The （overdamped）Langevin Equation is written as: $$dX_t = -V(X_t)dt + \sqrt{2}dW_t$$ for some function $V(X)$ and $W_t$ is the Brown motion. We can use this SDE to define a continuous Markov chain.
A continuous Markov chain is characterized by its generator operator, which is defined as $$L(g) = \lim_{t \to 0} \frac{P_t(g) - g}{t}$$ for every function $g$ which make sense. Here $P_t(g) = \int_{R^n} g(y)p_t(x,dy)$, and $p_t(x,Y)$ is the Markov kernel for time $t$.
I want to calculate the generator for Langevin dynamics. By definition, I need to know the Markov kernel for this process, that is, given a starting point $x$ and a set $Y \subset R^n$, I need to know the probability of after time $t$, a point evolves with Langevin equation starting from $x$ and arrive $Y$. How can I calculate this probability? Or there is a better way to calculate generator?