Stochastic Hamiltonian Systems Research Articles

Forward–backward stochastic evolution equations in infinite dimensions and application to LQ optimal control problems

This paper focuses on the study of forward–backward stochastic evolution equations (FBSEEs), which are a class of nonlinear fully coupled forward–backward stochastic differential equations (FBSDEs), in infinite dimensions. Drawing inspiration from various linear-quadratic (LQ) optimal control problems, we apply a set of domination-monotonicity conditions that are more relaxed compared to general conditions. Within this framework, we employ the method of continuation to establish the unique solvability result and provide a pair of solution estimates. Notably, to address the challenges posed by the infinite-dimensional setting, we introduce a class of approximating equations, as Itô’s formula is not directly applicable. Conversely, these results find application in various LQ problems, where the stochastic Hamiltonian systems precisely correspond to the FBSEEs satisfying the aforementioned domination-monotonicity conditions. Consequently, by solving the corresponding stochastic Hamiltonian systems, we can obtain explicit expressions for the optimal controls.

Strong backward error analysis of symplectic integrators for stochastic Hamiltonian systems

Backward error analysis is a powerful tool in order to detect the long-term conservative behavior of numerical methods. In this work, we present a long-term analysis of symplectic stochastic numerical integrators, applied to Hamiltonian systems with multiplicative noise. We first compute and analyze the associated stochastic modified differential equations. Then, suitable bounds for the coefficients of such equations are provided towards the computation of long-term estimates for the Hamiltonian deviations occurring along the aforementioned numerical dynamics. This result generalizes Benettin-Giorgilli Theorem to the scenario of stochastic symplectic methods. Finally, specific numerical methods are considered, in order to provide a numerical evidence confirming the effectiveness of the theoretical investigation.

A new symplectic integrator for stochastic Hamiltonian systems on manifolds

This paper presents a significant development in the field of geometric numerical stochastic integration schemes. Specifically, the Geometric Symplectic Itô-Taylor 1.0 strong numerical integration scheme, tailored for Hamiltonian systems evolving on 2-sphere or S2 manifold is proposed. The core of this proposed algorithm centers around a new weak symplectic condition. This condition ensures the numerical stability for long duration time integration that can be readily achieved through numerical steps rather than relying solely on analytical satisfaction of the condition. This proposed advancement over the state-of-the-art caters to improved accuracy and stability of numerical simulations in manifold-based dynamical systems for long duration simulations. Through extensive analysis and numerical experiments, the effectiveness and reliability of the proposed scheme and symplectic criterion are validated. The findings of this study promises to offer valuable insights for researchers and practitioners working in the field of symplectic numerical integration methods.

Exponential ergodicity for damping Hamiltonian dynamics with state-dependent and non-local collisions

In this paper, we investigate the exponential ergodicity in a Wasserstein-type distance for a damping Hamiltonian dynamics with state-dependent and non-local collisions, which indeed is a special case of piecewise deterministic Markov processes that is very popular in numerous modelling situations including stochastic algorithms. The approach adopted in this work is based on a combination of the refined basic coupling and the refined reflection coupling for non-local operators. In a certain sense, the main result developed in the present paper is a continuation of the counterpart in (Stochastic Process. Appl. (2022) 146 114–142) on exponential ergodicity of stochastic Hamiltonian systems with Lévy noises and a complement of (Ann. Inst. Henri Poincaré Probab. Stat. 58 (2022a) 916–944) upon exponential ergodicity for Andersen dynamics with constant jump rate functions.

Numerical integration of stochastic contact Hamiltonian systems via stochastic Herglotz variational principle

In this work, we establish a stochastic contact variational integrator and its discrete version via stochastic Herglotz variational principle for stochastic contact Hamiltonian systems. A general structure-preserving stochastic contact method is provided to seek the stochastic contact variational integrators. Numerical experiments are performed to verify the validity of this approach.

Long-Term Analysis of Stochastic Hamiltonian Systems Under Time Discretizations

In this paper, we focus our investigation on providing long-term estimates of the Hamiltonian deviation computed along numerical approximations to the solutions of stochastic Hamiltonian systems, of both Itô and Stratonovich types. It is well known that the expected Hamiltonian of an Itô Hamiltonian system with additive noise exhibits a linear drift in time [C. Chen et al., Adv. Comput. Math., 46 (2020), 27], while the Hamiltonian function is conserved along the exact flow of a Stratonovich Hamiltonian system [T. Misawa, Japan J. Indust. Appl. Math., 17 (2000), pp. 119–128; T. Misawa, Math. Probl. Eng., 2010 (2010), 384937]. Here, we focus our attention on providing modified differential equations associated to suitable discretizations for the above problems, by means of weak backward error analysis arguments [K. C. Zygalakis, SIAM J. Sci. Comput., 33 (2011), pp. 102–130]. Then, long-term estimates are provided for both Itô and Stratonovich Hamiltonian systems, revealing the presence of parasitic terms affecting the overall conservation accuracy. Finally, selected numerical experiments are provided to confirm the theoretical analysis.

