two jumps are exhibited.