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Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

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Abstract

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.

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Authors and Affiliations

  1. Departamento Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain

    Tomás Caraballo

  2. Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, D-60054 Frankfurt am Main, Germany

    Peter E. Kloeden

  3. Fakultät 5 – Mathematik, Universität Paderborn, D-33095 Paderborn, Germany

    Björn Schmalfuß

Authors
  1. Tomás Caraballo
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  2. Peter E. Kloeden
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  3. Björn Schmalfuß
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Correspondence to Tomás Caraballo, Peter E. Kloeden or Björn Schmalfuß.

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Caraballo, T., Kloeden, P. & Schmalfuß, B. Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation. Appl Math Optim 50, 183–207 (2004). https://doi.org/10.1007/s00245-004-0802-1

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  • DOI: https://doi.org/10.1007/s00245-004-0802-1

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