Infinite-dimensional Differential Equation Research Articles

A mild approach to spatial discretization for backward stochastic differential equations in infinite dimensions

In this paper, we present the stability result of a spatial semi-discrete scheme to backward stochastic differential equations taking values in a Hilbert space. Under suitable assumptions of the final value and the drift, a convergence rate is established.

Pseudo almost periodicity for stochastic differential equations in infinite dimensions

In this article, we introduce the concept of p-mean θ-pseudo almost periodic stochastic processes, which is slightly weaker than p-mean pseudo almost periodic stochastic processes. Using the operator semigroup theory and stochastic analysis theory, we obtain the existence and uniqueness of square-mean θ-pseudo almost periodic mild solutions for a semilinear stochastic differential equation in infinite dimensions. Moreover, we prove that the obtained solution is also pseudo almost periodic in path distribution. It is noteworthy that the ergodic part of the obtained solution is not only ergodic in square-mean but also ergodic in path distribution. Our main results are even new for the corresponding stochastic differential equations (SDEs) in finite dimensions.

Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturbation. Moreover, we provide conditions to obtain uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. We also present an example of evolution process in a Banach space that admits nonuniform exponential dichotomy and study the permanence of the nonuniform hyperbolicity under perturbation. Finally, we prove persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.

Supports for degenerate stochastic differential equations with jumps and applications

In the paper, we are concerned with degenerate stochastic differential equations with jumps. We first establish two theorems about supports for the solution laws of the degenerate stochastic differential equations, under different (sufficient) conditions. We then apply one of our results to a class of degenerate stochastic evolution equations (that is, stochastic differential equations in infinite dimensions) with jumps to obtain a characterisation of path-independence for the densities of their Girsanov transformations.

Applying Heath-Jarrow-Morton Model to Forecasting the US Treasury Daily Yield Curve Rates

The Heath-Jarrow-Morton (HJM) model is a powerful instrument for describing the stochastic evolution of interest rate curves under no-arbitrage assumption. An important feature of the HJM approach is the fact that the drifts can be expressed as functions of respective volatilities and the underlying correlation structure. Aimed at researchers and practitioners, the purpose of this article is to present a self-contained, but concise review of the abstract HJM framework founded upon the theory of interest and stochastic partial differential equations in infinite dimensions. To illustrate the predictive power of this theory, we apply it to modeling and forecasting the US Treasury daily yield curve rates. We fit a non-parametric model to real data available from the US Department of the Treasury and illustrate its statistical performance in forecasting future yield curve rates.

Concept to Evaluate and Quantify Field Inhomogeneities in Coaxial TEM-Cells

The quality of a field probe calibration directly affects the uncertainty statement of radiated immunity tests. Therefore, precise field calibration is required. An often-used field generator is a transversal electromagnetic (TEM) cell. The IEEE Std 1309 provides three methods to calibrate a field probe. Method B, also known as the standard field method, uses a calculated reference field based on the geometry of the measurement environment and its input parameters. However, in reality, mechanical tolerances and misalignment can decrease the field quality. The results of the calculated model and the actual system may differ. Hence, this contribution proposes an approach for computing electromagnetic (EM) fields inside a TEM-cell considering mechanical tolerances and unknown input parameters. It is accomplished by projecting Maxwell’s equations onto eigenfunctions resulting in an infinite-dimensional differential equation system, the so-called generalized telegraphist’s equations (GTEs). Since this method starts with Maxwell’s equations, it can be used for a variety of applications. The proposed concept is applied on a coaxial TEM-cell with a circular cross-section with random imperfections. Based on the semi-analytical method and an input–output model for the uncertainty propagation, the combined uncertainty can be calculated following the guide to the expression of uncertainty in measurement (GUM).

Robustness of Exponential Dichotomy in a Class of Generalised Almost Periodic Linear Differential Equations in Infinite Dimensional Banach Spaces

In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (Dichotomies in stability theory. Lecture notes in mathematics, Springer, Berlin, 1970). Based in Rodrigues (Invariância para sistemas de equacoes diferenciais com retardamento e aplicacoes, Tese de Mestrado, Universidade de Sao Paulo, Sao Carlos, 1970) and in Kloeden and Rodrigues (Nonlinear Anal 74:2695–2719, 2011), Rodrigues et al. (Stability problems in non autonomous linear differential equations in infinite dimensions. arXiv:1906.04642, 2019) we use the class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.

Stability problems in nonautonomous linear differential equations in infinite dimensions

In this paper we study the robustness of the stability in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel [3]. Based in Rodrigues [11] and in Kloeden & Rodrigues [10] we introduce a class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.As counterparts, e show first in another example that it is possible to stabilize an unstable system by using a perturbation with a large period and a small mean value, and finally we give an example where we stabilize an unstable linear ODE with a small perturbation in infinite dimensions by using some ideas developed in Rodrigues & Solà-Morales [21] after an example due to Kakutani (see [13]).

Distributions and Analytical Measures on Infinite-Dimensional Spaces

Spaces of test functions and spaces of distributions (generalized measures) on infinite-dimensional spaces are constructed, which, in the finite-dimensional case, coincide with classical spaces $$\mathscr{D}$$ and $$\mathscr{D}'$$ . These distribution spaces contain generalized Feynman measures (but do not contain a generalized Lebesgue measure, which is not considered in this paper). For broad classes of infinite-dimensional differential equations in distribution spaces, the Cauchy problem has fundamental solutions. These results are much more definitive than those of A.Yu. Khrennikov’s and A.V. Uglanov’s pioneering works.

Book Review: Yosida approximations of stochastic differential equations in infinite dimensions and applications

Book Review: Yosida approximations of stochastic differential equations in infinite dimensions and applications

Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions

The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler–Maruyama, Milstein, Crank–Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory.

