Equations In Infinite Dimensions Research Articles

Index Concepts for Linear Differential-Algebraic Equations in Infinite Dimensions

Different index concepts for regular linear differential-algebraic equations are defined and compared in the general Banach space setting. For regular finite dimensional linear differential-algebraic equations, all these indices exist and are equivalent. For infinite dimensional systems, the situation is more complex. It is proven that although some indices imply others, in general they are not equivalent. The situation is illustrated with a number of examples.

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.

Optimal Feedback for Stochastic Linear Quadratic Control and Backward Stochastic Riccati Equations in Infinite Dimensions

It is a longstanding unsolved problem to characterize the optimal feedbacks for general SLQs (i.e., stochastic linear quadratic control problems) with random coefficients in infinite dimensions; while the same problem but in finite dimensions was just addressed very recently. This paper is devoted to giving a solution to this problem under some assumptions which can be verified for interesting concrete models. More precisely, under these assumptions, we establish the equivalence between the existence of optimal feedback operator for infinite dimensional SLQs and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. A key contribution of this work is to introduce a suitable notion of solutions (i.e., transposition solutions to the aforementioned Riccati equations), which plays a crucial role in both the statement and the proof of our main results.

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.

Logarithmic Schrödinger equations in infinite dimensions

We study the logarithmic Schrödinger equation with a finite range potential on RZd. Through a ground-state representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinite-dimensional Cauchy problem.

Stochastic optimal control in infinite dimensions with state constraints

We consider the state-constrained stochastic optimal control problem in infinite-dimensional separable Hilbert spaces, where the state process is driven by the Q-Wiener process and the (possibly unbounded) linear operator. By applying the stochastic target theory and the backward reachability approach, we show that the original (possibly discontinuous) value function can be represented by the zero-level set of the auxiliary (continuous) value function. The auxiliary value function is obtained from the penalized unconstrained stochastic control problem (in infinite dimensions) that includes an additional control variable as a consequence of the (infinite-dimensional) martingale representation theorem. We then prove that the auxiliary value function is a unique (continuous) viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation in infinite dimensions. Note that the viscosity analysis developed in our paper generalizes that presented in the existing literature, since the corresponding infinite-dimensional HJB equation includes an additional operator-valued control variable in the Hamiltonian maximization and depends on an additional initial state variable.

Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations

This paper is devoted to investigating the well-posedness and asymptotic behavior of a class of stochastic nonlocal partial differential equations driven by nonlinear noise. First, the existence of a weak martingale solution is established by using the Faedo--Galerkin approximation and an idea analogous to Da Prato and Zabczyk [Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992]. Second, we show the uniqueness and continuous dependence on initial values of solutions to the above stochastic nonlocal problem when there exist some variational solutions. Third, the asymptotic local stability of steady-state solutions is analyzed either when the steady-state solution of the deterministic problem is also a solution of the stochastic one or when this does not happen. Next, to study the global asymptotic behavior, namely, the existence of attracting sets of solutions, we consider an approximation of the noise given by Wong and Zakai's technique using the so called colored noise. For this model, we can use the power of the theory of random dynamical systems and prove the existence of random attractors. Eventually, particularizing in the cases of additive and multiplicative noise, it is proved that the Wong--Zakai approximation models possess random attractors which converge upper-semicontinuously to the respective random attractors of the stochastic equations driven by standard Brownian motions. This fact justifies the use of this colored noise technique to approximate the asymptotic behavior of the models with general nonlinear noises, although the convergence of attractors and solutions is still an open problem.

Semigroups and evolutionary equations

We show how strongly continuous semigroups can be associated with evolutionary equations. For doing so, we need to define the space of admissible history functions and initial states. Moreover, the initial value problem has to be formulated within the framework of evolutionary equations, which is done by using the theory of extrapolation spaces. The results are applied to two examples. First, differential-algebraic equations in infinite dimensions are treated and it is shown, how a C_{0}-semigroup can be associated with such problems. In the second example we treat a concrete hyperbolic delay equation.

