What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?

I know that the Onsager-Machlup function for $dX(t)=f(X(t))dt+dB(t)$ is $$L(x,v)=\frac12\left[v-f(x)\right]^2+\frac12f'(x)$$

But what if instead of $f(X)$ it is $f(B)$? I have a strong suspicion that it is

$$L(x,v)=\frac12 E[(v-f(B(t)+x))^2]$$

I have tried finding the SDE for $X(t)$ but it is quite difficult. I tried doing this through infinitesimal generator, but I can't find it.

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