Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.

Probability Density Evolution Algorithm for Stochastic Dynamical Systems Based on Fractional Calculus

With the increasing load and speed of trains, the problems caused by various random excitations (such as safety and passenger comfort) have become more prominent and thus arises the necessity to analyze stochastic dynamical systems, which is important in both academic and engineering circles. The existing analysis methods are inadequate in terms of computational accuracy, computational efficiency, and applicability in solving complex problems. For that, a new efficient and accurate method is used in this paper, suitable for linear and nonlinear random vibration analysis of large structures as well as static and dynamic reliability assessment. It is the direct probability integration method, which is extended and applied to the random vibration reliability analysis of dynamical systems. Dynamical models of the dynamic system and coupled system “three-car vehicle-rail-bridge” are established, the time-varying differential equations of motion are derived in detail, and the dynamic response of the system is calculated using the explicit Newmark algorithm. The simulation results show the influence of the number of representative points on the smoothness of the image of the probability density function and the accuracy of the calculation results.

A software toolkit for modeling human sentence parsing: An approach using continuous-time, discrete-state stochastic dynamical systems

We present a new software toolkit for implementing a broad class oftheories of sentence processing. In this framework, processing a word ina sentence is viewed as a continuous-time random walk through a set ofdiscrete states that encode information about the emerging structure of thesentence so far. The state space includes one or more special absorbingstates, which, when reached, indicate the decision to move on to the nextword of the sentence. This setup allows us to ask how how long it takesto reach an absorbing state and what the probability of reaching this stateis. We summarize a number of important statistics that can be directlyrelated to human reading times and comprehension question performance.To illustrate the use of the toolkit, we model two types of garden paths,local coherence effects, and the ambiguity advantage using three qualitativelydifferent theories of sentence processing. While the modeler must still makedefensible theoretical and implementation choices, this framework representsan improvement over the descriptive, paper-pencil modeling that is thenorm in psycholinguistics by facilitating quantitative evaluations of modelperformance and laying the groundwork for Bayesian fitting of free parametersin a model. An open-source Python package is provided.