Stochastic Ordinary Differential Equations Research Articles

Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential Equations

Recently, Hairer et al. (2015) showed that there exist stochastic differential equations (SDEs) with infinitely often differentiable and globally bounded coefficient functions whose solutions fail to be locally Lipschitz continuous in the strong L p L^p -sense with respect to the initial value for every p ∈ ( 0 , ∞ ] p\in (0,\infty ] . In this article we provide conditions on the coefficient functions of the SDE and on p ∈ ( 0 , ∞ ] p \in (0,\infty ] that are sufficient for local Lipschitz continuity in the strong L p L^p -sense with respect to the initial value and we establish explicit estimates for the local Lipschitz continuity constants. In particular, we prove local Lipschitz continuity in the initial value for several nonlinear stochastic ordinary and stochastic partial differential equations in the literature such as the stochastic van der Pol oscillator, Brownian dynamics, the Cox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As an application of our estimates, we obtain strong completeness for several nonlinear SDEs.

Exponential Euler method for stiff stochastic differential equations with additive fractional Brownian noise

We discuss a system of stochastic differential equations with a stiff linear term and additive noise driven by fractional Brownian motions (fBms) with Hurst parameter H > 1 2 , which arise e.g. from spatial approximations of stochastic partial differential equations. For their numerical approximation, we present an exponential Euler scheme and show that it converges in the strong sense with an exact rate close to the Hurst parameter H. Further, based on E. Buckwar et al. [The numerical stability of stochastic ordinary differential equations with additive noise, Stoch. Dyn. 11 (2011), pp. 265–281], we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.

Genetic Network Design Automation with LOICA.

Genetic design automation (GDA) is the use of computer-aided design (CAD) in designing genetic networks. GDA tools are necessary to create more complex synthetic genetic networks in a high-throughput fashion. At the core of these tools is the abstraction of a hierarchy of standardized components. The components' input, output, and interactions must be captured and parametrized from relevant experimental data. Simulations of genetic networks should use those parameters and include the experimental context to be compared with the experimental results.This chapter introduces Logical Operators for Integrated Cell Algorithms (LOICA), a Python package used for designing, modeling, and characterizing genetic networks using a simple object-oriented design abstraction. LOICA represents different biological and experimental components as classes that interact to generate models. These models can be parametrized by direct connection to the Flapjack experimental data management platform to characterize abstracted components with experimental data. The models can be simulated using stochastic simulation algorithms or ordinary differential equations with varying noise levels. The simulated data can be managed and published using Flapjack alongside experimental data for comparison. LOICA genetic network designs can be represented as graphs and plotted as networks for visual inspection and serialized as Python objects or in the Synthetic Biology Open Language (SBOL) format for sharing and use in other designs.

Existence and Uniqueness Theorem of Fuzzy Stochastic Ordinary Differential Equations

     A fuzzy valued diffusion term, which in a fuzzy stochastic differential equation refers to one-dimensional Brownian motion, is defined by the meaning of the stochastic integral of a fuzzy process. In this paper, the existence and uniqueness theorem of fuzzy stochastic ordinary differential equations, based on the mean square convergence of the mathematical induction approximations to the associated stochastic integral equation, are stated and demonstrated.

Variable substitution methods in the solution of stochastic ordinary differential equations and their applications

Abstract In this paper, we first propose numerical solution methods for stochastic ordinary differential equations by using the two-step Maruyama method and Euler-Maruyama method in variable substitution, and analyze the mean-square compatibility, mean-square convergence and mean-square linear stability of the corresponding numerical methods, respectively. Finally, 10,000 times value experiments are conducted to verify the convergence accuracy and stability of the variable substitution methods. The results show that the Euler method simulates this equation when taking steps h = 0.05, β = 5 and σ = 2 for numerical experiments, and the numerical results obtained are convergent but unstable. On the other hand, the Euler-Maruyama method with variable substitution is consistent with the real solution. It shows that variable substitution is an important method for solving stochastic ordinary differential equations with high convergence and stability, which is of great significance for the solution of stochastic ordinary differential equations.

