# From the infinitesimal generator of a stochastic process, can we find the finite dimensional distribution of it?

Let $$X=(X_t)_t$$ and $$Y=(Y_t)_t$$ a stochastic processes. I know that if $$X$$ and $$Y$$ has the same infinitesimal generator, then the finite dimensional distribution (fdd) of $$X$$ and $$Y$$ coincide, i.e. $$\mathbb P\{X_{t_1}\in B_{t_1},...,X_{t_n}\in B_{t_n}\}=\mathbb P\{Y_{t_1}\in B_{t_1},...,Y_{t_n}\in B_{t_n}\}$$ for all $$t_1<... and all Borel set $$B_{t_i}$$.

Now, given an infinitesimal generator of a stochastic process $$X$$, let say : $$L=f(t,x)\frac{\partial }{\partial t}+g(t,x)\frac{\partial ^2}{\partial x^2},$$

Is it possible to find the fdd's of $$X$$ ? In somehow, I don't really understand in what these infinitesimal generator are useful if we can't find the fdd from it...