609 research outputs found
Random attractors for stochastic evolution equations driven by fractional Brownian motion
The main goal of this article is to prove the existence of a random attractor
for a stochastic evolution equation driven by a fractional Brownian motion with
. We would like to emphasize that we do not use the usual
cohomology method, consisting of transforming the stochastic equation into a
random one, but we deal directly with the stochastic equation. In particular,
in order to get adequate a priori estimates of the solution needed for the
existence of an absorbing ball, we will introduce stopping times to control the
size of the noise. In a first part of this article we shall obtain the
existence of a pullback attractor for the non-autonomous dynamical system
generated by the pathwise mild solution of an nonlinear infinite-dimensional
evolution equation with non--trivial H\"older continuous driving function. In a
second part, we shall consider the random setup: stochastic equations having as
driving process a fractional Brownian motion with . Under a
smallness condition for that noise we will show the existence and uniqueness of
a random attractor for the stochastic evolution equation
Random attractors for a class of stochastic partial differential equations driven by general additive noise
The existence of random attractors for a large class of stochastic partial
differential equations (SPDE) driven by general additive noise is established.
The main results are applied to various types of SPDE, as e.g. stochastic
reaction-diffusion equations, the stochastic -Laplace equation and
stochastic porous media equations. Besides classical Brownian motion, we also
include space-time fractional Brownian Motion and space-time L\'evy noise as
admissible random perturbations. Moreover, cases where the attractor consists
of a single point are considered and bounds for the speed of attraction are
obtained.Comment: 30 page
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
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
Phase transitions driven by L\'evy stable noise: exact solutions and stability analysis of nonlinear fractional Fokker-Planck equations
Phase transitions and effects of external noise on many body systems are one
of the main topics in physics. In mean field coupled nonlinear dynamical
stochastic systems driven by Brownian noise, various types of phase transitions
including nonequilibrium ones may appear. A Brownian motion is a special case
of L\'evy motion and the stochastic process based on the latter is an
alternative choice for studying cooperative phenomena in various fields.
Recently, fractional Fokker-Planck equations associated with L\'evy noise have
attracted much attention and behaviors of systems with double-well potential
subjected to L\'evy noise have been studied intensively. However, most of such
studies have resorted to numerical computation. We construct an {\it
analytically solvable model} to study the occurrence of phase transitions
driven by L\'evy stable noise.Comment: submitted to EP
- …