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.

Stochastic PDEs on graphs as scaling limits of discrete interacting systems

Stochastic partial differential equations (SPDE) on graphs were recently introduced by Cerrai and Freidlin (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 865–899). This class of stochastic equations in infinite dimensions provides a minimal framework for the study of the effective dynamics of much more complex systems. However, how they emerge from microscopic individual-based models is still poorly understood, partly due to complications near vertex singularities. In this work, motivated by the study of the dynamics and the genealogies of expanding populations in spatially structured environments, we obtain a new class of SPDE on graphs of Wright–Fisher type which have nontrivial boundary conditions on the vertex set. We show that these SPDE arise as scaling limits of suitably defined biased voter models (BVM), which extends the scaling limits of Durrett and Fan (Ann. Appl. Probab. 26 (2016) 3456–3490). We further obtain a convergent simulation scheme for each of these SPDE in terms of a system of Itô SDEs, which is useful when the size of the BVM is too large for stochastic simulations. These give the first rigorous connection between SPDE on graphs and more discrete models, specifically, interacting particle systems and interacting SDEs. Uniform heat kernel estimates for symmetric random walks approximating diffusions on graphs are the keys to our proofs. Some open problems are provided as further motivations of our study.

Fokker–Planck equation for dissipative 2D Euler equations with cylindrical noise

After a short review of recent progresses in 2D Euler equations with random initial conditions and noise, some of the recent results are improved by exploiting a priori estimates on the associated infinite dimensional Fokker-Planck equation. The regularity class of solutions investigated here does not allow energy- or enstrophy-type estimates, but only bounds in probability with respect to suitable distributions of the initial conditions. This is a remarkable application of Fokker-Planck equations in infinite dimensions. Among the example of random initial conditions we consider Gibbsian measures based on renormalized kinetic energy.

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.

Generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations associated to the number operator

By means of the Laplace transform, we give the solution of the generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations in infinite dimensions associated to the number operator. These solutions are given in terms of the Mittag-Leffler function and the convolution product.

Second Order Necessary Conditions for Optimal Control Problems of Stochastic Evolution Equations

In this paper, we obtain some second order necessary conditions for optimal control problems of stochastic evolution equations in infinite dimensions. The control acts on both the drift and diffusion terms while the control region is assumed to be convex. The concepts of relaxed and $V$-transposition solutions (introduced in our previous works) to operator-valued backward stochastic evolution equations are employed to formulate these optimality conditions. The correction part of the second order adjoint process, which does not appear in the Pontryagin-type stochastic maximum principle, plays a fundamental role in our results.

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]).

On approximations of the Euler-Peano scheme for multivalued stochastic differential equations

ABSTRACTIt is known that a unique strong solution exists for multivalued stochastic differential equations under the Lipschitz continuity and linear growth conditions. In this paper we apply the Euler-Peano scheme to show that existence of weak solution and pathwise uniqueness still hold when the coefficients are random and satisfy one-sided locally Lipschitz continuous and an integral condition (i.e. Krylov's conditions put forward in On Kolmogorov's equations for finite-dimensional diffusions, Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), Lecture Notes in Math., 1715, Springer, Berlin, 1999, pp. 1–63). When the coefficients are nonrandom and possibly discontinuous but only satisfy some integral conditions, the sequence of solutions of the Euler-Peano scheme converges weakly, and the limit is a weak solution of the corresponding MSDE. As a particular case, we obtain a global semi-flow for stochastic differential equations reflected in closed, convex domains.

Path-dependent Hamilton–Jacobi equations in infinite dimensions

We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and stability) for minimax solutions. A particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle. One of the main difficulties, the lack of compactness in infinite-dimensional Hilbert spaces, is circumvented by working with suitable compact subsets of our path space. As an application, our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of (delay) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskiĭ–Subbotin approach similarly as in finite dimensions.

On differential–algebraic equations in infinite dimensions

We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.