We provide a new approach to strong error analysis of the spatial-spectral Galerkin and temporal exponential Euler scheme for a family of second-order parabolic stochastic partial differential equations (SPDEs) driven by multiplicative noise. Applying these results to the stochastic advection-diffusion-reaction equation with a gradient term driven by white noise indicates that this scheme achieves optimal strong convergence order exactly 1/2 in space, which removes an infinitesimal factor in the literature, and 1/4 in time. Numerical experiments support our theoretical analysis.

This paper studies densities for solutions of the stochastic functional differential equation (SFDE) and of its Euler-type discretizations. First, by means of the Malliavin calculus, we prove the existence of densities for the exact solution and its discretizations. Then we establish the L1(Rd)-convergence for the density of discretizations by implementing a dimensionality reduction argument and a localization argument. Further, we prove that the pointwise convergence rate of the density is 1 when the noise is of additive type. The convergence results indicate that the total variation distance between laws of solutions for the SFDE and its discretizations vanishes to zero as the discretization parameter diminishes, while that between laws of functional solutions fails to vanish due to the high degeneracy of the equation. This finding highlights one of the main distinctions in asymptotic behaviors of the corresponding discretized systems when compared to stochastic ordinary (partial) differential equations.

For a stochastic autoparametric block-and-pendulum system, the long-time dynamics exhibit two fundamental features: the almost-sure stability of the single mode solution, characterized by its Lyapunov exponent, and the global asymptotic dynamics when this single mode solution loses stability. This naturally raises the question of whether these dynamical features are preserved under discretization, since such preservation is essential for the resulting discrete system to faithfully capture the qualitative behavior of the continuous system. To address this question, we first establish the existence of a random attractor for the continuous system subject to multiplicative stochastic excitation, providing a rigorous characterization of the global asymptotic dynamics. We then propose a numerical discretization that induces a discrete random dynamical system and prove the convergence of its random attractor to the continuous one as the step size tends to zero. In addition, we show that the numerical Lyapunov exponent of the single mode solution has the same sign as its continuous counterpart for sufficiently small step sizes, thus preserving the corresponding almost-sure stability or instability classification. These results demonstrate that the proposed discretization captures both the global asymptotic dynamics and the stability characteristics of the underlying stochastic autoparametric system.

In this work, we study the stochastic Schrödinger–KdV equation driven by additive noise from both analytical and numerical viewpoints. We first establish the evolution laws for the averaged plasmon number, momentum, and energy, together with the conservation of the averaged particle number. Motivated by these intrinsic structures, we develop two temporal discretizations. One is constructed based on the splitting strategy and Crank–Nicolson scheme, and is shown to preserve the discrete evolution laws of the averaged plasmon number and momentum, as well as the discrete conservation law of the averaged particle number. The other is proposed within the constant scalar auxiliary variable framework, in which the nonlinear energy functional is reformulated so that a modified averaged energy law can be preserved at the discrete level. Combining these temporal discretizations with a local discontinuous Galerkin approximation in space yields structure-preserving full discretizations inheriting the corresponding discrete physical laws. Numerical experiments are presented to validate the theoretical results and to demonstrate the accuracy, robustness, and effectiveness of the proposed methods.

In this paper, we study stochastic logarithmic Schr & ouml;dinger (SLogS) equations subject to random dispersion and external driving noise. The singularity near zero density induced by the logarithmic nonlinearity is handled via an energy regularization method. Using the resulting regularized approximation, we establish well-posedness for the SLogS with white-noise dispersion driven by both additive and multiplicative noise. We further prove the stochastic translation invariance and gauge invariance for the SLogS equation and analyze the impact of random perturbations on standing waves. In addition, we develop a regularized Lie splitting approximation and derive strong error bounds that depend explicitly on the random dispersion coefficient, the deterministic dispersion coefficient, and the driving-noise amplitude. These results provide a unified analytical and numerical framework for SLogS with logarithmic nonlinearities under combined random effects. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

. This paper studies the asymptotic error distributions of several symplectic and non-symplectic methods for stochastic Hamiltonian systems. Focusing on stochastic Hamiltonian systems driven by additive noise, we obtain the asymptotic limit of the normalized error distribution of the 9-method (9 is an element of [0, 1]) that is symplectic if and only if 9 = 21. The upper bound for the second moment of the asymptotic error distribution suggests that the midpoint method may minimize the error constant of the 9-method over a large time horizon T. Furthermore, we take the linear stochastic oscillator as a test equation and investigate exact asymptotic error constants of several symplectic and nonsymplectic methods. Our result implies that in the long-time computation, the probability that the error deviates from zero decays exponentially faster for the symplectic methods than for the non-symplectic ones.