Geometry preserving Ito-Taylor formulation for stochastic hamiltonian dynamics on manifolds

Naturally occurring systems often have inherent uncertainties and often evolve on complicated smooth and differentiable hypersurfaces that are not necessarily Euclidean. The dynamics of such systems can be potentially described using a stochastic Hamiltonian formulation on differentiable manifold. Solution of these stochastic Hamiltonian systems using the current state-of-the-art methods, either fails to preserve the geometry of the manifold, or, ignore the stochastic nature of the differential equations. To address this, we develop a framework which preserves geometry while accounting for stochasticity in input and states. A new development for higher order geometry preserving Ito-Taylor expansion based stochastic integration scheme is demonstrated. Detailed mathematical development of the scheme is laid down and numerically demonstrated through the application of this method on a number of practical physical systems such as spherical pendulum on a cart and two quadrotors transporting a mass point. The accuracy of the method is established by comparing the global error of the solution with the existing geometric Euler-Maruyama integration scheme, geometric Milstein scheme and the non-geometric Ito-Taylor based stochastic integration schemes.

NySALT: Nyström-type inference-based schemes adaptive to large time-stepping

Large time-stepping is important for efficient long-time simulations of deterministic and stochastic Hamiltonian dynamical systems. Conventional structure-preserving integrators, while being successful for generic systems, have limited tolerance to time step size due to stability and accuracy constraints. We propose to use data to innovate classical integrators so that they can be adaptive to large time-stepping and are tailored to each specific system. In particular, we introduce NySALT, Nyström-type inference-based schemes adaptive to large time-stepping. The NySALT has optimal parameters for each time step learnt from data by minimizing the one-step prediction error. Thus, it is tailored for each time step size and the specific system to achieve optimal performance and tolerate large time-stepping in an adaptive fashion. We prove and numerically verify the convergence of the estimators as data size increases. Furthermore, analysis and numerical tests on the deterministic and stochastic Fermi-Pasta-Ulam (FPU) models show that NySALT enlarges the maximal admissible step size of linear stability, and quadruples the time step size of the Störmer–Verlet and the BAOAB when maintaining similar levels of accuracy.

A moderate deviation principle for stochastic Hamiltonian systems

We prove a moderate deviation principle for stochastic differential equations (SDEs) with non-Lipschitz conditions. As an application of our result, we also study the stochastic Hamiltonian systems.

Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

<p style='text-indent:20px;'>In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [<xref ref-type="bibr" rid="b12">12</xref>] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues <inline-formula><tex-math id="M1">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> and construct corresponding eigenfunctions. Moreover, the order of growth for these <inline-formula><tex-math id="M2">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> are obtained: <inline-formula><tex-math id="M3">\begin{document}$ \lambda_m\sim m^2 $\end{document}</tex-math></inline-formula>, as <inline-formula><tex-math id="M4">\begin{document}$ m\rightarrow +\infty $\end{document}</tex-math></inline-formula>. As applications, we give an explicit estimation formula about the statistic period of solutions of Forward-Backward SDEs. Besides, by a meticulous example we show the subtle situation in time-dependent case that some eigenvalues appear when the solution of the associated Riccati equation does not blow-up, which does not happen in time-invariant case.</p>

On the conservative character of discretizations to Itô-Hamiltonian systems with small noise

In this paper, we consider stochastic Hamiltonian systems of Itô type driven by a multiplicative small noise. It is well-known, indeed, that stochastic Hamiltonian problems are suitable candidates to reconcile the Hamiltonian classical mechanics with the non-differentiability of the Wiener process, which provides the innovative character of stochastic effects. In particular, in this work, we provide a characterization of the behavior of averaged Hamiltonians arised in such systems, with more emphasis on the separable and quadratic Hamiltonians. Next, we show that, in general, first order approximations to such systems are not able to retain the same behavior discovered for the exact averaged Hamiltonian. Hence, the analysis for the specific case of ϑ-methods is performed. Finally, numerical evidence is depicted in order to confirm theoretical results.

Homogenization of Dissipative Hamiltonian Systems Under Lévy Fluctuations

We study the small mass limit for a class of Hamiltonian systems with multiplicative non-Gaussian Lévy noise. Derivation of the limiting equation depends on the structure of the stochastic Hamiltonian systems, in which a discontinuous noise-induced drift term arises. Firstly, we show that the momentum in the stochastic Hamiltonian system converges to zero when the kinetic energy has polynomial growth. Then, we prove that the stochastic Hamiltonian system with classical kinetic energy converges to the limiting equation in probability, with respect to Skorokhod topology as the mass tends to zero.

Explicit pseudo-symplectic Runge-Kutta methods for stochastic Hamiltonian systems

We give conditions for stochastic Runge-Kutta methods to near preserve quadratic invariants, and we discuss the associated simplified order conditions. For stochastic Hamiltonian systems we propose a systematic approach to construct explicit stochastic Runge-Kutta pseudo-symplectic schemes. Our approach is based on colored trees and B-series. We construct some pseudo-symplectic stochastic Runge-Kutta methods with strong convergence order, and we illustrate numerically the long term performance of the proposed schemes.

