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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