This paper investigates how the structure of the underlying graph influences the behavior of stochastic partial differential equations (SPDEs) on finite tree graphs, where each edge is driven by space-time white noise. We first introduce a novel graph-based null decomposition approach to analyzing the strong Feller property of the Markov semigroup generated by SPDEs on tree graphs. By examining the positions of zero entries in eigenfunctions of the graph Laplacian operator, we establish a sharp upper bound on the number of noise-free edges that ensures both the strong Feller property and irreducibility. Interestingly, we find that the addition of noise to any single edge is sufficient for chain graphs, whereas for star graphs, at most one edge can remain noise-free without compromising the system's properties. Furthermore, under a dissipative condition, we prove the existence and exponential ergodicity of a unique invariant measure.

Stochastic partial differential equations (SPDEs) driven by L & eacute;vy noise naturally arise in physical systems exhibiting abrupt or discontinuous random effects, whose temporal H & ouml;lder continuity in Lp sense is known to be at most 1 resulting from the Burkholder-Davis-Gundy inequality. This p poses a challenge in the numerical analysis for achieving the uniform Lp-strong convergence orders of numerical schemes for p >= 2. In this paper, we develop a discretization framework for constructing fully discrete schemes tailored to different conditions of L & eacute;vy measures, whose spatial direction is based on the spectral Galerkin method and temporal direction employs the Euler-type method. For the case of finite L & eacute;vy measures, a jump-adapted time discretization is utilized for the equation that may involve multiplicative noise; while for the case that L & eacute;vy measures can be infinite, we introduce an approach based on the quantitative John-Nirenberg inequality for SPDEs driven by additive L & eacute;vy noise. We prove that proposed schemes converge in Lp sense with orders nearly 1/2 in both space and time for all p >= 2, which contributes novel results in the numerical analysis of the SPDE driven by L & eacute;vy noise. Numerical experiments are presented to illustrate our theoretical findings.

In this paper, we propose a new class of splitting methods to solve the stochastic Langevin equation, which can simultaneously preserve the ergodicity and exponential integrability of the original equation. The central idea is to extract a stochastic subsystem that possesses the strict dissipation from the original equation, which is inspired by the inheritance of the Lyapunov structure for obtaining the ergodicity. We prove that the exponential moment of the numerical solution is bounded, thus validating the exponential integrability of the proposed methods. Further, we show that under moderate verifiable conditions, the methods have the first-order convergence in both strong and weak senses, and we present several concrete splitting schemes based on the methods. The splitting strategy of methods can be readily extended to construct conformal symplectic methods and high-order methods that preserve both the ergodicity and the exponential integrability, as demonstrated in numerical experiments. Our numerical experiments also show that the proposed methods have good performance in the long-time simulation.

In this paper, we propose and analyze semi-implicit numerical schemes for the stochastic nonlinear Klein–Gordon equation (SNKGE) with multiplicative noise. These numerical schemes, called stochastic scalar auxiliary variable (SAV) schemes, are constructed by transforming the considered SNKGE into a higher dimensional stochastic system with a stochastic SAV. We prove that they can be solved explicitly, and preserve the modified energy evolution law and the regularity structure of the original system. These structure-preserving properties are the keys to overcoming the mutual effect of noise and nonlinearity. By providing new regularity estimates of the introduced SAV, we obtain the strong convergence rate of stochastic SAV schemes under Lipschitz conditions. Furthermore, based on the modified energy evolution laws, we derive the exponential moment bounds and sharp strong convergence rate of the proposed schemes for SNKGE with a non-globally Lipschitz nonlinearity in the additive noise case. To the best of our knowledge, this is the first result on the construction and strong convergence of semi-implicit schemes preserving averaged energy evolution law for SNKGEs.

In this paper, we analyze the stability and convergence orders of the time-splitting schemes for the deterministic and stochastic Gross-Pitaevskii equations with rotating action in the Sobolev space. The whole process heavily depends on the integral representation. Firstly, we derive the computational properties of the Laplace operator and the angular momentum operator, and prove the second-order convergence of the Strang-type splitting scheme for the deterministic equation. Secondly, we introduce the truncated equation so that the nonlinear term is globally Lipschitz continuous. This makes it possible to prove the first-order convergence of the splitting method, and then extend the conclusion to the original stochastic equation. Finally, several numerical experiments are presented to support our theoretical findings.