A symplectic homotopy perturbation method for stochastic and interval Hamiltonian systems and its applications in structural dynamic systems

A symplectic homotopy perturbation method for stochastic and interval Hamiltonian systems and its applications in structural dynamic systems

Stochastic Port-Hamiltonian Systems

In the present work we formally extend the theory of port-Hamiltonian systems to include random perturbations. In particular, suitably choosing the space of flow and effort variables we will show how several elements coming from possibly different physical domains can be interconnected in order to describe a dynamic system perturbed by general continuous semimartingale. Relevant enough, the noise does not enter into the system solely as an external random perturbation, since each port is itself intrinsically stochastic. Coherently to the classical deterministic setting, we will show how such an approach extends existing literature of stochastic Hamiltonian systems on pseudo-Poisson and pre-symplectic manifolds. Moreover, we will prove that a power-preserving interconnection of stochastic port-Hamiltonian systems is a stochastic port-Hamiltonian system as well.

How do Monte Carlo estimates affect stochastic geometric numerical integration?

In this work, we investigate the numerical conservation of characteristic properties of stochastic Hamiltonian problems driven by additive noise under time discretizations, in the spirit of stochastic geometric numerical integration. In particular, we aim to understand how Monte Carlo and multilevel Monte Carlo estimators for the expected values eventually affect the conservation properties of the numerical scheme. Specifically, our analysis is focused on the role of random number generation in the conservation of invariant laws for stochastic Hamiltonian problems under time discretization by the drift-preserving numerical scheme introduced in C. Chen, D. Cohen, R. D'Ambrosio and A. Lang, [Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math. 46(2) (2020), pp. 27.] and the stochastic perturbation of a symplectic Runge–Kutta method, introduced in K. Burrage and P.M. Burrage, [Low rank Runge–Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math. 236(16) (2012), pp. 3920–3930.]. The numerical evidence confirms the aforementioned theoretical analysis.

Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises

We establish exponential ergodicity for the stochastic Hamiltonian system (Xt,Vt)t≥0 on R2d with Lévy noises dXt=(aXt+bVt)dt,dVt=U(Xt,Vt)dt+dLt,where a≥0, b>0, U:R2d→Rd and (Lt)t≥0 is an Rd-valued pure jump Lévy process. The approach is based on a new refined basic coupling for Lévy processes and a Lyapunov function for stochastic Hamiltonian systems. In particular, we can handle the case that U(x,v)=−v−∇U0(x) with double well potential U0 which is super-linear growth at infinity such as U0(x)=c1(1+|x|2)l−c2|x|2 with l>1 or U0(x)=c1e(1+|x|2)l−c2|x|2 with l>0 for any c1,c2>0, and also deal with the case that the Lévy measure ν of (Lt)t≥0 is degenerate in the sense that ν(dz)≥c|z|d+θ01{0<z1≤1}dzfor some c>0 and θ0∈(0,2), where z1 is the first component of the vector z∈Rd.

Inverse optimal control of regime-switching jump diffusions

&lt;p style='text-indent:20px;'&gt;This paper studies the inverse optimal control using Legendre-Fenchel (in short, LF) translation method for regime-switching jump diffusions. Our approach is to first design inverse pre-optimal stabilization controllers and then obtain inverse optimal stabilizers, which avoids solving a Hamilton-Jacobi-Bellman equation. Finally, an application to stochastic Hamiltonian systems with Markov regime-switching is studied in detail for illustration.&lt;/p&gt;

Simultaneous exponential stabilization for stochastic port-controlled Hamiltonian systems with actuator saturations, fading channels and delays

This paper is concerned with the simultaneous exponential stabilization problem for a set of stochastic port-controlled Hamiltonian (PCH) systems. Due to the limited bandwidth of the channels, the phenomena of fading channels and transmission delays which are described by a time-varying stochastic model always occur in the communication channels from the controller to the actuator. Meanwhile, actuator saturation constraint is taken into account. On the basis of dissipative Hamiltonian structural and saturating actuator properties, those stochastic PCH systems are combined to generate an augmented system. By utilizing the stochastic analysis theory, sufficient criterions are given for the simultaneous stabilization controller design ensuring that the closed-loop system is simultaneously exponentially mean-square stable (SEMSS). For the case that there exist external disturbances in the systems, some results on stability analysis and controller design are given. The developed controller design scheme is proved by a three-helicopter model simulation example.

Structure-Preserving Numerical Methods for Stochastic Poisson Systems

We propose a class of numerical integration methods for stochastic Poisson systems (SPSs) of arbitrary dimensions. Based on the Darboux-Lie theorem, we transform the SPSs to their canonical form, the generalized stochastic Hamiltonian systems (SHSs), via canonical coordinate transformations found by solving certain PDEs defined by the Poisson brackets of the SPSs. An a-generating function approach with \alpha\in [0,1] is then used to create symplectic discretizations of the SHSs, which are then transformed back by the inverse coordinate transformation to numerical integrators for the SPSs. These integrators are proved to preserve both the Poisson structure and the Casimir functions of the SPSs. Applications to a three-dimensional stochastic rigid body system and a three-dimensional stochastic Lotka-Volterra system show efficiency of the proposed methods.