Opuscula Math. 35, no. 4 (2015), 499-515
http://dx.doi.org/10.7494/OpMath.2015.35.4.499

 
Opuscula Mathematica

On potential kernels associated with random dynamical systems

Mohamed Hmissi
Farida Mokchaha
Aya Hmissi

Abstract. Let \((\theta,\varphi)\) be a continuous random dynamical system defined on a probability space \((\Omega,\mathcal{F},\mathbb{P})\) and taking values on a locally compact Hausdorff space \(E\). The associated potential kernel \(V\) is given by \[ Vf(\omega ,x)= \int\limits_{0}^{\infty} f(\theta_{t}\omega,\varphi(t,\omega)x)dt, \quad \omega \in \Omega, x\in E.\] In this paper, we prove the equivalence of the following statements: 1. The potential kernel of \((\theta,\varphi)\) is proper, i.e. \(Vf\) is \(x\)-continuous for each bounded, \(x\)-continuous function \(f\) with uniformly random compact support. 2. \((\theta ,\varphi)\) has a global Lyapunov function, i.e. a function \(L:\Omega\times E \rightarrow (0,\infty)\) which is \(x\)-continuous and \(L(\theta_t\omega, \varphi(t,\omega)x)\downarrow 0\) as \(t\uparrow \infty\). In particular, we provide a constructive method for global Lyapunov functions for gradient-like random dynamical systems. This result generalizes an analogous theorem known for deterministic dynamical systems.

Keywords: dynamical system, random dynamical system, random differential equation, stochastic differential equation, potential kernel, domination principle, Lyapunov function.

Mathematics Subject Classification: 37H99, 37B25, 37B35, 47D07.

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  • Mohamed Hmissi
  • Université de Tunis Elmanar, Faculté des Sciences de Tunis, Département de Mathématiques, TN-2092 Elmanar, Tunis, Tunisia
  • Al-Imam Muhammad Ibn Saud Islamic University, College of Science, Department of Mathematics and Statistics, P.O. Box 90950, Riyadh 11623, Saudi Arabia
  • Farida Mokchaha
  • Université de Tunis Elmanar, Faculté des Sciences de Tunis, Département de Mathématiques, TN-2092 Elmanar, Tunis, Tunisia
  • Al-Imam Muhammad Ibn Saud Islamic University, College of Science, Department of Mathematics and Statistics, P.O. Box 90950, Riyadh 11623, Saudi Arabia
  • Aya Hmissi
  • Université de Tunis Elmanar, Faculté des Sciences de Tunis, Département de Mathématiques, TN-2092 Elmanar, Tunis, Tunisia
  • Communicated by Palle E.T. Jorgensen.
  • Received: 2014-05-19.
  • Revised: 2014-08-20.
  • Accepted: 2014-10-24.
  • Published online: 2015-02-06.
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Cite this article as:
Mohamed Hmissi, Farida Mokchaha, Aya Hmissi, On potential kernels associated with random dynamical systems, Opuscula Math. 35, no. 4 (2015), 499-515, http://dx.doi.org/10.7494/OpMath.2015.35.4.499

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