Definition 1.2.1 (Random Differential Operator). Let $(\Omega, \cf, \bp)$ be a filtered probability space, $s \ge 0$, and $k \in \natp$. For each multi-index $\alpha \in \nat_{0}^{d}$ with $\abs{\alpha}\le k$, let $a_{\alpha}: [s, \infty) \times \Omega \to \real$ be measurable function. For each $f \in C^{k}(\real^{d})$ and $t \ge s$, let
\[[L_{t}(\omega)f](x) = \sum_{\abs{\alpha} \le k}a_{\alpha}(t, \omega) \partial^{\alpha} f(x)\]
then $L = \sum_{|\alpha| \le k}a_{\alpha} \partial^{\alpha}$ is a random differential operator of order $k$ with coefficients $\bracs{a_\alpha}_{|\alpha| \le k}$.