Given an integer $d > 0$, let $h: \mathbb{R}^d \rightarrow \mathbb{R}^d$ be a Lipschitz function on $\mathbb{R}^d$. Let $\Phi: \mathbb{R}^d \times [0, \infty) \rightarrow \mathbb{R}^d$, $(x, t) \mapsto \Phi_t(x)$ be the semiflow generated by the ODE $\dot{x}(t) = h(x(t))$. Given a continuous random process $\bar{x}(\cdot)$ with values in $\mathbb{R}^d$, we assume that: H1 :The function $\bar{x}(\cdot)$ is almost surely bounded, APT: For all $T > 0$, H2 : $\sup_{s \in [0,T]} \| \bar{x}(t + s) - \Phi_s(\bar{x}(t)) \| \xrightarrow{\text{p.s.}}_{t \rightarrow \infty} 0$
For $t > 0$, let $\nu_t$ be the random measure on $\mathbb{R}^d$ defined by the equation $ \int_{\mathbb{R}^d} f(x) \nu_t(dx) = \frac{1}{t} \int_{0}^{t} f(\bar{x}(s)) \, ds $ (we admit that this equation defines a measure). The family $(\nu_t)_{t > 0}$ is called the family of empirical measures or occupation measures of the process $\bar{x}(\cdot)$.) ($f$ is a continuous function of $\mathbb{R}^d$ to $\mathbb{R}$
Show that, thanks to hypothesis H1, there exists an event $A \subset \Omega$ of probability one on which the family $(\nu_t)$ is tight.
I fail to answer this question. I start like this. $f(\bar{x})$ is bounded. I would like to use the Markov inequality : for all $r > 0$
$\nu_t(\{ (f(\bar{x}) \leq r) \cap A \})^c) \leq \frac{1}{r} \int f d \nu_t$ but I am not sure {$\{ f(\bar{x}) \leq r \}$} $\cap A$ is compact.
Any help ? Thanks