Stochastic Forcing Term Research Articles

Long-run analysis of the stochastic replicator dynamics in the presence of random jumps

The effect of anomalous events on the replicator dynamics with aggregate shocks is considered. The anomalous events are described by a Poisson integral, where this stochastic forcing term is added to the fitness of each agent. Contrary to previous models, this noise is assumed to be correlated across the population. A formula to calculate a closed form solution of the long run behavior of a two strategy game will be derived. To assist with the analysis of a two strategy game, the stochastic Lyapunov method will be applied. For a population with a general number of strategies, the time averages of the dynamics will be shown to converge to the Nash equilibria of a relevant modified game. In the context of the modified game, the almost sure extinction of a dominated pure strategy will be derived. As the dynamic is quite complex, with respect to the original game a pure strict Nash equilibrium and an interior evolutionary stable strategy will be considered. Respectively, conditions for stochastically stability and the positive recurrent property will be derived. This work extends previous results on the replicator dynamics with aggregate shocks.

Some properties of non-linear fractional stochastic heat equations on bounded domains

We consider the following fractional stochastic partial differential equation on a bounded, open subset B of Rd for d ≥ 1∂tut(x)=Lut(x)+ξσ(ut(x))F˙(t,x),where ξ is a positive parameter and σ is a globally Lipschitz continuous function. The stochastic forcing term F˙(t,x) is white in time but possibly colored in space. The operator L is fractional Laplacian which is the infinitesimal generator of a symmetric α-stable Lévy process in Rd. We study the behaviour of the solution with respect to the parameter ξ.We show that under zero exterior boundary conditions, in the long run, the pth-moment of the solution grows exponentially fast for large values of ξ. However when ξ is very small we observe eventually an exponential decay of the pth-moment of this same solution.

A note on stochastic Navier–Stokes equations with not regular multiplicative noise

We consider the Navier–Stokes equations in \({\mathbb {R}}^d\) (\(d=2,3\)) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so Ito calculus cannot be applied in the space of finite energy vector fields. We prove existence of weak solutions for \(d=2,3\) and pathwise uniqueness for \(d=2\).

Ensemble Data Assimilation Using a Unified Representation of Model Error

Abstract A natural way to set up an ensemble forecasting system is to use a model with additional stochastic forcing representing the model error and to derive the initial uncertainty by using an ensemble of analyses generated with this model. Current operational practice has tended to separate the problems of generating initial uncertainty and forecast uncertainty. Thus, in ensemble forecasts, it is normal to use physically based stochastic forcing terms to represent model errors, while in generating analysis uncertainties, artificial inflation methods are used to ensure that the analysis spread is sufficient given the observations. In this paper a more unified approach is tested that uses the same stochastic forcing in the analyses and forecasts and estimates the model error forcing from data assimilation diagnostics. This is shown to be successful if there are sufficient observations. Ensembles used in data assimilation have to be reliable in a broader sense than the usual forecast verification methods; in particular, they need to have the correct covariance structure, which is demonstrated.

Modeling Long-term Vaccination Strategies With MenAfriVac in the African Meningitis Belt.

Background. The introduction of MenAfriVac in campaigns targeting people aged 1–29 years across the African meningitis belt has successfully reduced meningitis incidence and carriage due to Neisseria meningitidis group A (MenA). It is important to consider how best to sustain population protection in the long term.Methods. We created a mathematical model of MenA transmission and disease to investigate the potential impact of a range of immunization strategies. The model is age structured; includes classes of susceptible, carrier, ill, and immune people (who may be vaccinated or unvaccinated); and incorporates seasonal transmission and a stochastic forcing term that models between year variation in rates of transmission. Model parameters were primarily derived from African sources. The model can describe the typical annual incidence of meningitis in the prevaccine era, with irregular epidemics of varying size. Parameter and structural uncertainty were explored in sensitivity analyses.Results. Following MenAfriVac introduction at high uptake, the model predicts excellent short-term disease control. With no subsequent immunization, strong resurgences in disease incidence were predicted after approximately 15 years (assuming 10 years’ average vaccine protection). Routine immunization at 9 months of age resulted in lower average annual incidence than regular mass campaigns of 1- to 4-year-olds, provided coverage was above approximately 60%. The strategy with the lowest overall average annual incidence and longest time to resurgence was achieved using a combination strategy of introduction into the Expanded Programme on Immunization at 9 months, 5 years after the initial mass campaigns, with a catch-up targeting unvaccinated 1- to 4-year-olds.Conclusions. These results can be used to inform policy recommendations for long-term vaccination strategies with MenAfriVac.

Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction–diffusion equations on an unbounded domain

In this paper we study pullback attractors of reaction–diffusion equations on an unbounded domain with non-autonomous deterministic as well as stochastic forcing terms for which the uniqueness of solutions need not hold. We first present the existence and structure of pullback attractors of multi-valued non-compact random dynamical systems. Then we prove the existence of pullback attractors in L2(Rn), Lp(Rn) and H1(Rn) for multi-valued non-compact random dynamical systems associated with the reaction–diffusion equations on Rn, and the identical relation of pullback attractors in different spaces is also provided. In particular, the measurability of pullback attractors is established for reaction–diffusion equations on Rn.

Mathematical issues in eternal inflation

In this paper, we consider the problem of the existence and uniqueness of solutions to the Einstein field equations for a spatially flat Friedmann–Lemaître–Robertson–Walker universe in the context of stochastic eternal inflation, where the stochastic mechanism is modelled by adding a stochastic forcing term representing Gaussian white noise to the Klein–Gordon equation. We show that under these considerations, the Klein–Gordon equation actually becomes a stochastic differential equation. Therefore, the existence and uniqueness of solutions to Einstein’s equations depend on whether the coefficients of this stochastic differential equation obey Lipschitz continuity conditions. We show that for any choice of , the Einstein field equations are not globally well-posed, hence, any solution found to these equations is not guaranteed to be unique. Instead, the coefficients are at best locally Lipschitz continuous in the physical state space of the dynamical variables, which only exist up to a finite explosion time. We further perform Feller’s explosion test for an arbitrary power-law inflaton potential and prove that all solutions to the Einstein field equations explode in a finite time with probability one. This implies that the mechanism of stochastic inflation thus considered cannot be described to be eternal, since the very concept of eternal inflation implies that the process continues indefinitely. We therefore argue that stochastic inflation based on a stochastic forcing term would not produce an infinite number of universes in some multiverse ensemble. In general, since the Einstein field equations in both situations are not well-posed, we further conclude that the existence of a multiverse via the stochastic eternal inflation mechanism considered in this paper is still very much an open question that will require much deeper investigation.

Numerical Schemes for Stochastic Backscatter in the Inverse Cascade of Quasigeostrophic Turbulence

Backscatter is the process of energy transfer from small to large scales in turbulence; it is crucially important in the inverse energy cascades of geophysical turbulence, where the net transfer of energy is from small to large scales. One approach to modeling backscatter in underresolved simulations is to add a stochastic forcing term. This study, set in the idealized context of the inverse cascade of two-layer quasigeostrophic turbulence, focuses on the importance of spatial and temporal correlation in numerical stochastic backscatter schemes when used with low-order finite-difference spatial discretizations. A minimal stochastic backscatter scheme is developed as a stripped-down version of stochastic superparameterization [Grooms and Majda, J. Comput. Phys., 271, (2014), pp. 78--98]. This simplified scheme allows detailed numerical analysis of the spatial and temporal correlation structure of the modeled backscatter. Its essential properties include a local formulation amenable to implementation in finite difference codes and nonperiodic domains, and tunable spatial and temporal correlations. Experiments with this scheme in the idealized context of homogeneous two-layer quasigeostrophic turbulence demonstrate the need for stochastic backscatter to be smooth at the coarse grid scale when used with low-order finite-difference schemes in an inverse-cascade regime. In contrast, temporal correlation of the backscatter is much less important for achieving realistic energy spectra. It is expected that the spatial and temporal correlation properties of the simplified backscatter schemes examined here will inform the development of stochastic backscatter schemes in more realistic models.

Simulating bistable perception with interrupted ambiguous stimulus using self-oscillator dynamics with percept choice bifurcation.

