Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by...

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Bibliographic Details
Published in:Applied mathematics & optimization 2004-10, Vol.50 (3), p.183-207
Main Authors: Caraballo, Tom s, Kloeden, Peter E., Schmalfu, Bj rn
Format: Article
Language:English
Subjects:
ATTRACTORS
Chaos theory
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Differential equations
Dynamical systems
Hilbert space
MATHEMATICAL EVOLUTION
MATHEMATICAL SOLUTIONS
PARTIAL DIFFERENTIAL EQUATIONS
PERTURBATION THEORY
Random variables
RANDOMNESS
STABILITY
Stochastic models
STOCHASTIC PROCESSES
Theory
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Summary:We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities. [PUBLICATION ABSTRACT]
ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-004-0802-1