Smoothing properties of nonlinear stochastic equations in Hilbert spaces

1996 ◽  
Vol 3 (4) ◽  
pp. 445-464 ◽  
Author(s):  
Marco Fuhrman
Keyword(s):  
Hilbert Spaces ◽  
Stochastic Equations ◽  
Nonlinear Stochastic Equations

scholarly journals Projective Stochastic Equations and Nonlinear Long Memory

2014 ◽  
Vol 46 (4) ◽  
pp. 1084-1105
Author(s):  
Ieva Grublytė ◽  
Donatas Surgailis
Keyword(s):  
Long Memory ◽  
Volterra Series ◽  
Bernoulli Shift ◽  
Moving Average ◽  
Stochastic Equations ◽  
Partial Sums ◽  
Martingale Transform ◽  
New Class ◽  
Moving Average Processes ◽  
Nonlinear Stochastic Equations

A projective moving average {Xt, t ∈ ℤ} is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of Xt on ‘intermediate’ lagged innovation subspaces with given coefficients αi and βi,j. The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution Xt. We show that, under certain conditions on Q, αi, and βi,j, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

scholarly journals Almost Automorphic Solutions to Nonautonomous Stochastic Functional Integrodifferential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Li Xi-liang ◽  
Han Yu-liang
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Stochastic Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Banach Contraction ◽  
Stability Results ◽  
Stochastic Functional

This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.

Approximate controllability of stochastic equations in a Hilbert space with fractional Brownian motions

2014 ◽  
Vol 15 (01) ◽  
pp. 1550005 ◽  
Author(s):  
Mo Chen
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Additive Noise ◽  
Sufficient Conditions ◽  
Stochastic Equations ◽  
Hurst Parameter ◽  
Banach Fixed Point Theorem ◽  
Fractional Brownian Motions

In this paper, the approximate controllability for semilinear stochastic equations in Hilbert spaces is studied. The additive noise is the formal derivative of a fractional Brownian motion in a Hilbert space with the Hurst parameter in the interval (½, 1). Sufficient conditions are established. The results are obtained by using the Banach fixed point theorem.

Asymptotic Analysis of Nonlinear Stochastic Equations With Rapidly Oscillating and Decaying Components

2003 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Roessel
Keyword(s):  
Dynamical System ◽  
Markov Process ◽  
Martingale Problem ◽  
Nonlinear Dynamical System ◽  
Stochastic Equations ◽  
Limiting Behavior ◽  
Nonlinear Dynamical ◽  
Nonlinear Stochastic Equations ◽  
Lower Dimensional ◽  
Quadratic And Cubic Nonlinearities

We consider a noisy n-dimensional nonlinear dynamical system containing rapidly oscillating and decaying components. We extend the results of Papanicolaou and Kohler and Namachchivaya and Lin; these results state that as the noise becomes smaller, a lower dimensional Markov process characterizes the limiting behavior. Our approach springs from a direct consideration of the martingale problem and considers both quadratic and cubic nonlinearities.

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