A behavioral stochastic self-oscillator model is used for simulating interrupted ambiguous stimulus-induced percept reversals. The results provide further support for a dynamical systems foundation of cognitive and psychological problems as discussed in detail within the context of Gestalt psychology by Wagemans et al. (Concept Theor Found Psychol Bull 138(6):1218-1252, 2012), and for coordination dynamics of the brain (Kelso in Philos Trans R Soc B 367:906-918, 2012). Statistical evaluation of simulated reversal time series predicts a maximum of the percept reversal rate that conforms with a number of results in the literature. The macroscopic model is based on two inhibitorily coupled sets of three coupled nonlinear equations, one triplet for each percept. The derivation of our specific dynamics equations is based on a drastically simplified field theoretical approach using well-known phase synchronization for explaining brain dynamics on the macroscopic EEG level. The degree of coherence (contrast μ, 0≤μ≤1) of the superimposed fields required for onset of bistable dynamics is related to a phase synchronization index of EEG fields, and it is used in the present context as ambiguity control parameter. For quantitative agreement with the experimental data, the addition of a stochastic Langevin force term in the attention equation proved essential. Formal analysis leads to a quantification of well-known "cognitive inertia" and supports the interplay between percept choice (bifurcation) dynamics during stimulus onset and adaptive gain (attention fatigue) driven quasiperiodic percept reversals.

Pullback attractors for reaction-diffusion delay equations on unbounded domains with non-autonomous deterministic and stochastic forcing terms

The existence of a pullback attractor in a space of higher regularity is established for the non-autonomous non-compact dynamical system associated with the reaction-diffusion delay equation with both non-autonomous deterministic and stochastic forcing terms defined on all n-dimensional space. The asymptotical compactness of the solutions is demonstrated by using the uniform estimates for far-field values of solutions and a new method.

Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems

We study pullback attractors of non-autonomous non-compact dynamical systems generated by differential equations with non-autonomous deterministic as well as stochastic forcing terms. We first introduce the concepts of pullback attractors and asymptotic compactness for such systems. We then prove a sufficient and necessary condition for existence of pullback attractors. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors. For random systems containing periodic deterministic forcing terms, we show the pullback attractors are also periodic under certain conditions. As an application of the abstract theory, we prove the existence of a unique pullback attractor for Reaction–Diffusion equations on Rn with both deterministic and random external terms. Since Sobolev embeddings are not compact on unbounded domains, the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions.

POD-based Monte Carlo approach for the solution of regional scale groundwater flow driven by randomly distributed recharge

We present a methodology conducive to the application of a Galerkin model order reduction technique, Proper Orthogonal Decomposition (POD), to solve a groundwater flow problem driven by spatially distributed stochastic forcing terms. Typical applications of POD to reducing time-dependent deterministic partial differential equations (PDEs) involve solving the governing PDE at some observation times (termed snapshots), which are then used in the order reduction of the problem. Here, the application of POD to solve the stochastic flow problem relies on selecting the snapshots in the probability space of the random quantity of interest. This allows casting a standard Monte Carlo (MC) solution of the groundwater flow field into a Reduced Order Monte Carlo (ROMC) framework. We explore the robustness of the ROMC methodology by way of a set of numerical examples involving two-dimensional steady-state groundwater flow taking place within an aquifer of uniform hydraulic properties and subject to a randomly distributed recharge. We analyze the impact of (i) the number of snapshots selected from the hydraulic heads probability space, (ii) the associated number of principal components, and (iii) the key geostatistical parameters describing the heterogeneity of the distributed recharge on the performance of the method. We find that our ROMC scheme can improve significantly the computational efficiency of a standard MC framework while keeping the same degree of accuracy in providing the leading statistical moments (i.e. mean and covariance) as well as the sample probability density of the state variable of interest.

Nonequilibrium forces between atoms and dielectrics mediated by a quantum field

In this paper we give a first principles microphysics derivation of the nonequilibrium forces between an atom, treated as a three dimensional harmonic oscillator, and a bulk dielectric medium modeled as a continuous lattice of oscillators coupled to a reservoir. We assume no direct interaction between the atom and the medium but there exist mutual influences transmitted via a common electromagnetic field. By employing concepts and techniques of open quantum systems we introduce coarse-graining to the physical variables - the medium, the quantum field and the atom's internal degrees of freedom, in that order - to extract their averaged effects from the lowest tier progressively to the top tier. The first tier of coarse-graining provides the averaged effect of the medium upon the field, quantified by a complex permittivity (in the frequency domain) describing the response of the dielectric to the field in addition to its back action on the field through a stochastic forcing term. The last tier of coarse- graining over the atom's internal degrees of freedom results in an equation of motion for the atom's center of mass from which we can derive the force on the atom. Our nonequilibrium formulation provides a fully dynamical description of the atom's motion including back action effects from all other relevant variables concerned. In the long-time limit we recover the known results for the atom-dielectric force when the combined system is in equilibrium or in a nonequilibrium stationary state.

