Definition 2.4.1 (Submarkov Property).label Let $\bracs{\bp_t|t \ge 0}$ be a semigroup on $L^{2}$, then $\bracs{\bp_t|t \ge 0}$ has the submarkov property if for any $f \in L^{2}$ with $0 \le f \le 1$ (a.e.), $0 \le \bp_{t} f \le 1$ for all $t > 0$.
Definition 2.4.1 (Submarkov Property).label Let $\bracs{\bp_t|t \ge 0}$ be a semigroup on $L^{2}$, then $\bracs{\bp_t|t \ge 0}$ has the submarkov property if for any $f \in L^{2}$ with $0 \le f \le 1$ (a.e.), $0 \le \bp_{t} f \le 1$ for all $t > 0$.