Existence of Invariant Manifolds for Stochastic Equations in Infinite Dimension
نویسندگان
چکیده
We provide a Frobenius type existence result for finite-dimensional invariant submanifolds for stochastic equations in infinite dimension, in the spirit of Da Prato and Zabczyk [5]. We recapture and make use of the convenient calculus on Fréchet spaces, as developed by Kriegl and Michor [16]. Our main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application we characterize all finite-dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates.
منابع مشابه
Invariant Manifolds for Stochastic Partial Differential Equations
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite dimensional random ...
متن کاملar X iv : m at h / 04 09 48 5 v 1 [ m at h . D S ] 2 4 Se p 20 04 INVARIANT MANIFOLDS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
Annals of Probability 31(2003), 2109-2135. Invariant man-ifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invarian...
متن کاملSmooth Stable and Unstable Manifolds for Stochastic Partial Differential Equations
Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron’s method. Th...
متن کاملOn time-dependent neutral stochastic evolution equations with a fractional Brownian motion and infinite delays
In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory
متن کاملThe Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations∗
The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2. In Part 1, we prove...
متن کامل