Analysis of signal propagation in an elastic-tube flow model

We combine linear and nonlinear signal analysis techniques to investigate the transmission of pressure signals along a one-dimensional model of fluid flow in an elastic tube. We derive a simple, generally applicable measure for the robustness of a simulated vessel against in vivo pressure fluctuations, based on quantifying the degree of synchronization between proximal and distal pressure pulses. The practical use of this measure will be in its application to simulated pulses generated in response to a stochastic forcing term mimicking biological variations of root pressure in arterial blood flow.

Stochastically forced water waves in a circular basin

An analysis of wave motion under stochastic excitation is presented. The starting point is a Hamiltonian model of surface wave motion. This model is augmented with linear damping and stochastic forcing terms. An asymptotic scaling parameter is then introduced to show that the problem has three time scales. Integrability of the system is established for the case of two wave modes near 1:1 resonance. Stochastic averaging is used to reduce the dimensions of the system from four to two and the evolution of surface waves is now described, over long time scales and small amplitude stochastic forcing, by the evolution of the integrals of motion, as a random process. The domain of the generator of this random process is characterized and it includes a ‘gluing’ boundary condition. The adjoint of the generator yields the Fokker–Planck partial-differential equation (FPE). This equation governs the time evolution of the joint probability density of the two integrals of motion. The coefficients of the FPE are calculated numerically and the steady-state solution is found using the finite-element method. Results of the analysis show a distinct peak in the probability density along one edge of the reduced domain.

On the evolution of mean motion resonances through stochastic forcing: fast and slow libration modes and the origin of HD 128311

Aims. We clarify the response of extrasolar planetary systems in a 2:1 mean motion commensurability with masses ranging from the super Jovian range to the terrestrial range to stochastic forcing that could result from protoplanetary disk turbulence. The behaviour of the different libration modes for a wide range of system parameters and stochastic forcing magnitudes is investigated. The growth of libration amplitudes is parameterized as a function of the relevant physical parameters. The results are applied to provide an explanation of the configuration of the HD 128311 system. Methods. We first develop an analytic model from first principles without making the assumption that both eccentricities are small. We also perform numerical N-body simulations with additional stochastic forcing terms to represent the effects of putative disk turbulence. Results. We isolate two distinct libration modes for the resonant angles. These react to stochastic forcing in a different way and become coupled when the libration amplitudes are large. Systems are quickly destabilized by large magnitudes of stochastic forcing but some stability is imparted should systems undergo a net orbital migration. The slow mode, which mostly corresponds to motion of the angle between the apsidal lines of the two planets, is converted to circulation more readily than the fast mode which is associated with oscillations of the semi-major axes. This mode is also vulnerable to the attainment of small eccentricities which causes oscillations between periods of libration and circulation. Conclusions. Stochastic forcing due to disk turbulence may have played a role in shaping the configurations of observed systems in mean motion resonance. It naturally provides a mechanism for accounting for the HD 128311 system for which the fast mode librates and the slow mode is apparently near the borderline between libration and circulation.

