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

# On potential kernels associated with random dynamical systems

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.