Let $\{X_t\}_{t\geq 0}$ be a Markov process with state space in $[0,1]$ and generator $L$ defined on functions $f:[0,1]\to\mathbb R$ as follows \begin{align} Lf(x)=\int_0^1J(x,y)(f(y)-f(x))dx, \end{align} where $J:[0,1]^2\to\mathbb R$ is a smooth and symmetric function such that $\int_0^1 J(x,y)dy=1$.
I would like to prove that $P(X_t\leq x)=\int_0^xu_t(y)dy$ where \begin{align} \frac{d}{dt}u_t(x)=\int_0^1J(x,y)(u_t(y)-u_t(x))dy. \end{align} Is this a consequence of the fact that I know explicitly the generator? Could the Kolmogorov equations help me in some way? Do I need to prove something else like the absolutely continuity of the measure?
Thank you