Definition 5.2.1 (Mean Value Property).label Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^{d})$. For each $0 \le s \le t \le T$ and $x \in \real^{d}$, let $P_{s, t}(x, \cdot)$ be its transition distribution, and $h: (0, \infty) \times \real^{d} \to [0, \infty)$ be a measurable function, then $h$ satisfies the mean-value property with respect to $\bp$ if for any $0 \le s \le t \le T$ and $x \in \real^{d}$,