30 research outputs found
Stochastic Shell Models driven by a multiplicative fractional Brownian--motion
We prove existence and uniqueness of the solution of a stochastic
shell--model. The equation is driven by an infinite dimensional fractional
Brownian--motion with Hurst--parameter , and contains a
non--trivial coefficient in front of the noise which satisfies special
regularity conditions. The appearing stochastic integrals are defined in a
fractional sense. First, we prove the existence and uniqueness of variational
solutions to approximating equations driven by piecewise linear continuous
noise, for which we are able to derive important uniform estimates in some
functional spaces. Then, thanks to a compactness argument and these estimates,
we prove that these variational solutions converge to a limit solution, which
turns out to be the unique pathwise mild solution associated to the
shell--model with fractional noise as driving process.Comment: 23 page
Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients
In this paper we study the longtime dynamics of mild solutions to retarded
stochastic evolution systems driven by a Hilbert-valued Brownian motion. As a
preparation for this purpose we have to show the existence and uniqueness of a
cocycle solution of such an equation. We do not assume that the noise is given
in additive form or that it is a very simple multiplicative noise. However, we
need some smoothing property for the coefficient in front of the noise. The
main idea of this paper consists of expressing the stochastic integral in terms
of non-stochastic integrals and the noisy path by using an integration by
parts. This latter term causes that in a first moment only a local mild
solution can be obtained, since in order to apply the Banach fixed point
theorem it is crucial to have the H\"older norm of the noisy path to be
sufficiently small. Later, by using appropriate stopping times, we shall derive
the existence and uniqueness of a global mild solution. Furthermore, the
asymptotic behavior is investigated by using the {\it Random Dynamical Systems
theory}. In particular, we shall show that the global mild solution generates a
random dynamical system that, under an appropriate smallness condition for the
time lag, have associated a random attractor
Stochastic lattice dynamical systems with fractional noise
This article is devoted to study stochastic lattice dynamical systems driven
by a fractional Brownian motion with Hurst parameter . First of
all, we investigate the existence and uniqueness of pathwise mild solutions to
such systems by the Young integration setting and prove that the solution
generates a random dynamical system. Further, we analyze the exponential
stability of the trivial solution
Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in
This paper addresses the exponential stability of the trivial solution of
some types of evolution equations driven by H\"older continuous functions with
H\"older index greater than . The results can be applied to the case of
equations whose noisy inputs are given by a fractional Brownian motion
with covariance operator , provided that and is
sufficiently small.Comment: 19 page