Novel Fractional Order and Stochastic Formulations for the Precise Prediction of Commercial Photovoltaic Curves

To effectively represent photovoltaic (PV) modules while considering their dependency on changing environmental conditions, three novel mathematical and empirical formulations are proposed in this study to model PV curves with minimum effort and short timing. The three approaches rely on distinct mathematical techniques and definitions to formulate PV curves using function representations. We develop our models through fractional derivatives and stochastic white noise. The first empirical model is proposed using a fractional regression tool driven by the Liouville-Caputo fractional derivative and then implemented by the Mittag-Leffler function representation. Further, the fractional-order stochastic ordinary differential equation (ODE) tool is employed to generate two effective generic models. In this work, multiple commercial PV modules are modeled using the proposed fractional and stochastic formulations. Using the experimental data of the studied PV panels at different climatic conditions, we evaluate the proposed models’ accuracy using two effective statistical indices: the root mean squares error (RMSE) and the determination coefficient (R2). Finally, the proposed approaches are compared to several integer-order models in the literature where the proposed models’ precisely follow the real PV curves with a higher R2 and lower RMSE values at different irradiance levels lower than 800 w/m2, and module temperature levels higher than 50 °C.

Exact soliton solutions and soliton diffusion of two kinds of stochastic KdV equations with variable coefficients

In this paper, we consider the exact solutions and soliton diffusion phenomenon of two kinds of stochastic KdV equations with variable coefficients. Firstly, according to symmetric reduction, stochastic KdV equations with variable coefficients are transformed into a coupled system of a deterministic KdV-type equation with variable coefficients and a solvable stochastic ordinary differential equation. Then, by the generalized wave transformation and the Clarkson-Kruskal direct method, we obtain the exact solutions of the deterministic KdV-type equation with variable coefficients. By coupling with the exact solutions of the stochastic ordinary differential equation, the exact solutions of stochastic KdV equations with variable coefficients are obtained. Compared to Wick-type stochastic KdV equations, our research work does not require additional inverse transformations, but solves stochastic partial differential equations more concisely, systematically, and directly. Secondly, two examples are given to verify the correctness of the theoretical analysis, and the soliton diffusion phenomenon of the system is discussed. Finally, by the Zabusky-Kruskal finite difference scheme, numerical simulations are provided to demonstrate the effectiveness of the analytic methods. The results indicate that the soliton diffusion phenomenon is subject to noise influence. In particular, the wave speed accelerates the soliton diffusion over time in the multiplicative noise background, and the wave speed slows down the soliton diffusion over time in the additive noise background.

High frequency attenuation of elastic waves transmitted at an angle through a randomly-fluctuating horizontally-layered slab

This paper is concerned with the modeling of elastic waves traveling at small incidence angles through a randomly-fluctuating horizontally-layered slab, in regimes where the wavelength is small compared to the thickness of the slab. The wave propagation problem is reset in a frame following the coherent front, which propagates in a homogenized medium. This homogenized medium is anisotropic because of the layering, and the equations obtained account explicitly for the coupling of quasi-P and quasi-S waves. The resulting model is governed by a set of coupled stochastic ordinary differential equations that can be approximated numerically very efficiently, and yields in particular estimates of the transmission and reflections coefficients of the slab. The latter compare favorably to the coefficients obtained in a full scale numerical simulation of the (micro-scale) wave equation, for a fraction of the cost.

XSPDE3: Extensible software for stochastic ordinary and partial differential equations

The xSPDE toolbox treats stochastic partial and ordinary differential equations, with applications in biology, chemistry, engineering, medicine, physics and quantum technologies. It computes statistical averages, including time-step and/or sampling error estimation. xSPDE can provide higher order convergence, Fourier spectra and probability densities. The toolbox has graphical output and \chi^{2}χ2 statistics, as well as weighted, projected, or forward-backward equations. It can generate input-output quantum spectra. All equations may have independent periodic, Dirichlet, and Neumann or Robin boundary conditions in any dimension, for any vector field component, and at either end of any interval.

On the Unraveling of Open Quantum Dynamics

It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.

Model reduction of Brownian oscillators: quantification of errors and long-time behavior

A procedure for model reduction of stochastic ordinary differential equations with additive noise was recently introduced in Colangeli et al (2022 J. Phys. A: Math. Theor. 55 505002), based on the Invariant Manifold method and on the Fluctuation–Dissipation relation. A general question thus arises as to whether one can rigorously quantify the error entailed by the use of the reduced dynamics in place of the original one. In this work we provide explicit formulae and estimates of the error in terms of the Wasserstein distance, both in the presence or in the absence of a sharp time-scale separation between the variables to be retained or eliminated from the description, as well as in the long-time behavior.

