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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 $t_1<...<t_n$ 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...

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