Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

Journal Article · Fri Oct 15 04:00:00 EDT 2004 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245-004-0802-1· OSTI ID:21064221

Caraballo, Tomas ^[1]; Schmalfuss, Bjoern ^[2]

---

1. Departamento Ecuaciones Diferenciales y Analisis Numerico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla (Spain)
2. Fakultaet 5 - Mathematik, Universitaet Paderborn, D-33095 Paderborn (Germany)

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 anon-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.

• OSTI ID: 21064221
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 3 Vol. 50; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ATTRACTORS
MATHEMATICAL EVOLUTION
MATHEMATICAL SOLUTIONS
PARTIAL DIFFERENTIAL EQUATIONS
PERTURBATION THEORY
RANDOMNESS
STABILITY
STOCHASTIC PROCESSES