Asymptotic mean square stability of Predictor-Corrector methods for stochastic delay ordinary and partial differential equations

In this paper, we focus on the asymptotic mean square stability of stochastic delay ordinary and partial differential equations. By virtue of root locus technique, the sufficient and necessary conditions of asymptotic mean square stability for Predictor-Corrector methods are obtained, which are suitable for solving above two types of problems. Furthermore, by regulating the value of parameter θ in drift term of the schemes, we could investigate a series of schemes to explore different size of stable regions, this amounts to provide a road to obtain the optimal stable regions. Several theorems prove that the Predictor-Corrector schemes could inherit the asymptotic mean square stability for above two original problems. Numerical experiments verify the stability theorems, precision and the parallelism.

Seasonal Evolution of the Arctic Sea Ice Thickness Distribution

AbstractThe Thorndike et al. (1975, https://doi.org/10.1029/jc080i033p04501) theory of the ice thickness distribution, g(h), treats the dynamic and thermodynamic aggregate properties of the ice pack in a novel and physically self‐consistent manner. Therefore, it has provided the conceptual basis of the treatment of sea‐ice thickness categories in climate models. The approach, however, is not mathematically closed due to the treatment of mechanical deformation using the redistribution function ψ, the authors noting “The present theory suffers from a burdensome and arbitrary redistribution function ψ.” Toppaladoddi and Wettlaufer (2015, https://doi.org/10.1103/physrevlett.115.148501) showed how ψ can be written in terms of g(h), thereby solving the mathematical closure problem and writing the theory in terms of a Fokker‐Planck equation, which they solved analytically to quantitatively reproduce the observed winter g(h). Here, we extend this approach to include open water by formulating a new boundary condition for their Fokker‐Planck equation, which is then coupled to the observationally consistent sea‐ice growth model of Semtner (1976, https://doi.org/10.1175/1520-0485(1976)006<0379:amfttg>2.0.co;2) to study the seasonal evolution of g(h). We find that as the ice thins, g(h) transitions from a single‐ to a double‐peaked distribution, which is in agreement with observations. To understand the cause of this transition, we construct a simpler description of the system using the equivalent Langevin equation formulation and solve the resulting stochastic ordinary differential equation numerically. Finally, we solve the Fokker‐Planck equation for g(h) under different climatological conditions to study the evolution of the open‐water fraction.

On limit measures and their supports for stochastic ordinary differential equations

This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin–Wentzell or Dembo–Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This illustrates that limit measures are concentrated on Liapunov stable compact invariant sets. Applications are made to the Morse–Smale systems, the Axiom A systems including structural stability systems and separated star systems, the gradient or gradient-like systems, those systems possessing the Poincaré–Bendixson property with a finite number of limit sets to obtain that limit measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria and Liapunov stable limit sets including equilibria, limit cycles and saddle or trap cycles, respectively. Four nontrivial examples admitting a unique limit measure are provided, two of which possess infinite equivalent classes and their supports are Liapunov stable periodic orbits, one is supported on saddle-nodes.

Uncertainty in the Flexural Behavior of Soft-Core Sandwich Beams

AbstractThe impact of uncertainty on the flexural response of sandwich beams is studied in this paper. The study focuses on the unique features of typical soft-core sandwich beams, including local effects and stresses concentrations. Those are critical for the integrity and performance of the structural system and, to the best knowledge of the authors, were never considered from a stochastic viewpoint. Several structural features of the beam are considered, separately, as uncertain, and the stochastic characteristics of the distributions of the displacements and stresses are investigated. To capture the unique local effects, the extended high-order sandwich panel theory is adopted. Numerical solutions of the stochastic ordinary differential equations use the finite element (FE) method. The parametric uncertainty and its spatial distribution are modeled by random fields. The stochastic analysis uses both the Karhunen–Loeve polynomial-chaos and the perturbation-based stochastic finite element methods. The two approaches are compared and comprehensively validated against Monte Carlo simulations. The numerical results quantify the impact of uncertainty on the response, but show significant disagreements between the two stochastic approaches, and the perturbation-based stochastic finite element method is chosen as the most suitable one. The effects of the coefficient of variation of the uncertain input parameter and its correlation length are also studied. Highly amplified levels of uncertainty are revealed by the stochastic analysis near points of local stress concentration. These localized yet significant uncertainties, reported here for the first time, shed new light on the design, analysis, and safety of such sandwich beams.