Thin film models with constant source: model selection in a stochastic setting

Dunn and Tichenor [Atmospheric Environment, 22:885--894, 1988] proposed a class of differential equation models to describe the phenomenon of transient sink behaviour for organic emissions exhibited by interior surface films in state-of-the-art emission test chambers. The proposed model selection scheme embeds the derived models within a class of stochastic differential equations. The quality of model fit varies inversely with the strength of the stochastic forcing term; that is, if the model is adequate the stochastic forcing term should be small. Data from a particular application where the source can be considered to be constant demonstrates the approach. The approach can be applied to any phenomenon that is modelled by a class of linear differential equations where different models are embedded within a full model. References B. D. Anderson and J. B. Moore Optimal Filtering, Prentice--Hall, Inc., Englewood Cliffs, N. J., 1979. C. F. Ansley and R. Kohn A geometrical derivation of the fixed interval smoothing algorithm. Biometrika, 69:486--487. J. E. Dunn and B. A. Tichenor. Compensating for sink effects in emissions test chambers by mathematical modeling. Atmospheric Environment, 22:885--894, 1988. M. R. Osborne and T. Prvan. Smoothness and conditioning in generalised smoothing spline calculations. J. Austral. Math. Soc. Ser. B, 30:43--56, 1988. T. Prvan and M. R. Osborne. Model selection in a stochastic setting. ANZIAM J., 45(E) ppC787--C799, 2004. http://anziamj.austms.org.au/V45/CTAC2003/Prva/home.html H. E. Rauch, F. Tung and C. T. Striebel. Maximum Likelihood Estimates of Linear Dynamic Systems. AIAA J., 3: 1445--1450, 1965. G. Wahba. Improper priors, spline smoothing and the problem of guarding against model errors in regression. J. R. Statist. Assoc., 40:364--372, 1978. G. Wahba. A comparison of GCV and GML for choosing the smoothing parameter in the generalised spline smoothing problem. Ann. Statist., 13: 1378--1402, 1985. W. Wecker and C. F. Ansley. Signal extraction approach to non linear regression and spline smoothing. J. Amer. Statist. Assoc., 78: 81--89, 1983.

Stochastic immersed boundary method incorporating thermal fluctuations

AbstractWe give a brief introduction to the stochastic immersed boundary method which allows for simulation of small length‐scale physical systems in which elastic structures interact with a fluid flow in the presence of thermal fluctuations. The conventional immersed boundary method is extended to account for thermal fluctuations by introducing stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE's for which standard numerical approaches perform poorly. We discuss a numerical method derived using stochastic calculus to overcome the stiff features of the equations. We then discuss results which indicate that the method captures physical features predicted by statistical mechanics for small length‐scale systems. The stochastic immersed boundary method holds promise as a numerical approach in simulating microscopic mechanical systems in which thermal fluctuations play a fundamental role. A more detailed discussion of this work is given in [1, 2, 3]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Fluid-structure analysis of microparticle transport in deformable pulmonary alveoli

Micron-sized particles inhaled into the respiratory system can traverse the airway tree and deposit on the walls of the pulmonary alveoli. The fate of these particles can be measured by their propagation depth, wall deposition rate, and time to impaction. These three quantities depend on various physical parameters including particle size, alveolar geometry, tissue properties, and breathing patterns. In this study, we develop a 2-D fluid-structure computational model of alveolar dynamics to quantify how these parameters influence particle transport in the deep lung. The computational model simulates negative pressure breathing conditions in which applied tissue forces deform the lung parenchyma and produce oscillatory flow within the alveoli. Alveolar flow patterns are used to calculate the trajectories of micron-sized particles (i.e. 0.1 – 5 μ m ) by solving the Langevin equation, which contains a stochastic Brownian force term. As a result, these fluid-structure models can be used to investigate how tissue mechanical properties and breathing patterns influence deep-lung flow fields and particle dynamics. Results indicate that Brownian diffusion dominates the transport of small particles ( d p < 0.5 μ m ) , while gravitational sedimentation dominates the transport of large particles ( d p > 1 μ m ) . Mid-size particles ( 0.5 ⩽ d p ⩽ 1 μ m ) experience slower diffusion and slower sedimentation transport, so their time to impaction is maximized. Stiffer tissues produce higher particle impaction rates under microgravity conditions while tissue viscosity has negligible effects on particle dynamics. We also compare our computational results to previously published experimental studies and conclude that targeting inhaled particles to the deep lungs increases whole-lung deposition efficiencies because diffusion and sedimentation are efficient transport mechanisms in the small, distal structures.

Algorithm refinement for the stochastic Burgers’ equation

In this paper, we develop an algorithm refinement (AR) scheme for an excluded random walk model whose mean field behavior is given by the viscous Burgers’ equation. AR hybrids use the adaptive mesh refinement framework to model a system using a molecular algorithm where desired while allowing a computationally faster continuum representation to be used in the remainder of the domain. The focus in this paper is the role of fluctuations on the dynamics. In particular, we demonstrate that it is necessary to include a stochastic forcing term in Burgers’ equation to accurately capture the correct behavior of the system. The conclusion we draw from this study is that the fidelity of multiscale methods that couple disparate algorithms depends on the consistent modeling of fluctuations in each algorithm and on a coupling, such as algorithm refinement, that preserves this consistency.