Definition (Probabilistic Safety over finite time horizon)

Let $\epsilon \in(0,1)$, and $T\in \mathbb{N}$. We say that the policy $\gamma:\mathbb{X}\to \mathbb{U}$ is $\epsilon$-safe over a time horizon $T$ if for any $x_{0}\in \mathcal{C}$ — where $\mathcal{C}$ is some Safe Set — the iterates of $$x_{t+1}=f(x_{t},u_{t},w_{t})$$ under $u_{t}=\gamma(x_{t})$ are such that $$\mathbb{P}\left( \bigcap_{t=0}^{T}\{ x_{t}\in \mathcal{C} \} \right)\ge 1-\epsilon$$

Definition (Probabilistic CBF)

Let $\delta \in(0,1)$. The function $h:\mathbb{X}\to \mathbb{R}$ is a $\delta$-probabilistic CBF if $\exists\alpha \in[0,1]$ s.t. $\forall x\in \mathcal{C}$, $\exists u\in \mathbb{U}$ s.t. $$\mathbb{P}(h(f(x,u,w))\ge\alpha h(x))\ge{1}-\delta$$ where $w\sim\mu_{w}$. Whenever the parameter $\delta$ is clear from the context we simply refer to $h$ as a probabilistic CBF.