TY  - JOUR
ID  - Hmissi2015
LB  - Hmissi2015
AU  - Hmissi, Mohamed
AU  - Mokchaha, Farida
AU  - Hmissi, Aya
TI  - On potential kernels associated with random dynamical systems
ST  - On potential kernels associated with random dynamical systems
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2015
VL  - 35
IS  - 4
SP  - 499
EP  - 515
DO  - http://dx.doi.org/10.7494/OpMath.2015.35.4.499
UR  - http://dx.doi.org/10.7494/OpMath.2015.35.4.499
AB  - 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.
KW  - dynamical system
KW  - random dynamical system
KW  - random differential equation
KW  - stochastic differential equation
KW  - potential kernel
KW  - domination principle
KW  - Lyapunov function
SN  - 1232-9274
ER  -