Real-time simulation of H–P noisy Schrödinger equation and Belavkin filter

The Hudson---Parthasarathy noisy Schrodinger equation is an infinite-dimensional differential equation where the noise operators--Creation, Annihilation and Conservation processes--take values in Boson Fock space. We choose a finite truncated basis of exponential vectors for the Boson Fock space and obtained the unitary evolution in a truncated orthonormal basis using the Gram---Schmidt orthonormalization process to the exponential vectors. Then, this unitary evolution is used to obtained the approximate evolution of the system state by tracing out over the bath space. This approximate evolution is compared to the exact Gorini---Kossakowski---Sudarshan---Lindblad equation for the system state. We also perform a computation of the rate of change of the Von Neumann entropy for the system assuming vacuum noise state and derive condition for entropy increase. Finally, by taking non-demolition measurement in the sense of Belavkin, we simulate the Belavkin quantum filter and show that the Frobenius norm of the error observables $$j_t(X)-\pi _t(X)$$jt(X)-?t(X) becomes smaller with time for a class of observable X. Here $$j_t(X)$$jt(X) is the H---P equation observable and $$\pi _t(X)$$?t(X) is the Belavkin filter output observable. In last, we have derived an approximate expression for the filtered density and entropy of the system after filtering.

Existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions

In this paper, we present a general framework for solving stochastic functional differential equations in infinite dimensions in the sense of martingale solutions, which can be applied to a large class of SPDE with finite delays, e.g. d-dimensional stochastic fractional Navier–Stokes equations with delays, d-dimensional stochastic reaction–diffusion equations with delays, d-dimensional stochastic porous media equations with delays. Moreover, under local monotonicity conditions for the nonlinear terms we obtain the existence and uniqueness of strong solutions to SPDE with delays.

Invariant Manifolds for Steady Boltzmann Flows and Applications

We develop a theory of invariant manifolds for the steady Boltzmann equation and apply it to the study of boundary layers and nonlinear waves. The steady Boltzmann equation is an infinite dimensional differential equation, so the standard center manifold theory for differential equations based on spectral information does not apply here. Instead, we employ a time-asymptotic approach using the pointwise information of Green’s function for the construction of the linear invariant manifolds. At the resonance cases when the Mach number at the far field is around one of the critical values of −1, 0 or 1, the truly nonlinear theory arises. In such a case, there are wave patterns combining the fast decaying Knudsen-type and slow varying fluid-like waves. The key Knudsen manifolds consisting of only Knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical Mach numbers. Around Mach number ± 1, the fluidlike waves are compressive and expansive waves; and around the Mach number 0, they are linear thermal layers. The quantitative analysis of the fluid-like waves is done using the reduction of dimensions to the center manifolds.Two-scale nonlinear dynamics based on those on the Knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns. There are striking bifurcations in the transition of evaporation to condensation and in the transition of the Milne’s problem with a subsonic far field to one with a supersonic far field. The analysis of these wave patterns allows us to understand the Sone Diagram for the study of the complete condensation boundary value problem. The monotonicity of the Boltzmann shock profiles, a problem that initially motivated the present study, is shown as a consequence of the quantitative analysis of the nonlinear fluid-like waves.

The Yamada-Watanabe theorem for mild solutions to stochastic partial differential equations

We prove the Yamada-Watanabe theorem for semilinear stochastic partial differential equations with path-dependent coefficients. The so-called method of the moving frame allows us to reduce the proof to the Yamada-Watanabe theorem for stochastic differential equations in infinite dimensions.

On backward stochastic differential equations in infinite dimensions

In the present paper we present a result in which probabilistic methods are used to prove existence and uniqueness of a backward partial differential equation in a Hilbert space. This equation is of the form (7) in Theorem 1.1 below. In particular semi-linear conditions on the coefficient $f$ are imposed.

An Infinite-Dimensional Fractional Linear Quadratic Regulator Problem

We consider a controlled linear stochastic infinite-dimensional differential equation with an additive fractional Brownian motion as noise input. An optimal closed-loop control is determined in the case of complete state information and a quadratic goal functional.

Systems of essentially infinite-dimensional differential equations

We investigate systems of differential equations with essentially infinite-dimensional elliptic operators (of the Laplace–Levy type). For nonlinear systems, we prove theorems on the existence and uniqueness of solutions. For a linear system, we give an explicit formula for the solution.

An improved result of exponential stability of stochastic semilinear evolution equations

Sufficient conditions for the exponential stability of a class ofnonlinear, non-autonomous stochastic differential equations in infinitedimensions are studied. The analysis consists of introducing a suitableapproximating solution systems and using a limiting argument to pass onstability of strong solutions to mild ones. As a consequence, the classicalcriteriaof stability in A. Ichikawa [8] are improved and extended to cover a class ofnon-autonomous stochastic evolution equations.Two examples are investigated to illustrate our theory.

Balanced POD for model reduction of linear PDE systems: convergence theory

We consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems. The algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations. The algorithm is related to the proper orthogonal decomposition, and it was first proposed for systems of ordinary differential equations by Rowley (Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(3):997–1013, 2005). For the convergence analysis, we consider the algorithm in terms of the Hankel operator of the system, rather than the product of the system Gramians as originally proposed by Rowley. For exponentially stable systems with bounded finite rank input and output operators, we prove that the balanced realization can be expressed in terms of balancing modes, which are related to the Hankel operator. The balancing modes are required to be smooth, and this can cause computational difficulties for PDE systems. We show how this smoothness requirement can be lessened for parabolic systems, and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems. We prove entrywise convergence of the matrices in the approximate reduced order models in both cases, and present numerical results for two example PDE systems.