Chaotic phases in stochastic differential equations are characterized by two essential long-time dynamical features: a random attractor capturing asymptotic geometry and a Sinai-Ruelle-Bowen (SRB) measure describing statistical information. This paper investigates whether the stochastic Hopf bifurcation under discretization could inherit both features. We establish that the stochastic Hopf bifurcation under discretization induces a discrete random dynamical system. Further, we prove that this discrete system possesses a random attractor, and then derive the existence of an SRB measure by demonstrating a strictly positive numerical Lyapunov exponent. Numerical experiments visualize the retained random attractor and SRB measure for the discrete random dynamical system, revealing structures consistent with the theoretical chaotic phase.

In this paper, we investigate the asymptotic error distributions of symplectic methods for stochastic Hamiltonian systems and further provide Hamiltonian-specific analysis that clarifies the superiority of symplectic methods. Our contribution is threefold. First, we derive the asymptotic error distributions of symplectic methods for stochastic Hamiltonian systems with multiplicative noise and additive noise, respectively, and show that the obtained limiting stochastic processes satisfy equations retaining the Hamiltonian formulations. Second, we propose a new approach for calculating the asymptotic error distribution, revealing the connection between the stochastic modified equation and the asymptotic error distribution. Third, we characterize the limiting distribution of the normalized Hamiltonian deviation, thereby illustrating through test equations the superiority of symplectic methods for long-time simulations of the Hamiltonians, even in the limit as the step size tends to zero.

This paper aims to investigate the asymptotic error distribution of several numerical methods for stochastic partial differential equations (SPDEs) with multiplicative noise. Firstly, we give the limit distribution of the normalized error process of the exponential Euler method in $\dot{H}^η$ for some $η>0$. A key finding is that the asymptotic error in distribution of the exponential Euler method is governed by a linear SPDE driven by infinitely many independent $Q$-Wiener processes. This characteristic represents a significant difference from numerical methods for both stochastic ordinary differential equations and SPDEs with additive noise. Secondly, as applications of the above result, we derive the asymptotic error distribution of a full discretization based on the temporal exponential Euler method and the spatial finite element method. As a concrete illustration, we provide the pointwise limit distribution of the normalized error process when the exponential Euler method is applied to a specific class of stochastic heat equations. Finally, by studying the asymptotic error of the spatial semi-discrete spectral Galerkin method, we demonstrate that the actual strong convergence speed of spatial semi-discrete numerical methods may be highly problem-dependent, rather than universally predictable.

In this paper, several energy-conserving numerical schemes are constructed for solving the two-dimensional Maxwell equations. Initially, the original problem is decomposed into two one-dimensional subproblems using operator splitting techniques. Subsequently, for spatial discretization, we employ the local radial basis function (LRBF) method, while for temporal discretization, three different splitting composition methods are selected, including the Lie-Trotter method, the Strang method, and the three-stage fourth-order splitting method. Through an analysis of the structural characteristics of the spatial differential matrices generated by the LRBF method, the unconditional stability and energy conservation properties of the fully discretized schemes are further proved. Numerical experiments validate the temporal convergence accuracy of the three numerical schemes and the preservation of their relevant properties.

This paper studies the existence and uniqueness of the invariant measure for a class of stochastic Maxwell equations and proposes a novel kind of ergodic numerical approximations to inherit the intrinsic properties. The key to proving the ergodicity lies in the uniform regularity estimates of the exact and numerical solutions with respect to time, which are established by analyzing some important physical quantities. By introducing an auxiliary process, we show that the mean-square convergence order of the discontinuous Galerkin full discretization is 12 in the temporal direction and 12 in the spatial direction, which provides the convergence order of the numerical invariant measure to the exact one in L2-Wasserstein distance. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The law of the iterated logarithm (LIL) for the time-homogeneous Markov process with a unique invariant measure characterizes the almost sure maximum possible fluctuation of time averages around the ergodic limit. Whether a numerical approximation can preserve this asymptotic pathwise behavior remains an open problem. In this work, we give a positive answer to this question and establish the LIL for the numerical approximation of such a process under verifiable assumptions. The Markov process is discretized by a decreasing time-step strategy, which yields the non-homogeneous numerical approximation but facilitates a martingale-based analysis. The key ingredient in proving the LIL for such numerical approximation lies in extracting a quasi-uniform time-grid subsequence from the original non-uniform time grids and establishing the LIL for a predominant martingale along it, while the remainder terms converge to zero. Finally, we illustrate that our results can be flexibly applied to numerical approximations of a broad class of stochastic systems, including SODEs and SPDEs.