Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space.
Suppose that $(T_t)_{t\geq 0}$ is a continuous semi-group on $L^2(\mathcal{F},\mathbb{P})$; that is:
- $T_0(f)=f$ for every $f \in L^2(\mathcal{F},\mathbb{P})$,
- $T_{s+t}=T_s\circ T_t$,
- $t\mapsto T_t(f)$ is continuous for every $f \in L^2(\mathcal{F},\mathbb{P})$,
and $T_t=e^{tA}$ for some closed operator $A:L^2\rightarrow L^2$. Upong rescaling $A$, can we necessarily find a Markov process $X_t$ on $\mathbb{R}$ whose law is determined by $T_t$?