Universal Out-of-Equilibrium Dynamics of 1D Critical Quantum Systems Perturbed by Noise Coupled to Energy

We consider critical one dimensional quantum systems initially prepared in their groundstate and perturbed by a smooth noise coupled to the energy density. By using conformal field theory, we deduce a universal description of the out-of-equilibrium dynamics. In particular, the full time-dependent distribution of any $2$--pt chiral correlation function can be obtained from solving two coupled ordinary stochastic differential equations. In contrast with the general expectation of heating, we demonstrate that the system reaches a non-trivial and universal stationary state characterized by broad distributions. As an example, we analyse the local energy density: while its first moment diverges exponentially fast in time, the stationary distribution, which we derive analytically, is symmetric around a negative median and exhibits a fat tail with $3/2$ decay exponent. We obtain a similar result for the entanglement entropy production associated to a given interval of size $\ell$. The corresponding stationary distribution has a $3/2$ right tail for all $\ell$, and converges to a one-sided Levy stable for large $\ell$. Our results are benchmarked via analytical and numerical calculations for a chain of non-interacting spinless fermions with excellent agreement.

Modeling study of the impact of actomyosin contractility on cell proliferation in the Drosophila wing imaginal disc.

A fundamental open question in developmental biology is deciphering how the regulation of cell proliferation and the actomyosin cytoskeleton coordinate proper formation and maintenance of tissue shape and size. To address this question, a close integration between experimental work and simulations from a novel multi-scale computational model are used. Specifically, a subcellular element (SCE) model of the Drosophila wing imaginal disc cross-section along the anterior-posterior axis is developed and calibrated using experimental data. This model accounts for actomyosin contractility in different cell regions and incorporates a detailed cell division sub model capable of capturing interkinetic nuclear migration, a fundamental process for proper cell division. Coupled model simulations and experiments reveal that anisotropic patterning of actomyosin contractility can regulate basal local curvature of the wing disc and positioning of nuclei. Moreover, it was found that cell proliferation also plays an important role in regulating basal curvature. However, this model has a simplified representation of actomyosin contractility which is not able to capture the micro level interactions between actin filaments and myosin motors that make up the actomyosin network and thus, the direction of the contractile forces cannot be accounted for explicitly. Therefore, in order to have a more biological representation of actomyosin, a coarse-grained stochastic model of subcellular level actomyosin network dynamics in two-dimensions is being developed. This model describes the actomyosin network via a large system of nodes with their dynamics represented by stochastic ordinary differential equations. The multi-scale SCE model, coupled with the stochastic model, will be used to gain insight into the detailed mechanism of how actomyosin contractility facilitates interkinetic nuclear migration, cell proliferation and formation of the wing shape.

Accuracy analysis of numerical simulations and noisy data assimilations in two-dimensional stochastic neural fields with infinite signal transmission speed

This study addresses the accuracy of stochastic simulations performed in Two-Dimensional Stochastic Neural Fields (2D-SNFs) with the infinite signal transmission speed and in the presence of external stimuli input. The numerical method in use belongs to the family of Galerkin-kind spectral approximations to Two-Dimensional Stochastic Neural Field Equations (2D-SNFEs). It translates the partial integro-differential fashion of such models into a large system of ordinary Stochastic Differential Equations (SDEs). Eventually, these SDEs are integrated approximately by the Euler–Maruyama scheme of the strong convergence order 0.5. In this paper, we devise a different-order approximate solution to the SNFE models at hand and look at the difference of such stochastic simulations on average for evaluating the consistency of the Euler–Maruyama-based numerical solution derived. The error committed in the 2D-SNFE-numerical-integration-scheme under study becomes available in our research. The other issue of particular attention and interest is hidden state reconstructions rooted in the 2D-SNFE approximations and incomplete noisy measurements of the membrane potential fulfilled at some user-assigned space positions and time instants. This statement leads to high-dimensional prediction and filtering problems to be solved. Here, we implement the Extended Kalman Filtering (EKF) approach, but accommodate it to our 2D-SNFE-oriented data assimilation scheme of huge size because of the two-dimensional manner of the stochastic process models in use. A sound performance of the newly-devised hidden state estimation technique is observed and exposed on a challenging 2D-SNFE example of computational neuroscience in Matlab.

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Kr\"uger and Stannat for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.

Carleman Estimates of Refined Stochastic Beam Equations and Applications

This paper is devoted to establishing global Carleman estimates for refined stochastic beam equations. First, by establishing a fundamental weighted identity, two Carleman estimates are derived with different weight functions for the refined stochastic beam equation, which is a coupled system consisting of a stochastic ordinary differential equation and a stochastic partial differential equation. As applications of these Carleman estimates, the exact controllability of the refined system is proved by the least controls in some sense. Different from classical stochastic beam equations, the refined one is exactly controllable at any time. Meanwhile, the uniqueness of an inverse source problem for refined stochastic beam equations is obtained without any requirement on the initial and terminal data.