Wong–Zakai approximations with convergence rate for stochastic partial differential equations

ABSTRACT

The goal of this paper is to prove a convergence rate for Wong–Zakai approximations of semilinear stochastic partial differential equations driven by a finite-dimensional Brownian motion. Several examples, including the HJMM equation from mathematical finance, illustrate our result.     

Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and super-linear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models.     

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.     

Ginzburg–Landau equations and their generalizations

The Ginzburg–Landau equations were proposed in the superconductivity theory to describe mathematically the intermediate state of superconductors in which the normal conductivity is mixed with the superconductivity. It turned out that these equations have interesting and non-trivial generalizations. First of all, they can be extended to arbitrary compact Riemann surfaces. Next, they can be generalized to dimension 3 as dynamical (or hyperbolic) Ginzburg–Landau equations. They also have a 4-dimensional extension provided by Seiberg–Witten equations. In this review we describe all these interesting topics together with some unsolved problems.     

Generalized Ginzburg–Landau equations in high dimensions

In this work, we study critical points of the generalized Ginzburg–Landau equations in dimensions \(n\ge 3\) which satisfy a suitable energy bound, but are not necessarily energy-minimizers. When the parameter in the equations tend to zero, such solutions are shown to converge to singular n-harmonic maps into spheres, and the convergence is strong away from a finite set consisting (1) of the infinite energy singularities of the limiting map, and (2) of points where bubbling off of finite energy n-harmonic maps could take place. The latter case is specific to dimensions \({>}2\). We also exhibit a criticality condition satisfied by the limiting n-harmonic maps which constrains the location of the infinite energy singularities. Finally we construct an example of non-minimizing solutions to the generalized Ginzburg–Landau equations satisfying our assumptions.     

The Wong–Zakai approximations of invariant manifolds and foliations for stochastic evolution equations

In this paper, we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated dynamics of a class of stochastic evolution equations with a multiplicative white noise. We prove that the solutions of Wong–Zakai approximations almost surely converge to the solutions of the Stratonovich stochastic evolution equation. We also show that the invariant manifolds and stable foliations of the Wong–Zakai approximations converge to the invariant manifolds and stable foliations of the Stratonovich stochastic evolution equation, respectively.     

Upper semicontinuity of random attractors of stochastic discrete complex Ginzburg–Landau equations with time-varying delays in the delay

A system of stochastic discrete complex Ginzburg–Landau equations with time-varying delays is considered. We first prove the existence and uniqueness of random attractor for these equations. Then, we analyze the convergence properties of the solutions as well as the attractors as the length of time delay approaches zero.     

Error estimate of Fourier pseudo-spectral method for multidimensional nonlinear complex fractional Ginzburg–Landau equations

In this paper, we discuss the error estimate of Fourier pseudo-spectral method for multidimensional nonlinear complex space fractional Ginzburg-Landau equations. The continuous mass and energy inequalities as well as their discrete versions are presented. Moreover, by the discrete mass and energy inequalities, the error estimate of the Fourier pseudo-spectral scheme is established, and the scheme is proved to have the spectral accuracy.     

Asymptotic expansions and convergence rates for Euler’s approximations of semigroups

We provide optimal bounds for errors in Euler’s approximations of semigroups in Banach algebras and of semigroups of operators in Banach spaces. Furthermore, we construct asymptotic expansions for such approximations with optimal bounds for remainder terms. The sizes of errors are controlled by smoothness properties of semigroups. In this paper we use Fourier–Laplace transforms and a reduction of the problem to the convergence rates and asymptotic expansions in the Law of Large Numbers. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09. This paper was written in 2004. In the interim, several related articles were published; let us mention [14, 13, 15].     

Existence of strong solutions for Itô's stochastic equations via approximations

Summary Given strong uniqueness for an Itô's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on any probability space by using, for example, Euler's polygonal approximations. Stochastic equations in ^d and in domains in ^d are considered.Research supported by the Hungarian National Foundation of Scientific Research No. 2990.Supported in part by NSF Grant DMS-9302516     

Exponential stability of equidistant Euler–Maruyama approximations of stochastic differential delay equations

Our aim is to study under what conditions the exact and numerical solution (based on equidistant nonrandom partitions of integration time-intervals) to a stochastic differential delay equation (SDDE) share the property of mean-square exponential stability. Our approach is trying to avoid the use of Lyapunov functions or functionals. We show that under a global Lipschitz assumption an SDDE is exponentially stable in mean square if and only if for some sufficiently small stepsize Δ$\Delta$ the Euler–Maruyama (EM) method is exponentially stable in mean square. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same “if and only if” result. The important feature of this result is that it transfers the asymptotic problem into a finite-time convergence problem. Replacing the exact and EM numerical solution with stochastic processes, we also discuss whether a family of stochastic processes share the stability property. This new approach allows us to discuss (i) whether a family of SDDEs share the stability property, and (ii) whether an SDDE with variable time lag shares stability property with the modified EM method. As another application of this general approach we consider a linear SDDE with variable time lag and establish an “if and only if” result. It should also be mentioned that the questions addressed, results proved, as well as style of analysis borrow heavily from [14] but the computations involved to cope with time delay are nontrivial.     

Stability of symmetric vortices for two-component Ginzburg–Landau systems

We study Ginzburg–Landau equations for a complex vector order parameter Ψ=(_ψ+,_ψ−)∈C²$\Psi =({\psi }_{+},{\psi }_{-})\in {\mathbb{C}}^2$. We consider the Dirichlet problem in the disk in R²${\mathbb{R}}^2$ with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find that the stability of the degree-one equivariant solution depends on the Ginzburg–Landau parameter as well as the sign of the interaction term in the energy.     

Homogenization of a Ginzburg–Landau functional

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that sizes and distances between inclusions are of the same order ?, we obtain a limiting functional as $?\to 0$. We generalize the method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We give computational formulas for material characteristics of an effective medium. To cite this article: L. Berlyand et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Wong–Zakai approximations and attractors for stochastic reaction–diffusion equations on unbounded domains

In this paper, we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of the stochastic reaction–diffusion equation driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong–Zakai approximations of stochastic reaction–diffusion equation. Then, we show that the attractors of Wong–Zakai approximations converges to the attractor of the stochastic reaction–diffusion equation for both additive and multiplicative noise.     

Adiabatic Limit for Hyperbolic Ginzburg–Landau Equations

We study the adiabatic limit in hyperbolic Ginzburg–Landau equations which are Euler–Lagrange equations for the Abelian Higgs model. Solutions of Ginzburg–Landau equations in this limit converge to geodesics on the moduli space of static solutions in the metric determined by the kinetic energy of the system. According to heuristic adiabatic principle, every solution of Ginzburg–Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some geodesic. A rigorous proof of this result was proposed recently by Palvelev.